Keywords: nombres premiers, théorie des cribles, groupes des classes, théorie prime numbers, sieve theory, class groups, class eld theory . . . . . . . . . .

des corps de classes Résumé : Nous démontrons deux nouveaux résultats de densité à propos du 16-rang des groupes des classes de corps de nombres quadratiques: le groupe des classes de Q( √ -p) a un élément d'ordre 16 pour un quart des nombres premiers p de la forme a 2 + 16c 4 ; et deuxième, le groupe des classes de Q( √ -2p) a un élément d'ordre 16 pour un huitième des nombres premiers p ≡ -1 mod 4. Ces résultats sont les premiers résultats de densité non triviaux sur le 16-rang des groupes des classes dans une famille de corps de nombres quadratiques, et ils prouvent une instance des conjectures de Cohen et Lenstra. En plus, leurs preuves impliquent de nouvelles applications des cribles développés par Friedlander et Iwaniec. Nous démontrons un très bon terme d'erreur pour une fonction de comptage des nombres premiers qui est liée au 16-rang du groupe des classes de Q( √ -2p), donnant ainsi des indications fortes contre une conjecture de Cohn et Lagarias que le 16-rang est contrôlé par un critère de type Chebotarev.

√

-p) has an element of order 16 for one-fourth of prime numbers p of the form a 2 + 16c 4 ; and, the class group of Q( √ -2p) has an element of order 16 for one-eighth of prime numbers p ≡ -1 mod 4. These are the rst non-trivial density results about the 16-rank of class groups in a family of quadratic number elds, and they prove an instance of the Cohen-Lenstra conjectures. Moreover, both of their proofs involve new applications of powerful sieving techniques developed by Friedlander and Iwaniec. Finally, we prove a power-saving error term for a prime-counting function related to the 16-rank of the class group of Q( Chapter 1

Introduction

The main object of study in number theory is the ring of rational integers Z.

The ring Z can be studied from several dierent perspectives. One of them is to study the distribution of prime numbers, the building blocks of its multiplicative structure, and functions theoreof. More precisely, given a suciently well-behaved sequence of complex numbers {a n } n indexed by natural numbers n ∈ N, one might ask for an estimate of the sum p prime a p .

(1.1) For instance, if the sequence {a n } n is dened by

a n = 1 if n ≤ X 0 otherwise, then the statement that p prime a p - X 2 dt log t ≤ X 1 2 log X 8π
for all suciently large real numbers X is equivalent to the famous Riemann Hypothesis [START_REF] Schoenfeld | Sharper Bounds for the Chebyshev Functions θ(x) and ψ(x)[END_REF]Corollary 1,p. 339].

Another way to study arithmetic of the ring Z is to study solutions of polynomial equations over the integers. One such equation is the negative Pell equation: given a positive integer d, one might ask when the equation

x 2 -dy 2 = -1

(P -)
has solutions x, y ∈ Z. For example, if d = 2016, then (P -) has no solutions, while if d = 2017, then (P -) has innitely many solutions, the smallest of which is (x, y) = (106515299132603184503844444, 2371696115380807559791481).

An area of number theory that naturally combines the above two perspectives of studying the integers is the study of arithmetic statistics of 2-parts of class groups of quadratic number elds.

Quadratic rings and arithmetic statistics

When solving polynomial equations over Z, it is often useful to view these equations inside rings that are slightly larger than Z. One natural generalization of Z that is particularly conducive to studying the negative Pell equation is a quadratic ring, i.e., a commutative ring with unity that is a free Z-module of rank 2. An example of a quadratic ring is Z[

√ -6] = Z ⊕ Z • √ - 6
.

(1 + √ -6) • (-3 - √ -6) = 3 -4 √ - 6
There are many quadratic rings. In fact, their isomorphism classes are in one-to-one correspondence with the set of integers congruent to 0 or 1 modulo 4, where a quadratic ring corresponds to its discriminant (see for instance A quadratic ring whose discriminant is not a square is an integral domain and in fact an order in the quadratic number eld that is its eld of fractions.

We will call such a ring a quadratic domain. If R is a quadratic domain of discriminant D, then there exists an isomorphism of rings

R ∼ = Z[(D + √ D)/2].
Among quadratic domains, a special role is played by those that are maximal orders in quadratic number elds. The discriminant of a quadratic number eld is dened to be the discriminant of the maximal order in the quadratic number eld. Such a discriminant is called a fundamental discriminant. Fundamental discriminants are exactly the integers of the form d, where d = 1 is squarefree and d ≡ 1 mod 4, and 4d, where d is squarefree and d ≡ 2 or 3 mod 4.

We now introduce two arithmetic invariants of quadratic domains that are relevant to the negative Pell equation.

Class groups

The arithmetic of quadratic domains can be more complicated than that of the ring Z. The fundamental theorem of arithmetic states that Z is a unique factorization domain, that is, a domain in which every non-zero element has a factorization into irreducible elements that is unique up to reordering and multiplication by units. In a quadratic domain, this need not be the case. For example, in Z[

√ -6], 2 • 5 and (2 + √ -6) • (2 - √ -6)
are two distinct factorizations of the element 10 into irreducible elements. An integral domain which is not a unique factorization domain cannot be a principal ideal domain. Hence one obstruction to unique factorization in a quadratic domain is the failure of ideals to be principal. One way to measure this obstruction is via an algebraic invariant called the class group.

Let R be a quadratic domain, and let D and K denote its discriminant and its eld of fractions, respectively. Then the (ordinary) class group Cl of R is dened as the quotient Cl = I/P, where I is the group of invertible fractional ideals of R and P is the subgroup of I consisting of principal invertible fractional ideals. Since a discriminant determines a quadratic ring up to isomorphism, we will sometimes denote the class group of R by Cl(D). A closely related group is the narrow class group

Cl + , dened as the quotient Cl + = I/P + , where now P + is the subgroup of I consisting of principal invertible fractional ideals that can be generated by a totally positive element (i.e., an element α ∈ K such that σ(α) > 0 for all real embeddings σ : K → R). The study of the narrow class group precedes that of the ordinary class group the narrow class group was introduced by Gauss [START_REF] Friedrich | Disquisitiones arithmeticae[END_REF], albeit in the language of binary quadratic forms.

We recall that Cl is a nite abelian group. We also note that if a quadratic domain is the maximal order in a quadratic number eld, then it is a unique factorization domain if and only if it is a principal ideal domain, and so the class group is in fact the only obstruction to unique factorization. For example, the ring Z[ √ -6] from above is the maximal order in the quadratic number eld Q( √ -6), its class group Cl (-24) is isomorphic to Z/2Z, and the ideal generated by 2 and √ -6 is not principal. A standard reference for these denitions and facts is [START_REF] Janusz | Algebraic number elds[END_REF].

As a fairly simple algebraic object that nonetheless carries very important information about the arithmetic of the corresponding quadratic domain, the class group is one of the most important and widely studied invariants in number theory.

What do class groups look like?

We already mentioned above that class groups are nite abelian groups. Given a nite abelian group G, a prime number , and a positive integer k, we dene the k -rank of G to be

rk k G = dim F k-1 G/ k G .
In other words, rk G is the number of cyclic -groups appearing in the decomposition of G as a direct sum of cyclic subgroups of prime-power order, and rk k G is the number of these cyclic -groups that have an element of order k .

Hence the -rank measures the width of the -part, while the k -rank as k increases measures the depth of the -part.

Knowing the k -rank of G for every prime power k is equivalent to knowing the isomorphism class of G. Therefore, as Cl(D) is a nite abelian group, we can study the average behavior of Cl(D) as D ranges over some family of discriminants by studying the distribution of rk k Cl(D) for various prime powers k .

Let D be a fundamental discriminant. The width of the 2-part of Cl(D)

is given by Gauss's genus theory [START_REF] Friedrich | Disquisitiones arithmeticae[END_REF]. More precisely, we have

rk 2 Cl(D) + = ω(D) -1, (1.2)
where ω(D) denotes the number of distinct primes dividing D.

Cohen and Lenstra [START_REF] Cohen | Heuristics on class groups of number elds[END_REF] developed a heuristic model to predict the behavior of the odd parts of class groups of maximal orders in quadratic number elds. Roughly, Cohen-Lenstra heuristics stipulate that an odd abelian group G occurs as the odd part of a class group with probability proportional to the inverse of the size of the automorphism group of G. These heuristics can be used to make many precise conjectures about the distribution of k -ranks for After more than 30 years, very few such conjectures have been proved. In fact, the only result for = 2 giving a precise asymptotic formula is that of Davenport and Heilbronn [START_REF] Davenport | On the density of discriminants of cubic elds[END_REF], for the average value of 3 rk3Cl(D) as D ranges over all positive (or negative) fundamental discriminants (their result actually predates the Cohen-Lenstra heuristics by more than 10 years; see also [START_REF] Bhargava | On the Davenport-Heilbronn theorems and second order terms[END_REF] and [START_REF] Taniguchi | Secondary terms in counting functions for cubic elds[END_REF] for subsequent renements). Their methods and results are still insucient to produce a positive proportion of D with rk 3 Cl(D) = 1.

Much more is known in the case that = 2. Rédei [START_REF] Rédei | Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper[END_REF] gave formulas for the 4-rank in terms of the individual primes dividing the discriminant (see also [START_REF] Clark | The 4-part of the class groups of a quadratic eld[END_REF]Theorem 1.2.3,p. 20]), and his work was sucient to deduce distribution results over discriminants with a xed 2-rank (see [START_REF] Gerth | Extension of conjectures of Cohen and Lenstra[END_REF]). Extending these distribution results to all discriminants was a much harder problem, resolved by Fouvry and Klüners [START_REF] Fouvry | On the 4-rank of class groups of quadratic number elds[END_REF]. They succeeded in proving that, for each integer k ≥ 0, the set of fundamental discriminants D such that rk 4 Cl(D) = k has the positive density predicted by Cohen and Lenstra (see [START_REF] Fouvry | On the 4-rank of class groups of quadratic number elds[END_REF] and also [START_REF] Fouvry | Cohen-Lenstra heuristics of quadratic number elds[END_REF]).

Fouvry and Klüners [START_REF] Fouvry | The parity of the period of the continued fraction of √ d[END_REF] proved certain distribution results about the 8-rank in a special family of positive discriminants, but under the constraint that the 4-rank is equal to 1. Perhaps the most general result concerning the 8-rank is due to Stevenhagen [START_REF] Stevenhagen | Ray class groups and governing elds[END_REF]. He proved that if d = 0 and k ≥ 0 are integers, then the set of primes p such that rk 8 Cl(dp) = k and such that dp is a fundamental discriminant has a density which is a rational number.

Density results appear to be far more dicult to obtain for the 16-rank than for the lower 2-power ranks (see [41, p. 16-18]). Our main goal is to prove density results about the 16-rank, albeit in certain particularly simple families of quadratic number elds. Before we state our results, we rst give further motivation coming from the study of the negative Pell equation.

Fundamental units and the negative Pell equation

Let R, D, K, I, P, and P + be dened as in Section 1.2. We say that R is imaginary if there are no real embeddings K → R, or, equivalently, if its discriminant D is negative. In this case, the narrow class group clearly coincides with the ordinary class group. Otherwise, if D > 0, we say that R is real. In this case, the relationship between the ordinary and the narrow class groups is more interesting.

Let R be a real quadratic domain. The group P + is an index-1 or -2 subgroup of P, depending on whether or not R has a unit of norm -1. Indeed, the norm of a totally positive element is clearly positive, while the norm of √ D ∈ R is negative, and so the principal ideal generated by √ D can be generated by a totally positive element if and only if R has a unit of norm -1.

The unit group of R is of the form

R × ∼ = -1 × ε ,
where ε is a unit of innite order (see for instance [START_REF] Janusz | Algebraic number elds[END_REF]Theorem 11.19,p. 61]).

We say that ε is a fundamental unit. The norm Norm(ε) does not depend on the choice of ε, and is thus an invariant of a real quadratic domain.

As the norm function is multiplicative, the real quadratic domain R has a unit of norm -1 if and only if Norm(ε) = -1. Hence the invariant Norm(ε) simply detects if the ordinary and the narrow class groups dier.

We now link the invariant Norm(ε) to a negative Pell equation. Let

d = D if D ≡ 1 mod 4 D/4 if D ≡ 0 mod 4. One can check that the unit group Z[ √ d] × is a subgroup of index 1 or 3 of the unit group R × . Hence Norm(ε) = -1 if and only if Z[ √ d]
has a unit of norm -1, and this happens if and only if (P -) is solvable. Hence

x 2 -dy 2 = -1 is solvable over Z ⇐⇒ Cl(D) = Cl(D) + . (1.3)
The odd parts of Cl and Cl + coincide, so the study of the 2-parts of the ordinary and the narrow class groups is closely related to the study of solutions of the negative Pell equation. In fact, the equation

x 2 -dy 2 = -1 is solvable over Z if and only if rk 2 k Cl(D) = rk 2 k Cl(D) + (1.4)
for all integers k ≥ 1. [START_REF] Fouvry | On the negative Pell equation[END_REF] and [START_REF] Fouvry | The parity of the period of the continued fraction of √ d[END_REF]).

The equation

x 2 -2py 2 = -1
To demonstrate the diculty of improving the upper and lower bounds of Fouvry and Klüners, we now restrict our attention to a certain subset of D, namely the set of integers of the form 2p, where p is a prime number congruent to 1 mod 4. The associated quadratic number elds are the elds Q( √ 2p) of discriminant 8p. The reason this family is relatively simple is given by Gauss's genus theory.

From (1.2), we see that the 2-part of Cl(D) + (and so also Cl(D)) is relatively simpler to study when D has fewer prime divisors. If D has only one prime divisor, however, then the 2-part of the narrow class group is trivial and there is nothing to be done. Therefore, if we wish to study how the 2-part of the class group varies in some family of quadratic number elds, the simplest non-trivial types of families to consider are those parametrized by fundamental discriminants of the form qp, where ±q is an odd prime, 4, or 8, and where p varies over the set of prime numbers in some xed congruence classes modulo 4. For instance, if we take q = 8 and allow p to vary over the set of prime numbers congruent to 1 modulo 4, we recover the family {Q( √ 2p)} p≡1 mod 4 that we mentioned above.

For details of the following discussion, see [START_REF] Stevenhagen | The number of real quadratic elds having units of negative norm[END_REF]. Given a real number X > 5, let ρ(X) denote the proportion of primes p ≡ 1 mod 4 less than X for which the negative Pell equation x 2 -2py 2 = -1 is solvable. Stevenhagen's conjectural framework predicts that ρ(X) → 2 3 as X → ∞. However, the best known bounds are

5 8 ≤ lim inf X→∞ ρ(X) ≤ lim sup X→∞ ρ(X) ≤ 3 4 .
(1.5)

These bounds are obtained in the following way. Gauss's genus theory implies that the 2-part of Cl(8p) + is cyclic, so that the 2-part of Cl(8p) + is completely determined by the largest integer k such that rk 2 k Cl(8p) + = 1. As Cl(8p) is a quotient of Cl(8p) + by a subgroup of order 1 or 2, we deduce that

rk 2 k Cl(8p) + -1 ≤ rk 2 k Cl(8p) ≤ rk 2 k Cl(8p) + (1.6)
for all integers k ≥ 1. The condition p ≡ 1 mod 4 ensures that rk 2 Cl(8p) = rk 2 Cl(8p) + = 1.

By (1.6) and (1.3), we have the implications (1.9)

rk 4 Cl(8p) + = 0 =⇒ Cl(8p) = Cl(8p) + =⇒ x 2 -2py 2 = -
Again, one can show that this holds for one-fourth of primes p ≡ 1 mod 8, which gives the lower bound in (1.5).

At this point, we emphasize the the best known bounds (1.5), although rst explicitly stated in [42, p. 127], can be readily deduced from algebraic criteria (1.8) and (1.9) that were already known to Rédei [START_REF] Rédei | Ein neues zahlentheoretisches Symbol mit Anwendungen auf die Theorie der quadratischen Zahlkörper[END_REF] and Scholz [START_REF] Scholz | Über die Lösbarkeit der Gleichung t 2 -Du 2 = -4[END_REF] in the 1930's. In other words, there has been no tangible progress on the bounds (1.5) in over 80 years.

If we wish to improve the bounds in (1.5) using the same general strategy that we employed above, we would have to be able to compute the density of primes p ≡ 1 mod 4 satisfying either (for an improvement of the upper bound) rk 8 Cl(8p) + = 1 and rk 8 Cl(8p) = 0

or (for an improvement of the lower bound) rk 8 Cl(8p) + = rk 8 Cl(8p) = 1 and rk 16 Cl(8p) = 0.

As we will soon see, these two problems are of a similar diculty. We focus on the second problem, namely the 16-rank of Cl(8p) + .

The elds Q( √ 2p) are real quadratic elds, so at rst sight there seems to be no relation to studying imaginary quadratic elds. Generally, studying class groups of real quadratic elds is much more dicult than studying class groups of imaginary quadratic elds, primarily because real quadratic domains have units of innite order. However, in this particular case, Stevenhagen established a connection between the 16-rank of Cl(8p) and the 16-ranks of Cl(-4p) and Cl(-8p) for primes p ≡ 1 mod 4 (see [START_REF] Stevenhagen | Divisibility by 2-powers of certain quadratic class numbers[END_REF]Theorem 3,p. 3]). One consequence of Stevenhagen's result (already known to Oriat [START_REF] Oriat | Sur la divisibilité par 8 et 16 des nombres de classes d'idéaux des corps quadratiques Q( √ 2p) et Q( √ -2)[END_REF]) that can be stated simply is

rk 16 Cl(8p) + = 1 =⇒ rk 16 Cl(-8p) = 1.
Hence we could improve the lower bound in

Statements of main results

The two main results of this thesis come from the articles [START_REF] Milovic | The innitude of Q( √ -p) with class number divisible by 16[END_REF] and [START_REF] Milovic | On the 16-rank of class groups of Q( √ -8p) for p ≡ -1 mod 4[END_REF],

which will comprise Chapter 2 and Chapter 3, respectively. In the following, p always denotes a prime number. The rst result concerns a subfamily of {Q( √ -p)} p≡1 mod 4 .

Theorem A. We have where {a n } n is a reasonably nice sequence of complex numbers indexed by natural numbers n and X is a positive real number tending to innity. For Theorem A, the relevant sequence is given by

a n = 1 if n = a 2 + c
a p =      1 if p ≡ -1 mod 4 and rk 16 Cl(-8p) = 1 -1 if p ≡ -1 mod 4 and rk 16 Cl(-8p) = rk 8 Cl(-8p) -1 = 0 0 otherwise.
As such, proving Theorem B consists of proving a density result for the 8-rank, which is classical, and proving that a p oscillates as p varies. In fact, with a p dened as above, we prove that there exists δ > 0 such that p≤X a p X 1-δ as X tends to innity. The power-saving in X in the estimate above has additional implications about the 16-rank which we discuss in the introduction to Chapter 3. The method we use to prove that a p oscillates can be traced back to the work of Vinogradov [START_REF] Vinogradov | The method of trigonometrical sums in the theory of numbers[END_REF] from the 1930's, but our application of this method is reminiscent of its use by Friedlander and Iwaniec, coincidentally again in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF].

Strategies for the 16-rank

We now describe one reason that density results about the 16-rank are dicult to prove, and we present our strategies to circumvent these diculties in case of the families from Theorem A and Theorem B. Before we can do so, we have to introduce some algebraic objects that allow us to interpret class groups as Galois groups.

Class groups as Galois groups

The following denitions and facts can be found in [START_REF] Janusz | Algebraic number elds[END_REF]. Let E/F be a nite abelian extension of number elds. Let I F denote the free abelian group generated by prime ideals of F that are unramied in E. In the two cases q = -4 and q = -8, we manage to overcome the diculty of explicitly generating the 8-Hilbert class eld as follows. In the case q = -4, instead of nding H 8 for all prime numbers p ≡ 1 mod 4, we are able to write down H 8 explicitly when p is a prime of the form a 2 + c 4 with c even. Thus, we trade the generality of working with the full family {Q( √ -p)} p≡1 mod 4 in exchange for an explicit understanding of the 8-Hilbert class eld of Q(

√ -p).

If p ≡ -1 mod 4, then rk 4 Cl(-8p) = 1 if and only if p ≡ -1 mod 8. In the case q = -8, the idea is to write down, for p ≡ -1 mod 8, both

• the 4-Hilbert class eld H 4 of Q( √ -2p
), and

• an ideal u generating a class of order 4 in Cl(-8p)

in terms of integers u and v satisfying p = u 2 -2v 2 , and then to characterize

those p such that u H 4 /Q( √ -2p) = 1.
(1.13)

The isomorphism (1.12) for n = 2 and the equality (1.13) then imply that the class of order 4 in Cl in fact belongs to Cl 4 , which proves that Cl has an element of order 16.

Without further ado, we now move to the main body of this thesis, which consists of two chapters. Chapter 2 is based on [START_REF] Milovic | The innitude of Q( √ -p) with class number divisible by 16[END_REF] and deals with Theorem A and related results. Since we deal with a family of quadratic number elds whose class groups have cyclic 2-parts, the 16-rank is 1 or 0 according to whether or not 16 divides the class number, i.e., the order of the class group.

We adopt this terminology in Chapter 2.

Chapter 3, dedicated to Theorem B and related results, is based on [START_REF] Milovic | On the 16-rank of class groups of Q( √ -8p) for p ≡ -1 mod 4[END_REF].

Chapter 2

Innitude of Q(

√ -4p) with class number divisible by 16

Let p be a prime number, and let Cl and h be the class group and the class number of Q( √ -4p), respectively. Since the discriminant of this eld is either -p or -4p, Gauss's genus theory implies that the 2-part of Cl is cyclic, and so the structure of the 2-part of the class group is entirely determined by the highest power of 2 dividing h. More precisely, Gauss's genus theory implies that 2|h ⇐⇒ p ≡ 1 mod 4.

The criterion 4|h ⇐⇒ p ≡ 1 mod 8 can be deduced easily from Rédei's work on the 4-rank of quadratic number elds [START_REF] Rédei | Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper[END_REF]. In [START_REF] Barrucand | Note on primes of type x 2 + 32y 2 , class number, and residuacity[END_REF], Barrucand and Cohn gave an explicit criterion for divisibility by 8 by successively extracting square roots of the class of order two. It states that 8|h ⇐⇒ p = x 2 + 32y 2 for some integers x and y.

This can be restated as 8|h ⇐⇒ p ≡ 1 mod 8 and 1 + i is a square modulo p

(2.1)
where i is a square root of -1 modulo p (see [1, (10), p.68]). In [START_REF] Stevenhagen | Ray class groups and governing elds[END_REF], Stevenhagen also obtained the criterion (2.1), albeit by a more abstract argument using class eld theory over the eld Q(i).

Given a subset S of the prime numbers, and a real number X ≥ 2, dene R(S, X) := #{p ≤ X prime : p ∈ S} #{p ≤ X prime } .

If the limit lim X→∞ R(S, X) exists, we denote it by ρ(S) and call it the natural density of S. Let S(n) = {p prime : n|h(-4p)} ;

here we write h(-4p) for the class number of Q( √ -4p) to emphasize its dependence on p. From the above, classical results about primes in arithmetic progressions imply that ρ(S(2)) = 1/2 and ρ(S(4)) = 1/4. From (2.1), we see that 8 divides h if and only if p splits completely in Q(ζ 8 , √ 1 + i), where ζ 8 is a primitive 8 th root of unity. Since this is a degree 8 extension of Q, ebotarev's density theorem implies that ρ(S(8)) = 1/8. For a discussion of these and similar density results, see [41, p.16-19].

The Cohen-Lenstra heuristics [START_REF] Cohen | Heuristics on class groups of number elds[END_REF] can be adapted to this situation to predict the density of primes p such that 2 k divides h for k ≥ 1. Cohen and Lenstra stipulate that an abelian group G occurs as the class group of an imaginary quadratic eld with probability proportional to the inverse of the size of the automorphism group of G. Under this assumption, the cyclic group of order 2 k-1 would occur as the 2-part of the class group of an imaginary quadratic number eld twice as often as the cyclic group of order 2 k . As we just saw above, ρ(S(2 k )) = 1 2 ρ(S(2 k-1 )) for k ≤ 3, so we are led to conjecture Conjecture 2.1. For all k ≥ 1, the limit lim X→∞ R(S(2 k ), X) exists and is equal to 2 -k .

While Conjecture 2.1 is true for k ≤ 3, it has not been proven for any k ≥ 4. In fact, proving the conjecture for k ≥ 4 would likely require signicant new ideas because a proof along the lines of the arguments for k ≤ 3 seems far out of reach (see [41, p. 16]). Although several criteria for divisibility by 16 have been found already (see [START_REF] Kaplan | Cycles d'ordre au moins 16 dans le 2-groupe des classes d'idéaux de certains corps quadratiques[END_REF], [START_REF] Williams | On the class number of Q( √ -p) modulo 16, for p ≡ 1 (mod 8) a prime[END_REF], and [START_REF] Leonard | On the divisibility of the class numbers of Q( √ -p) and Q( √ -2p) by 16[END_REF]), none of them appear to be sucient to produce even innitely many primes p for which the class number of Q( √ -4p) is divisible by 16. This is precisely our aim in this chapter we will show that there is an innite number of primes p for which 16|h and also an innite number of primes p for which 8|h but 16 h. We also derive some consequences for the fundamental unit p of the real quadratic number eld Q( √ p).

We tackle the question of innitude not by developing a new criterion for divisibility by 16 which handles all primes, but by focusing on a very special subset of primes. These are the primes of the form p = a 2 + c 4 , c even.

(2.

2)

The

a 2 +c 4 prime 1 = κ 2π X 3/4 log X 1 + O log log X log X , (2.3)
where a and c run over Z and

κ = 1 0 (1 -t 4 ) 1 2 dt ≈ 0.874 . . . .
In particular, there exist innitely many primes of the form a 2 + c 4 with a ≡ a 0 mod 16 and c ≡ c 0 mod 4.

Proposition 2.1 also implies the innitude of primes p of the form as in the statements (i) -(iv) Theorem 2.1. We have the following quantitative result: Corollary 2.1. For a prime p, let h(-4p) denote the class number of Q(

√ -4p).
Then, for suciently large X, we have

#{p ≤ X : h(-4p) ≡ 0 mod 16} ≥ X 3/
= T + U √ p denote the fundamental unit of Q( √ p).
Then, for suciently large X, we have

#{p ≤ X : p ≡ 1 mod 8, T ≡ 0 mod 16} ≥ X 3/4 8 log X and #{p ≤ X : p ≡ 1 mod 8, T ≡ 8 mod 16} ≥ X 3/4 8 log X .
The existence of innitely many p ≡ 1 mod 8 such that T ≡ T 0 mod 16 for a xed T 0 ∈ {0, 8} is not at all trivial. Hence Corollary 2.2 sheds some new light on the fundamental unit p of Q( √ p), one of the most mysterious quantities in number theory.

Hilbert class elds

Suppose p ≡ 1 (mod 4). Then there are two nite primes of Q which ramify in Q(

√ -4p), namely 2 and p. The prime p = ( √ -p) of Q( √ -4p) lying above
p is principal, and so its ideal class in Cl is the identity. Genus theory then implies that the class of the prime ideal t = (2, 1 +

√ -p) of Q( √ -4p
) lying above 2 is the unique element of order two in Cl. Assuming that h is divisible by 2 n for some non-negative integer n, to check that it is divisible by 2 n+1 , it would suce to check that the class of t belongs to Cl 2 n .

2 n -Hilbert class elds

Suppose that 2 n |h for some non-negative integer n. Then recall that (1.11) induces a canonical isomorphism of cyclic groups of order 2 n

• H 2 n /K : Cl/Cl 2 n -→ Gal(H 2 n /K).
(

2 n+1 |h ⇐⇒ [t] splits completely in H 2 n . (2.6)
The main idea of the proof of Theorem 2.1 is to write down explicitly the 8-

Hilbert class eld H 8 of Q( √ -4p
), and then to characterize those p such that t splits completely in H 8 . We remark here that although Cohn and Cooke [START_REF] Cohn | Parametric form of an eight class eld[END_REF] have already written down H 8 in terms of the fundamental unit p of the real quadratic number eld Q( √ p) and certain integer solutions u and v to p = 2u 2 -v 2 , not enough is known about either p or u and v to deduce anything about the distribution of primes p such that t splits completely in H 8 .

Generating 2 n -Hilbert class elds

We rst state and prove some lemmas which will prove to be useful in our quest to explicitly generate H 8 .

The 2-Hilbert class eld, also called the genus eld of Q(

√ -4p), is known to be H 2 = Q(i, √ p). Hence every 2 n -Hilbert class eld of Q( √ -4p) contains Q(i)
, and so we can study the splitting behavior of t in H 2 n by working over the quadratic subeld Q(i) of H 2 . With this in mind, we now state some wellknown generalities about the completion of Q(i) with respect to the prime ideal (1 + i) lying over 2.

This completion is Q 2 (i), and its ring of integers Z 2 [i] is a discrete valuation ring with uniformizer m = 1 + i and maximal ideal m = (m). Let U = (Z 2 [i]) × denote the group of units of Z 2 [i] and for each positive integer k, dene U (k) = 1 + m k . Then there is a ltration

U = U (1) ⊃ U (2) ⊃ • • • ⊃ U (k) ⊃ • • • . For any k ≥ 3, squaring gives an isomorphism U (k) ∼ -→ U (k+2) . Indeed, let 1 + m k+2 y ∈ U (k+2) . Hensel's lemma implies that there exists x ∈ m k-2 such that x 2 + x = -m k-2 y. Then (1 + 2x) 2 = 1 + m k+2 y and 1 + 2x ∈ U (k) . It is not hard to see that U = i × U (3) = i × 2 + i × U (4) , so that U 2 = -1 × U (5) . In other words, u ∈ U is a square in Q 2 (i) if and only if u ≡ ±1 (mod m 5 ). Moreover, if ω ≡ ±1 (mod m 4 ), then Q 2 (i, √ ω) is generated over Q 2 (i) by a root of the polynomial X 2 -X + (1 ∓ ω)/4, which reduces to X 2 + X or X 2 + X + 1 modulo m. We collect these observations into the following lemma. Lemma 2.1. Let ω be a unit in Z 2 [i]. Then Q 2 (i, √ ω) is unramied over Q 2 (i) if and only if ω ≡ ±1 (mod m 4 ). Moreover, Q 2 (i, √ ω) = Q 2 (i), i.e., ω is a square in Q 2 (i) if and only if ω ≡ ±1 (mod m 5 ).
Next, we state two lemmas which we will use to check that the extensions of

Q( √ -4p
) which we construct are normal and cyclic. First, in both Chapter 2

and Chapter 3, we will make extensive use of the following lemma from Galois theory (see [29, Chapter VI, Exercise 4, p.321]).

Lemma 2.2. Let F be a eld of characteristic dierent from

2, let E = F ( √ d),
where

d ∈ F × \ (F × ) 2 , and let L = E( √ x), where x ∈ E × \ (E × ) 2 . Let N = Norm E/F (x).
Then we have three cases:

1.

If N / ∈ (E × ) 2 ∩ F × = (F × ) 2 ∪ d • (F × ) 2 , then L/F has normal closure L( √ N ) and Gal(L( √ N )/F ) is a dihedral group of order 8. 2. If N ∈ (F × ) 2 , then L/F is normal and Gal(L/F ) is a Klein four-group. 3. If N ∈ d • (F × ) 2 , then L/F is normal and Gal(L/F ) is a cyclic group of order 4.
Lemma 2.3. Let K be a eld. Suppose M/K is a cyclic extension of degree 2m and let σ be a generator of Gal(M/K). Let L be the subeld of M xed by σ m . Suppose N/K is a Galois extension containing M such that N/L is cyclic of degree 4. Then N/K is cyclic of degree 4m.

Proof. Let σ 1 denote a lift of σ to Gal(N/K). The order of σ 1 is at least 2m since the order of σ is 2m. As σ m xes L, σ m 1 is an element of Gal(N/L)which is non-trivial on M and hence has order 4. Thus the order of σ 1 is 4m.

Finally, we arrive at the main lemma we will use to construct 2 n -Hilbert class elds from 2 n-1 -Hilbert class elds. This result is inspired by a theorem of Reichardt [START_REF] Reichardt | Zur Struktur der absoluten Idealklassengruppe im quadratischen Zahlkörper[END_REF]3. Satz,p.82]. His theorem proves the existence of generators

√

for H 2 n over H 2 n-1 with ∈ H 2 n-1 of a certain form. We prove sucient conditions for an element of a similar form to give rise to a generator, so that we can actually construct H 2 n .

Lemma 2.4. Let h be the class number of Q( √ -4p), let n ≥ 2, and suppose that 2 n divides h. Suppose that we have a sequence of eld extensions

Q = A 1 ⊂ Q(i) = A 2 ⊂ A 4 ⊂ • • • ⊂ A 2 n-1
such that:

• A 2 k is a degree 2 k extension of Q for 1 ≤ k ≤ n -1, • A 2 k ⊂ H 2 k for 1 ≤ k ≤ n -1, • A 2 k ∩ H 2 k-1 = A 2 k-1 for 2 ≤ k ≤ n -1, • (1 + i) is unramied in A 2 n-1 /Q(i), and
• there is a prime element in the ring of integers of A 2 n-1 such that: lies above p and its ramication and inertia indices over p are equal to 1, denoting the conjugate of over A 2 n-2 by , we have

H 2 n-1 = H 2 n-2 ( √ ) = A 2 n-1 ( √ ), (U 2 ): (1 + i) remains unramied in A 2 n = A 2 n-1 ( √ ),
and

(N ): H 2 n-1 ( √ ) is normal over Q. Then H 2 n = H 2 n-1 ( √ ).
Proof. The ramication index of over p is 1, so and are coprime in

A 2 n-1 .
First we check that

is not a square in H 2 n-1 . Since [A 2 n : A 2 n-1 ] = [H 2 n-1 : A 2 n-1 ] = 2 and A 2 n = A 2 n-1 ( √ ), we deduce that is a square in H 2 n-1 if and only if A 2 n = H 2 n-1 .
But this cannot happen because the ramication index of p in H 2 n-1 is 2, while has ramication index 1 over p and, as and are coprime,

remains unramied in A 2 n .
By assumption, H 2 n-1 ( √ ) is normal over Q, and hence also over Q( √ -4p) and H 2 n-2 . Since and are conjugates over A 2 n-2 , they are also conju- gates over

H 2 n-2 . As H 2 n-1 = H 2 n-2 ( √ ) and = • 1 2 , Lemma 2.2 implies that H 2 n-1 ( √ ) is degree 4 cyclic extension of H 2 n-2 . Moreover, H 2 n-1 is a degree 2 n-1 cyclic extension of Q( √ -4p), so Lemma 2.3 implies that H 2 n-1 ( √ ) is a degree 2 n cyclic extension of Q( √ -4p). It remains to show that H 2 n-1 ( √ )/Q( √ -4p
) is unramied. We will establish this by showing that each of the ramication indices of the primes 2 and p in H 2 n-1 (

√

) is at most 2.

The prime 2 ramies in Q(i), but by assumption (1

+ i) is unramied in A 2 n . As H 2 n-1 ( √ ) = A 2 n ( √ ) and p ≡ 1 mod 4, Lemma 2.1 ensures that (1 + i) is unramied in H 2 n-1 ( √ ). Hence the ramication index of 2 in H 2 n-1 ( √ ) is 2. Now note that [H 2 n-1 ( √ ) : A 2 n ] = 2
, the ramication index of the prime over p is 1, and does not ramify in A 2 n /A 2 n-1 . Hence the ramication index of p in H 2 n-1 (

√

) is at most 2, and this completes the proof. From now on, assume that 4 divides h, i.e. that p ≡ 1 (mod 8). We will now use Lemma 2.4 to construct the 4-Hilbert class eld of Q(

√ -4p).

A prime p ≡ 1 (mod 4) splits in Q(i), so that there exists π in Z[i] such that p = ππ; here π denotes the conjugate of π over A 1 := Q. If we write π as a + bi with a and b integers, then we see that p = a 2 + b 2 . We choose π so that b is even. As p ≡ 1 (mod 8), we see that b is in fact divisible by 4. Hence π = a + bi, b ≡ 0 mod 4.

(2.7) Now x a square root of π and denote it by

√ π. Recall that H 2 = Q(i, √ p) is the 2-Hilbert class eld of Q( √ -4p
). We claim that the hypotheses of Lemma 2.4 for n = 2 are satised with A 2 := Q(i) and = π.

All of the hypotheses other than (U 2 ) and (N ) are easy to check. Note that our choice of π ensures that π ≡ ±1 (mod 4), so that (U 2 ) follows from Lemma 2.1. To see that

(N ) is satised, note that H 2 ( √ π) is the splitting eld (over Q) of the polynomial f 4 (X) := (X 2 -π)(X 2 -π).
H 4 = H 2 ( √ π) = Q(i, √ p, √ π) (2.8)
with π as in (2.7). We remark that Fouvry and Klüners developed similar methods in [START_REF] Fouvry | The parity of the period of the continued fraction of √ d[END_REF], where they constructed an analogue of the 4-Hilbert class eld to deduce a criterion for the 8-rank of class groups in a family of real quadratic number elds. From now on, suppose that 8|h. Replacing π by -π if necessary, we assume that π ≡ 1 (mod m 5 ).

H 4 = Q(i, √ p, √ π) H 2 = Q(i, √ p) A 4 = Q(i, √ π) Q( √ -4p) A 2 = Q(i) Q Next,
(2.9) This means that a + b ≡ 1 (mod 8). Our choice of √ π above is only unique up to sign. By Hensel's lemma, we can now x this sign by imposing that √ π ≡ 1 (mod m 3 ).

(2.10)

In order to explicitly generate H 8 from H 4 using Lemma 2.4, we are led to the problem of nding a prime element in A 4 = Q(i, √ π) whose norm down to Q(i) is π, up to units. This is the problem that we cannot solve explicitly enough in general to answer questions about innitude or density.

However, for a very thin subset of primes, we can write down an element of A 4 of norm -π. These are primes p of the form p = a 2 + c 4 , c even, (2.11) that is, primes p of the form a 2 + b 2 with b a perfect square divisible by 4.

Suppose that p is a prime of the form (2.11). Set

0 = c(1 + i) + √ π.
(2.12)

For 1 ≤ m ≤ 3, set m = σ m ( ), where σ is a generator for Gal(H 4 /Q( √ -4p)). The restriction of σ to H 2 generates Gal(H 2 /Q( √ - 4p
)), so σ(i) = -i. Also, looking at the polynomial f 4 (X) above, we see that σ(

√ π) = - √ π. Hence 0 • 2 = (c(1 + i) + √ π)(c(1 + i) - √ π) = -π.
(2.13) and

1 • 3 = (c(1 -i) + σ( √ π))(c(1 -i) -σ( √ π)) = -π.
(2.14)

We can now prove the main result of this section.

Proposition 2.2. Let p be a prime of the form (2.11), let π be as in (2.9), let √ π be as in (2.10), and let 0 be as in (2.12). Let

√ 0 denote a square root of 0 . Then H 4 ( √ 0 ) is the 8-Hilbert class eld of Q( √ -4p).
Proof. We again use Lemma 2.4, but this time with n = 3, A 4 = Q(i, √ π) and = 0 . All of the hypotheses except for (U 2 ) and (N ) immediately follow from the identity (2.13).

H 8 = Q(i, √ p, √ π, √ 0 ) A 8 = Q(i, √ π, √ 0 ) A 8 = Q(i, √ π, √ 2 ) H 4 = Q(i, √ p, √ π) H 2 = Q(i, √ p) A 4 = Q(i, √ π) Q( √ -4p) A 2 = Q(i) Q We now prove hypothesis (N ). We claim that H 4 ( √ 0 ) is the splitting eld of the polynomial f 8 (X) = (X 2 -0 )(X 2 -1 )(X 2 -2 )(X 2 -3 ).
It is easy to see that 0 2 = -π and 1 3 = -π are squares in H 4 . To prove (N ), it now suces to show that 0 1 is a square in H 4 . Let

d = √ π + σ( √ π) 2 and e = √ π -σ( √ π) 2i ∈ H 4 . Then 0 • 1 = (c(1 + i) + √ π)(c(1 -i) + σ( √ π)) = 2c 2 + √ πσ( √ π) + c ((1 + i)σ( √ π) + (1 -i) √ π) = (c 2 + 2de) + (d 2 + e 2 ) + c(2d + 2e) = (c + d + e) 2 ,
which completes the proof of hypothesis (N ).

It remains to prove hypothesis (U 2 ). The assumption that π ≡ 1 (mod m 5 )

actually means that π is a square in Q 2 (i), i.e. that (1 + i) splits in A 4 . Hence it remains to show that Q 2 (i, √ 0 ) is unramied over Q 2 (i), and Lemma 2.1
implies that it is enough to prove that 0 ≡ ±1 (mod m 4 ).

Recall from (2.10) that √ π ≡ 1 (mod m 3 ), so that √ π ≡ 1 or 1+m 3 (mod m 4 ). Squaring, we nd that π ≡ 1 or 1 + m 5 (mod m 6 ), respectively. Also recall that a + b ≡ 1 mod 8, i.e., a + c 2 ≡ 1 (mod m 6 ). We now split our argument into two cases, the rst when c ≡ 0 mod 4 and the second when c ≡ 2 mod 4.

If c ≡ 0 (mod m 4 ), then c 2 ∈ m 6 , so a -1 ∈ m 6 as well. Then π = a + c 2 i ≡ 1 (mod m 6 ), which means that √ π ≡ 1 (mod m 4 ). Then 0 = c(1 + i) + √ π ≡ 1 (mod m 4 ). If c ≡ 2 (mod m 4 ), then c 2 ≡ -m 4 (mod m 6 ). In this case, we have a -1 + m 4 ∈ m 6 , so that π = a + c 2 i ≡ 1 -m 4 -m 4 i ≡ 1 + m 4 (-1 -i) ≡ 1 + m 5 (mod m 6
). This means that √ π ≡ 1 + m 3 (mod m 4 ), and hence

0 = √ π + c(1 + i) ≡ 1 + m 3 + m 3 ≡ ±1 (mod m 4 ).
This nishes the proof that Q 2 (i,

√ 0 ) is unramied over Q 2 (i).

Proof of Theorem 2.1

The proof of Theorem 2.1 will proceed in much the same way as the last part of the proof of Proposition 2.2. Now, instead of showing that Q 2 (i, √ 0 ) is unramied over Q 2 (i), we must decide when this extension is trivial (i.e. when t splits completely in H 8 ) and when it is unramied of degree 2 (i.e. when t does not split completely in H 8 ). This is equivalent to determining when 0 is a square in Q 2 (i).

We will distinguish between two cases as above. The rst case is when c ≡ 0 (mod 4), i.e., c ∈ m 4 . Recall from above that then a ≡ 1 (mod 8) and √ π ≡ 1 (mod m 4 ).

To check whether or not 0 is a square in Q 2 (i), we must compute 0 modulo m 5 . Since c ≡ 0 (mod 4), we deduce that 0 ≡ √ π modulo m 5 . Thus, we must determine conditions on a such that √ π ≡ ±1 (mod m 5 ), and for this, by Hensel's lemma, it is necessary to determine π modulo m 7 . Hence, assuming c ≡ 0 (mod 4),

16|h ⇐⇒ √ π ≡ ±1 (mod m 5 ) ⇐⇒ π ≡ 1 (mod m 7 ) ⇐⇒ a ≡ 1 (mod 16).
This proves parts (i) and (iii) of Theorem 2.1.

We handle the second case similarly. Now c ≡ 2 (mod 4), a ≡ 5 (mod 8)

and

√ π ≡ 1 + m 3 (mod m 4
). Then 0 ≡ 2m + √ π modulo m 5 and so we must determine conditions on a such that

√ π ≡ ±1 -2m (mod m 5 ). Under the current assumptions, 16|h ⇐⇒ √ π ≡ ±1 -2m (mod m 5 ) ⇐⇒ π ≡ 1 + m 5 + m 6 (mod m 7 ) ⇐⇒ a ≡ -3 (mod 16).
Note that because of the choice (2.9) we have actually shown the theorem for a ≡ 1 (mod 4). If p = a 2 + c 4 with a ≡ 3 (mod 4), then p = (-a) 2 + c 4 with -a ≡ 1 (mod 4), so that the other cases can be deduced immediately. This nishes the proof of Theorem 2.1.

Overview of the proof of Proposition 2.1

In [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF], Friedlander and Iwaniec prove an asymptotic formula for the number of primes of the form a 2 + c 4 , that is, primes of the form a 2 + b 2 where b itself is a square. For a summary of their proof, see the exposition in [START_REF] Friedlander | Opera de cribro[END_REF]Chapter 21]. They use a new sieve that they developed to detect primes in relatively thin sequences [START_REF] Friedlander | Asymptotic sieve for primes[END_REF]. This sieve has its roots in the work of Fouvry and Iwaniec [START_REF] Fouvry | Gaussian primes[END_REF], where they used similar sieve hypotheses to give an asymptotic formula for the number of primes of the form a 2 + b 2 where b is a prime.

The purpose of the following three sections is to demonstrate that the method of Friedlander and Iwaniec is robust enough to incorporate congruence conditions on a and c. While we are convinced that the appropriate analogue of Proposition 2.1 is true when a and c satisfy reasonable congruence conditions modulo any positive integers q 1 and q 2 , respectively, the technical obstacles necessary to insert the congruence condition for c are cumbersome. Hence we will restrict ourselves to the case q 2 = 4.

The proof of Proposition 2.1 involves certain alterations in the way that the sieve [START_REF] Friedlander | Asymptotic sieve for primes[END_REF] is used. For this reason, we rst briey recall the inputs and the output of the sieve.

Asymptotic sieve for primes

Suppose (a n ) (n ∈ N) is a sequence of non-negative real numbers. Then the asymptotic sieve for primes developed in [START_REF] Friedlander | Asymptotic sieve for primes[END_REF] yields an asymptotic formula for

S(x) = p≤x p prime a p log p,
provided that the sequence (a n ) satises several hypotheses, all but two of which are not dicult to verify. To state them, we rst need to x some terminology. For d ≥ 1, let

A d (x) = n≤x n≡0 mod d a n ,
and let A(x) = A 1 (x). Moreover, let g be a multiplicative function, and dene the error term r d (x) by the equality

A d (x) = g(d)A(x) + r d (x). (2.15)
The hypotheses which are not dicult to verify are listed in equations (2.1)-(2.8) in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]. We briey recall them here. We assume the bounds

A(x) A( √ x)(log x) 2 (H1)
and

A(x) x 1 3   n≤x a 2 n   1 2 . ( H2
)
We assume that the multiplicative function g satises

0 ≤ g(p 2 ) ≤ g(p) ≤ 1, (H3) g(p) p -1 , (H4) and g(p 2 ) p -2 . ( H5
)
We also assume that for all y ≥ 2,

p≤y g(p) = log log y + c + O((log y) -10 ), ( H6
)
where c is a constant depending only on g; this is the linear sieve assumption.

Finally, we assume the bound

A d (x) d -1 τ (d) 8 A(x) (H7) uniformly in d ≤ x 1 3
; here τ is the divisor function.

Now we state the two hypotheses which are more dicult to verify. The rst is a classical sieve hypothesis; it is a condition on the average value of the error terms r d (x). Let L = (log x) 2 24 .

Hypothesis (R). There exists x r > 0 and D = D(x) in the range

x 2 3 < D < x (2.16)
such that for all x ≥ x r , we have

d cubefree d≤DL 2 |r d (t)| ≤ A(x)L -2 (R)
uniformly in t ≤ x.

In our applications, D will be x 3/4-ε for a suciently small ε. This condition about remainders will be called condition (R).

The second is a complicated condition on bilinear forms in the elements of the sequence (a n ) weighed by truncated sums of the Möbius function

β(n, C) = µ(n) c|n, c≤C µ(c).
(2.17)

It is designed to make sure that the sequence (a n ) is orthogonal to the Möbius function; this is crucial in overcoming the parity problem. We now state this hypothesis, named (B) for bilinear. Hypothesis (B). Suppose (R) is satised for x r and D = D(x). Then there exists x b > x r such that for every x > x b , there exist δ, ∆, and P satisfying

2 ≤ δ ≤ ∆, 2 ≤ P ≤ ∆ 1/2 35 log log x ,
and such that for every C with

1 ≤ C ≤ xD -1 ,
and for every N with

∆ -1 √ D < N < δ -1 √ x,
we have

m N ≤n≤2N mn≤x (n,mΠ)=1 β(n, C)a mn ≤ A(x)(log x) -2 26 , (B)
where Π = p≤P p.

(2.18)

Note that establishing condition (R) for a larger D decreases the range of C and N for which we have to verify condition (B).

The main result of [START_REF] Friedlander | Asymptotic sieve for primes[END_REF] is Theorem 2.2. Assuming hypotheses (H1)-(H7), (R), and (B), we have

S(x) = HA(x) 1 + O log δ log ∆ ,
where H is the positive constant given by the convergent product

H = p (1 -g(p)) 1 - 1 p -1
and the constant implied in the O-symbol depends on the function g and the constants implicit in (H1), (H2), and (H7).

Preparing the sieve for Proposition 2.1

For our application, we will denote by v the analogue of a quantity v from the proof of Friedlander and Iwaniec in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]. We take (a n ) to be the following sequence. Suppose q 1 and q 2 are positive integers and let q denote the least common multiple of q 1 and q 2 . We say that a pair of congruence classes a 0 mod q 1 c 0 mod q 2 is admissible if for every pair of congruence classes a 1 mod q c 1 mod q such that a 1 ≡ a 0 mod q 1 and c 1 ≡ c 0 mod q 2 , the congruence class a2 1 + c 41 mod q is a unit modulo q.

Example. Suppose that a 0 ∈ {1, 3, 5, 7, 9, 11, 13, 15} and c 0 ∈ {0, 2}. Then the pair of congruence classes a 0 mod 16 and c 0 mod 4 is admissible.

Example. Suppose that a 0 = c 0 = 1. Then the pair of congruence classes a 0 mod 3 and c 0 mod 2 is not admissible. Indeed, 1 ≡ a 0 ≡ c 0 mod 6 but Henceforth, suppose q 1 and q 2 are positive integers, let q be the least common multiple of q 1 and q 2 , and suppose a 0 mod q 1 and c 0 mod q 2 is an admissible pair of congruence classes. We dene nally, set g(2) = 1 2 and g(4) = 1 4 . For our extension, we dene a multiplicative function g by setting

a n := a, b ∈ Z a 2 +b 2 =n a≡a0 mod q1 Z (b),
g (n) = g(n) if (n, q) = 1 0 otherwise.
Then, provided that (H1)-(H7), (R), and (B) are satised with δ a large power of log x and ∆ a small power of x, the asymptotic formula given by the sieve

(see Theorem 2.2) is S (x) := p≤x p prime a p log p = c(q 1 , q 2 ) 16κ π x 3/4 1 + O log log x log x (2.21) where c(q 1 , q 2 ) = 1 q 1 q 2 p|q (1 -g(p)) -1
and κ is the integral given in the statement of Proposition 2.1. Note that the sieve applied to the original sequence (a n ) from [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF], with

a n = a, b ∈ Z a 2 +b 2 =n Z(b), (2.22)
where

Z(b) = c∈Z c 2 =b 1, (2.23)
yields the asymptotic formula

S(x) = 16κ π x 3/4 1 + O log log x log x
(see [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]Theorem 1,p.946]). Thus c(q 1 , q 2 ) can be interpreted as the density of primes of the form a 2 + c 4 such that a ≡ a 0 mod q 1 and c ≡ c 0 mod q 2 within the set of all primes of the form a 2 + c 4 .

Remark. Throughout the following two sections, we regard q 1 and q 2 as xed constants, and so the implied constants in every bound we give may depend on q 1 and q 2 , even if this dependence is not explicitly stated. Thus, whenever we state the implied constant is absolute, the implied constant may actually depend on q 1 and q 2 . In our application q 1 = 16 and q 2 = 4, so we are not concerned with uniformity of the above asymptotic formula with respect to q 1 and q 2 .

It is obvious that our modied sequence (a n ) satises (H1)-(H7) for the same reasons as the original sequence (a n ). We will prove that (a n ) above satises condition (R) for general q 1 and q 2 . The congruence condition on c is more dicult to insert into the proof of condition (B), so we prove condition (B) only for the special case where q 2 = 4 and c 0 ∈ {0, 2}.

Proof of condition (R)

Here we closely follow and refer to the arguments laid out in [ We separate the case when d is not coprime to q because in this case A d (x) = 0. This follows because the pair of congruences a 0 mod q 1 and c 0 mod q 2 is admissible and hence a n is supported on n coprime to q. The lemma we wish to prove is now identical to [19, Lemma 3.1, p.956].

Lemma 2.5. For any D ≥ 1, any ε > 0, and any x ≥ 2, we have

d≤D |A d (x) -M d (x)| D 1 4 x 9 16 +ε ,
where the implied constant depends only on ε.

This result is useful because it is easy to obtain an asymptotic formula for M d (x) where the coecient of the leading term is, up to a constant, a nice multiplicative function of d. In fact, let h be the multiplicative function supported on cubefree integers dened in [19, (3.16), p.961] by

h(p)p = 1 + 2(1 + χ 4 (p)) h(p 2 )p 2 = p + 2(1 + χ 4 (p)), (2.25)
and dene a multiplicative function h by setting

h (n) = h(n) if (n, q) = 1 0 otherwise.
(2.26)

Then following the same argument as in the proof of [19, Lemma 3.4, p.961],

we get Lemma 2.6. For d cubefree we have

M d (x) = g (d) 4κx 3 4 q 1 q 2 + O h (d)x 1 2
, where κ is the integral given in the statement of Proposition 2.1 and the implied constant is absolute.

Combining Lemmas 2.5 and 2.6, we get, as in [19, Proposition 3.5, p.362],

Proposition 2.3. Let a 0 mod q 1 c 0 mod q 2 be an admissible pair of congruence classes, let a n be dened as in (2.19), and let r d (x) be dened as in (2.24). Then for every ε > 0 and every D ≥ 1, there exists an x 0 = x 0 (ε) > 0 and C = C(ε) > 0 such that for every x ≥ x 0 , we have

d cubefree d≤D |r d (t)| ≤ CD 1 4 x 9 16 +ε
uniformly for t ≤ x.

Choosing D = x 3 4 -8ε , we obtain hypothesis (R).

It remains to prove Lemma 2.5. We may assume that the sum is over d ≤ D with (d, q) = 1. For such d, we rst approximate the sum A d (x) by a smoothed sum

A d (f ) = n≡0 mod d a n f (n),
where f is a smooth function satisfying:

• f is supported on [0, x],

•

f (u) = 1 for 0 < u ≤ x -y, • f (j) (u) y -j for x -y < u < x,
where y = D 1 4 x 13 16 and the implied constants depend only on j (see [19, p.958]). Since a n is supported on integers of the form a 2 + c 4 , we trivially have

d≤D (d,q)=1 |A d (x) -A d (f )| yx -1 4 +ε ,
where the implied constant depends only on ε.

A d (f ) = b Z (b) α mod d α 2 +b 2 ≡0 mod d a≡α mod d a≡a0 mod q1 f (a 2 + b 2 ).
(2.27) Since (d, q) = 1, so also (d, q 1 ) = 1, and the two conditions a ≡ α mod d and a ≡ a 0 mod q 1 can be combined into one condition a ≡ α mod dq 1 . In fact, xing an integer d that is an inverse of d modulo q 1 and an integer q1 that is an inverse of q 1 modulo d, we can dene α as α = αq 1 q1 + a 0 d d.

We apply Poisson's summation formula to the sum over a to obtain

a≡α mod dq1 f (a 2 + b 2 ) = 1 dq 1 k e α k dq 1 ∞ -∞ f (t 2 + b 2 )e -tk dq 1 dt.
Here and henceforth, we use the standard notation e(t) := e 2πit .

Substituting this into (2.27) we get

A d (f ) = 2 dq 1 b Z (b) k ρ (k, b; d)I(k, b; dq 1 )dt, where ρ (k, b; d) = α mod d α 2 +b 2 ≡0 mod d e α k dq 1 ,
and where

I(k, b; dq 1 ) = ∞ 0 f (t 2 + b 2 ) cos(2πtk/dq 1 )dt
is dened exactly the same as on [19, p.959]. We dene M d (f ) to be the main term in this expansion, i.e. the term corresponding to k = 0,

M d (f ) = 2 dq 1 b Z (b)ρ(b; d)I(0, b; dq 1 ).
Since I(0, b; dq 1 ) = I(0, b; q 1 ), the argument on page 959 shows that Lemma 2.7. For any D, K, and L ≥ 1, for any complex numbers ξ(k, l), and for any ε > 0, we have the inequality

d≤D (d,q)=1 |M d (f ) -M d (x)| yx -1 4 (log x)
d≤D 0<k≤K 0<l≤L ξ(k, l)ρ (k, l; d) (D + √ DKL)(DKL) ε ξ
where

ξ 2 = 0<k≤K 0<l≤L |ξ(k, l)| 2 ,
and the implied constant depends only on ε. Lemma 2.8. Let D, N ≥ 1 and let α n be any complex numbers. For integers d such that (d, q 1 ) = 1, let ν be an integer in the unique residue class modulo dq 1 that reduces to ν modulo d and a 0 modulo q 1 . Then there exists an absolute constant C = C(q 1 ) such that for all D and N suciently large, we have

d≤D (d,q1)=1 ν mod d ν 2 +1≡0 mod d n≤N α n e ν n dq 1 ≤ CD 1 2 (D + N ) 1 2 α . (2.29)
Inequality (2.28) is a consequence of a large sieve inequality applied to the rationals ν/d mod 1 with ν ranging over the roots of ν 2 + 1 ≡ 0 mod d for d in a range around D. The large sieve inequality can be applied because these rationals ν/d are well-spaced modulo 1 for d in a certain range around D (i.e. pairwise dierences are uniformly bounded from below by about 1/D instead of 1/D 2 ). This is a key ingredient in the work of [START_REF] Fouvry | Gaussian primes[END_REF]. In our analogue, however, it is not clear that ν /dq 1 are also well-spaced modulo 1 for d in a similar range around D. Nonetheless, we can reduce Lemma 2.8 to inequality (2.28) as follows.

We rst split the sum over n into congruence classes modulo q 1 to get

n0 mod q1 n≤N n≡n0 mod q1 α n e ν n dq 1 = n0 mod q1 m≤(N -n0)/q1 α m,n0 e ν m d e ν n 0 dq 1 ,
where α m,n0 = α mq1+n0 .

Since e (ν n 0 /dq 1 ) does not depends on m, the sum on the left-hand-side of

ν mod d ν 2 +1≡0 mod d n≤N α n e ν n dq 1 q 1 D 1/2 (D + N/q 1 ) 1/2 α .
This nishes the proof of (2.8) and thus also the proof of condition (R).

Proof of condition (B)

Many of the upper bound estimates carried out in sections 4 and 5 of [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF] require no changes since 0 ≤ a n ≤ a n (compare (2. [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]) and (2.22)). In most cases, we now sum over fewer non-negative terms.

Recall that we established condition (R) with D = x Proposition 2.4. Let c 0 ∈ {0, 2}, let q 2 = 4, and let a 0 mod q 1 c 0 mod q 2 be an admissible pair of congruence classes. Dene β(n, C) as in (2.17), Π as in (2.18), and a n as in (2.19). Let x ≥ 3, η > 0, and A > 0. Let P be in the range

(log log x) 2 ≤ log P ≤ (log x)(log log x) -2 .
(2.30)

Let B = 4A + 2 20 .
(2.31)

Then there exists x 0 = x 0 (η, A) such that for all x ≥ x 0 , for all N with

x 1 4 +η < N < x 1 2 (log x) -B , (2.32)
and for all C with

1 ≤ C ≤ N 1-η , (2.33)
we have

m N ≤n≤2N mn≤x (n,mΠ)=1 β(n, C)a mn ≤ A (x)(log x) 5-A .
(2.34)

From Propositions 2.3 and 2.4 to Proposition 2.1

Before proving Proposition 2.4, we deduce Proposition 2.1 from Propositions 2.3 and 2.4. Let a 0 ∈ {1, 3, 5, 7, 9, 11, 13, 15}, q 1 = 16, c 0 ∈ {0, 2}, and q 2 = 4.

Then a 0 mod q 1 c 0 mod q 2 is an admissible pair of congruences. We apply the asymptotic sieve for primes described in Section 2. for the sequence (a n ) with δ = (log x) B , ∆ = x η , and x b = max{x r , x 0 (η, A)}.

We then obtain the asymptotic formula (2.21) with c(q 1 , q 2 ) = 1 32 , which proves (2.3).

Proof of Proposition 2.4

Suppose that we are in the setting of Proposition 2.4. Now take A = 2A + 2

β(n) = β(n, C) = p(n)µ(n) c|n, c≤C µ(c) (2.37)
and the bound ≤ A (x)(log x) 5-A replaced by ≤ CϑθA (x)(log x)

(n, Π) = 1, (2.40)
where Π is dened in (2.18). Finally, let α(m) be any complex numbers supported on M < m ≤ 2M with |α(m)| ≤ 1, and dene

B * (M, N ) := (m,n)=1 α(m)β(n)a mn , (2.41)
where β(n) = β(n, C) is dened as in (2.37) (see [19, (4.20), p.966]). To establish condition (B) it then suces to prove Lemma 2.9. Let η > 0 and A > 0 and take B as in (2.31). Then there exists x 0 = x 0 (η, A) > 0 such that for all x ≥ x 0 , for all M and N satisfying (2.32) and (2.38), and for all C satisfying (2.33) we have

|B * (M, N )| ≤ ϑθ(M N ) 3 4 (log M N ) 5 . ( B
a mn = |w| 2 =m |z| 2 =n Imwz≡a0 mod q1 Z (Rewz),
where the sum over z is restricted to primary Gaussian integers, i.e. z satisfying z ≡ 1 mod 2(1 + i).

Recall from (2.20) that the congruence condition c ≡ c 0 mod q 2 is incorporated into the denition of Z . We now dene α w := α(|w| 2 ) and β z := β(|z| 2 ) as on 1 ) sums by restricting the support of α w to w in a xed residue class modulo q 1 and β z to z in a xed residue class z 0 modulo 64q 1 , such that z 0 ≡ 1 mod 2(1 + i). Now the residue class of Imwz modulo q 1 is xed, and so we can eliminate the condition Imwz ≡ a 0 mod q 1 . We further modify the support of β z as in equation [19, (5.13), p.969]. Let r(α) be a smooth periodic function of period 2π supported on ϕ < α ≤ ϕ+2πθ (where θ is as dened in (2.36)) for some -π < ϕ < π such that r (j) θ -j for j = 0, 1, 2, and let

β z = r(α)p(n)µ(n) c|n, c≤C µ(c), (2.43)
where α = arg z and n = |z| 2 . Recall that by (2.39) and (2.40), β z = 0 if either τ (|z| 2 ) > τ or if |z| 2 is not coprime with Π. We remove the condition (ww, zz) = 1 from (2.42) at an acceptable cost as in [19, (5.10) We then apply Cauchy-Schwarz as in [19, (5.17), p.970] and introduce a smooth radial majorant f supported on the annulus

1 2 √ M ≤ |w| ≤ 2 √ M (see [19, p.970]) to get B (M, N ) M 1 2 D (M, N ) 1 2
,

where

D (M, N ) := w f (w) z β z Z (Rewz) 2 .
This eliminates the dependence on α w , so that the sum over w above is free.

After inserting a coprimality condition, we arrive at the sum

D * (M, N ) := (z1,z2)=1 β z1 β z2 C (z 1 , z 2 ) (2.45)
where

C (z 1 , z 2 ) := w f (w)Z (Rewz 1 )Z (Rewz 2 )
(see [19, (5.26), p.972] and [19, (5.27), p.972]). The coprimality condition was inserted at the cost

D * (M, N ) = D (M, N ) + O τ 2 (M 3 4 N 3 4 + P -1 M 1 2 N 3 2 )(log M N ) 516
(see [19, (5.22), p.972]). Recall that the congruence condition c ≡ c 0 mod q 2 is hidden in the denition of Z , while the congruence condition a ≡ a 0 mod q 1 has been removed by restricting the support of β z . To prove Lemma 2.9, we now have left to prove Lemma 2.10. Let η > 0 and A > 0, and take B as in (2.31). Then there exists x 0 = x 0 (η, A) such that for all x ≥ x 0 , for all M and N satisfying (2.32) and (2.38), and for all C satisfying (2.33), we have

|D * (M, N )| ≤ Cϑ 2 θ 4 M 1 2 N 3 2 (log M N ) 10 . (B )
Note the extra factor of θ coming from the restriction of support of β to a sector of angle θ.

Proof of Lemma 2.10

In order to obtain this upper bound, Friedlander and Iwaniec introduce a quantity they call the modulus

∆ = ∆(z 1 , z 2 ) = Im(z 1 z 2 ),
which is non-zero whenever (z 1 , z 2 ) = 1 and z 1 and z 2 are odd and primitive. The sum dening D * (M, N ) is split into several dierent sums depending on the size of the modulus ∆. Dierent techniques are used to treat each of these sums, but we will manage to avoid going into the details by reducing our sums to those already studied in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF].

The Fourier analysis carried out on [19, p.974] depends on the greatest common divisor of ∆ and q 2 . Using the Poisson summation formula similarly as on [19, p.974], equation (2.45) can now be written as

D * (M, N ) = δ|q2 (z1,z2)=1 (q2,|∆|)=δ β z1 β z2 C (z 1 , z 2 ),
where

C (z 1 , z 2 ) = (q 2 /δ) -2 |z 1 z 2 | -1/2 • h1 h2 F h1 |∆z2| 1/2 q2/δ , h2 |∆z1| 1/2 q2/δ G (h 1 , h 2 ), (2.46)
the Fourier integral

F (u 1 , u 2 ) = f z 2 |z 2 | t 2 1 - z 1 |z 1 | t 2 2 e(u 1 t 1 + u 2 t 2 )dt 1 dt 2
is the same as the one dened in [19, (6.8), p.974] and

G (h 1 , h 2 ) = 1 |∆| γ1,γ2 mod |∆| γ 2 1 z2≡γ 2 2 z1 mod |∆| γ1≡γ2≡c0 mod δ e γ 1 h 1 + γ 2 h 2 |∆|q 2 /δ
is an arithmetic sum similar to G(h 1 , h 2 ) dened in [19, (6.10), p.974], but now incorporating the congruence condition c ≡ c 0 mod q 2 ; here γ i is the solution (modulo |∆|q2 δ ) to the system of congruences

γ i ≡ γ i mod |∆| γ i ≡ c 0 mod q 2 .
Such a solution is guaranteed to exist because γ 1 ≡ γ 2 ≡ c 0 mod δ. Note that similarly as in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF], we omit in the notation the dependence of F and G on z 1 and z 2 .

The main term in the above expansion for C (z

F 0 (z 1 , z 2 ) := F (0, 0) = 2 f (0) log 2|z 1 z 2 /∆| + O(∆ 2 M 1 2 N -2 log N ). ( 2
G 0 (z 1 , z 2 ) := G (0, 0)
similar to the one in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]Lemma 8.4,p.980]. This is where we now specialize to the case q 2 = 4 and c 0 ∈ {0, 2}.

Recall that we restricted the support of β z to z in a xed congruence class modulo 64q 1 . Hence z 1 ≡ z 2 mod 64, so that ∆ = Im(z 1 z 2 ) ≡ 0 mod 64. This signicantly simplies our arguments since now δ = (4, |∆|) = 4.

The arithmetic sum G (h 1 , h 2 ) now simplies to

G (h 1 , h 2 ) = 1 |∆| γ1,γ2 mod |∆| γ 2 1 z2≡γ 2 2 z1 mod |∆| γ1≡γ2≡c0 mod 4 e γ 1 h 1 + γ 2 h 2 |∆| .
We rst prove a lemma analogous to [19, Lemma 8.1, p.978].

Lemma 2.11. Fix θ ∈ {2, 4} and let

G (h 1 , h 2 ; θ) = 1 |∆| γ1,γ2 mod |∆| γ 2 1 z2≡γ 2 2 z1 mod |∆| γ1≡γ2≡0 mod θ e γ 1 h 1 + γ 2 h 2 |∆| .
Then

|G (h 1 , h 2 ; θ)| ≤ 16τ 3 (∆)|∆| -1 (z 1 h 2 1 -z 2 h 2 2 , ∆).
(2.48)

Introducing a change of variables γ 1 = θω 1 and γ 2 = θω 2 , we get

G (h 1 , h 2 ; θ) = 1 |∆| ω1,ω2 mod |∆|/θ ω 2 1 z2≡ω 2 2 z1 mod |∆|/θ 2 e ω 1 h 1 + ω 2 h 2 |∆|/θ .
Proceeding in a similar fashion as on [19, p.977-978], we write

∆/θ = θ∆ 1 (∆ 2 ) 2 ,
with ∆ 1 squarefree. The condition ω 2 1 z 2 ≡ ω 2 2 z 1 mod |∆|/θ 2 implies that (ω 2 1 , ∆/θ 2 ) = (ω 2 2 , ∆/θ 2 ), so we can write

(ω 2 1 , ∆/θ 2 ) = (ω 2 2 , ∆/θ 2 ) = d 1 d 2 2 with d 1 squarefree. Then d 1 |∆ 1 , d 2 |∆ 2 , (d 1 , ∆ 2 /d 2 ) =
G (h 1 , h 2 ; θ) = 1 |∆| b1d1=|∆1| b2d2=∆2 (d1,b2)=1 η1,η2 mod θb1b 2 2 d2 (η1η2,b1b2)=1 η 2 1 z2≡η 2 2 z1 mod b1b 2 2 e((η 1 h 1 + η 2 h 2 )/θb 1 b 2 2 d 2 )
The innermost sum vanishes unless

h 1 ≡ h 2 ≡ 0 mod θd 2 , so G (h 1 , h 2 ) is equal to 1 |∆| b1d1=|∆1| (d1,b2)=1 b2d2=∆2 θd2|(h1,h2) θ 2 d 2 2 η1,η2 mod b1b 2 2 (η1η2,b1b2)=1 η 2 1 z2≡η 2 2 z1 mod b1b 2 2 e((η 1 h 1 + η 2 h 2 )/θb 1 b 2 2 d 2 ).
Performing the change of variables η 2 = ωη 1 , the analogue of equation [19, (8.3), p.978] becomes

1 |∆| b1d1=|∆1| (d1,b2)=1 b2d2=∆2 θd2|(h1,h2) θ 2 d 2 2 ω≡z2/z1 mod b1b 2 2 R((h 1 + ωh 2 )(θd 2 ) -1 ; b 1 b 2 2 ),
where R(h; b) is the classical Ramanujan sum dened on [19, p.978]. Now the same argument as on [19, p.978] yields the desired upper bound (2.48).

We now turn our attention back to G (h 1 , h 2 ). In case c 0 = 0, we're in the case of Lemma 2.11 and

|G (h 1 , h 2 )| = |G (h 1 , h 2 ; 4)| ≤ 16τ 3 (∆)|∆| -1 (z 1 h 2 1 -z 2 h 2 2 , ∆).
If, on the other hand, c 0 = 2, we note that G (h

1 , h 2 ) = G (h 1 , h 2 ; 2) - G (h 1 , h 2 ; 4) since ∆ ≡ 0 mod 16. Hence |G (h 1 , h 2 )| ≤ 32τ 3 (∆)|∆| -1 (z 1 h 2 1 -z 2 h 2 2 , ∆).
The same arguments as those in Section 9 of [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF] now suce to show that the main term in the Fourier expansion indeed comes from h 1 = h 2 = 0.

Specically, if we dene

D 0 (M, N ) := (z1,z2)=1 β z1 β z2 C 0 (z 1 , z 2 ),
where

C 0 (z 1 , z 2 ) = |z 1 z 2 | -1/2 F 0 (z 1 , z 2 )G 0 (z 1 , z 2 ), (2.49)
then the reader may easily check that the above estimates yield the following analogue of [19, (9.10), p.983].

Lemma 2.12. Let η > 0 and A > 0, and take B as in (2.31). Then there exists x 0 = x 0 (η, A) such that for all x ≥ x 0 , for all M and N satisfying (2.32) and (2.38), and for all C satisfying (2.33), we have

|D * (M, N ) -D 0 (M, N )| ≤ ϑ -1 τ 2 N 2 (log N ) η -1/η ,
where τ is dened in (2.39).

It now remains to estimate D 0 (M, N ). We turn to obtaining an exact formula for G 0 (z 1 , z 2 ). Recall, from top of [19, p.979], that

G 0 (z 1 , z 2 ) := 1 |∆| γ1,γ2 mod |∆| γ 2 1 z2≡γ 2 2 z1 mod |∆| 1 = N (z 2 /z 1 ; |∆|)/|∆|,
where N (a; r) denotes the number of solutions (γ 1 , γ 2 ) modulo r to aγ 2 1 ≡ γ 2 2 mod r.

Similarly,

G 0 (z 1 , z 2 ) = N (z 2 /z 1 ; |∆|)/|∆|,
where N (a; r) is the number of solutions (γ 1 , γ 2 ) modulo r to the congruences

aγ 2 1 ≡ γ 2 2 mod r γ 1 ≡ γ 2 ≡ c 0 mod 4.
Since z 2 /z 1 ≡ 1 mod 64 and ∆ ≡ 0 mod 64, we are only concerned with the case a ≡ 1 mod 64 and r ≡ 0 mod 64.

Computation of N (a; r)/r

Case c 0 = 0 First let us compute N (a; r)/r when c 0 = 0. Since γ 1 ≡ γ 2 ≡ 0 mod 4, we can make a change of variables γ 1 = 4ω 1 and γ 2 = 4ω 2 , where now ω i are congruence classes modulo r/4, to nd that N (a; r) = 16N (a; r/16), i.e. N (a; r)/r = N (a; r/16)/(r/16).

This leads to a formula of type [19, (8.16)

G 0 (z 1 , z 2 ) = ν 16d|∆ d odd ϕ(d) d z 2 /z 1 d .
Since ∆ ≡ 0 mod 64, we are only interested in the case ν ≥ 2, where this becomes Case c 0 = 2

G 0 (z 1 , z 2 ) = 2 64d|∆ ϕ(d) d z 2 /z 1 d , ( 2
When c 0 = 2 and 4|r, we can make a change of variables γ 1 = 2ω

G 0 (z 1 , z 2 ) = 2 16d|∆ ϕ(d) d z 2 /z 1 d -2 64d|∆ ϕ(d) d z 2 /z 1 d , (2.51)
which is the analogue of (2.50).

End of proof of of Lemma 2.10

We now turn back to estimating D 0 (M, N ). As in [19, (10.4)

D 0 (M, N ) = 2 f (0)N 1 2 T (β) + O (τ -1 + θ)Y (β)M 1 2 N -1 2 log N where T (β) := (z1,z2)=1 β z1 β z2 G 0 (z 1 , z 2 ) log 2|z 1 z 2 /∆| and Y (β) := (z1,z2)=1 |β z1 β z2 |τ (|z 1 | 2 )τ (|z 2 | 2 )τ 3 (∆).
Similarly as in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]Lemma 10.1,p.985], we can bound Y (β) by

Y (β) θ 4 N 2 (log N ) 2 19
, so that we are left with estimating the sum T (β). In each of the cases c 0 = 0 and c 0 = 2, we can use the formula for G 0 (z 1 , z 2 ) and F 0 (z 1 , z 2 ) to write T (β)

as a sum similar to [19, (10.13) Lemma 2.13. Fix ξ ∈ {16, 64}. Let η > 0, A > 0, and σ > 0, and take B as in (2.31). Then there exists x 0 = x 0 (η, A) and C 0 = C 0 (η, A, σ) > 0 such that for all x ≥ x 0 , for all N satisfying (2.32), and for all C satisfying (2.33), we have

T (β, ξ) := 2 d ϕ(d) d (z1,z2)=1 ∆(z1,z2)≡0 mod ξd β z1 β z2 z 2 /z 1 d log 2|z 1 z 2 /∆|, then T (β) = T (β, 64) if c 0 = 0 T (β, 16) -T (β, 64) if c 0 = 2
T (β, ξ) ≤ C 0 N 2 (log N ) -σ + P -1 N 2 log N,
where P is any number in the range (2.30).

We recall that N and P appear as parameters restricting the support of β z ; see (2.43).

Proof of Lemma 2.13: oscillations of characters and symbols

Although complicated, the proof of [

ϕ(∆) = b|∆ µ(b)b -1 (see [19, p. 1013]).
Moreover, the restriction on the support of β z to z in a xed primary congruence class modulo 64q 1 (where q 1 is as in (2.19)) as opposed to modulo 8 is handled in the same way as in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]. For sums over medium-size moduli, the estimation of β z is trivial and so the restriction on the support is irrelevant (see bottom of [19, p. 1003]). For sums over small moduli, i.e., d of size at most a large power of log N , the key sum to bound from above is the character sum [19, (17.8) for p ≡ -1 mod 4

S k χ (β) = z β z χ(z) z |z| k , ( 2
β z = i r-1 2 s |r| β z if z = r + is (see
Let D be a fundamental discriminant, i.e., a discriminant of a quadratic number eld, and let Cl(D) denote the (narrow) class group of the quadratic number eld Q( √ D). Although there are algorithms to compute Cl(D) for any particular discriminant D, very little has been proved about the average behavior of Cl(D) as D ranges over families of fundamental discriminants.

Rédei [START_REF] Rédei | Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper[END_REF], Gerth [START_REF] Gerth | Extension of conjectures of Cohen and Lenstra[END_REF], Fouvry and Klüners [START_REF] Fouvry | The parity of the period of the continued fraction of √ d[END_REF][START_REF] Fouvry | On the 4-rank of class groups of quadratic number elds[END_REF][START_REF] Fouvry | Cohen-Lenstra heuristics of quadratic number elds[END_REF], and Stevenhagen [START_REF] Stevenhagen | Ray class groups and governing elds[END_REF], among others, obtained many density results about 4and 8-ranks of class groups in various families of quadratic number elds.

Density results appear to be far more dicult to obtain for the 16-rank than for the lower 2-power ranks (see [41, p. 16-18]). Our main goal in this chapter is to prove a density result about the 16-rank, albeit in a particularly simple family of quadratic number elds. This family is indexed by fundamental discriminants of the form -8p. Although -8p is a fundamental discriminant for all odd prime numbers p, the 8-rank of Q( √ -8p) behaves dierently in the cases that p ≡ 1 mod 4 and p ≡ -1 mod 4. Hence it is natural to study the families {Q( √ -8p)} p≡-1(4) and {Q( √ -8p)} p≡1(4) separately.

Equation (1.2) implies that the 2-part of the class group Cl(-8p) is nontrivial and cyclic, so the structure of the 2-part is completely determined by its depth, i.e., the largest integer k such that rk 2 k Cl(-8p) = 1. This motivates the following denition. Given an integer k ≥ 0, a real number X ≥ 2, and ω ∈ {±1}, dene ρ(2 k ; ω) to be the limit

ρ(2 k ; ω) = lim X→∞ #{p ≤ X prime : p ≡ ω mod 4, rk 2 k Cl(-8p) = 1} #{p ≤ X prime} , if it exists.
We now suppose that p ≡ -1 mod 4. It follows from the work of Rédei [START_REF] Rédei | Arithmetischer Beweis des Satzes über die Anzahl der durch vier teilbaren Invarianten der absoluten Klassengruppe im quadratischen Zahlkörper[END_REF] that rk 4 Cl(-8p) = 1 ⇐⇒ p ≡ -1 mod 8.

Furthermore, Hasse [START_REF] Hasse | Über die Klassenzahl des Körpers P ( √ -2p) mit einer Primzahl p = 2[END_REF] proved that rk 8 Cl(-8p) = 1 ⇐⇒ p ≡ -1 mod 16.

Both congruence conditions on p in the criteria above can be interpreted as splitting conditions on p in the degree-8 cyclotomic extension Q(ζ 16 )/Q. Now the hebotarev's Density Theorem implies that ρ(2 k ; -1) = 2 -k for 1 ≤ k ≤ 3.

A simple splitting condition that determines the value of rk 16 Cl(D) has not been found, and in fact might not even exist. Nonetheless, numerics and heuristics both suggest that ρ(2 k ; -1) exists and is equal to 2 -k for all k ≥ 1. Indeed, Cohen-Lenstra heuristics [START_REF] Cohen | Heuristics on class groups of number elds[END_REF] suggest that the cyclic group of order 2 k-1 would occur as the 2-part of the class group of an imaginary quadratic number eld twice as often as the cyclic group of order 2 k . As we just saw above, ρ(2 k ; -1) = 1 2 ρ(2 k-1 ; -1) for k = 2, 3, so we are led to conjecture Conjecture 3.1. For all k ≥ 1, the limit ρ(2 k , -1) exists and is equal to 2 -k .

No progress had been made on Conjecture 3.1 since the case k = 3 was settled by Hasse in 1969. Our main result of this chapter, Theorem A, now proves that ρ(16; -1) = 1

16 .

Theorem B. The density of the set of prime numbers p ≡ -1 mod 4 for which rk 16 Cl(-8p) = 1 is equal to

lim X→∞ #{p ≤ X, p prime, p ≡ -1 mod 4, rk 16 Cl(-8p) = 1} #{p ≤ X, p prime} = 1 16 .
To the best of the author's knowledge, this is the rst density result about the 16-rank of class groups of quadratic number elds.

Prior to this work, the only method for obtaining a density result was to construct certain normal extensions of Q that govern the 2 k -rank and then to apply the ebotarev Density Theorem. To be more precise, given a non-zero integer d and an integer k ≥ 1, we say that a normal extension M/Q is a governing eld for the 2 k -rank in the family of quadratic number elds {Q( √ dp)} p (parametrized by primes p for which dp is a fundamental discriminant) if the value of rk 2 k Cl(dp) is determined by the Frobenius class of p in Gal(M/Q). Knowing explicitly a governing eld for the 2 k -rank makes it easy to study the density of primes p for which rk 2 k Cl(dp) = k. Indeed, by the ebotarev Density Theorem, the mere existence of a governing eld already guarantees that this density exists and is equal to a rational number.

Although Cohn and Lagarias [START_REF] Cohn | Is there a density for the set of primes p such that the class number of Q( √ -p) is divisible by 16?[END_REF][START_REF] Cohn | On the existence of elds governing the 2-invariants of the classgroup of Q( √ dp) as p varies[END_REF] conjectured that, for a family {Q( √ dp)} p as above, a governing eld for the 2 k -rank exists for every integer k ≥ 1, and although Stevenhagen [START_REF] Stevenhagen | Ray class groups and governing elds[END_REF] proved their conjecture for k ≤ 3, a governing eld has not been found for the 16or higher 2-power ranks in any family. This

1 2 1 + v u = 1 if rk 16 Cl(-8p) = 1, 0 if rk 16 Cl(-8p) = 0.
Hence Theorem B is a corollary of the following theorem:

Theorem 3.1. For every > 0, there is a constant C > 0 depending only on such that for every X ≥ 2, we have

p≤X p≡-1 mod 16 v u ≤ C X 149 150 + ,
where, for each p in the sum above, u and v are taken to be integers satisfying

u 2 + v 2 √ 2 = ε 8 (u 1 + v 1 √ 2).
Then

v 1 u 1 = v 2 u 2 .
In other words, we have the equality of Jacobi symbols

v 1 u 1 = 408u 1 + 577v 1 577u 1 + 816v 1 .
The rest of this section is devoted to proving Proposition 3.1 and Proposition 3.2.

Preliminaries

We will use the following lemma several times.

Lemma 3.1. Let E/F be an abelian extension of number elds, let L/F be a nite extension, and let

ι : Gal(EL/L) → Gal(E/F )
be the restriction-to-E map. Then for every prime ideal p of L that is coprime to Disc(E/F ), we have

ι p EL/L = Norm L/F (p) E/F . Proof. See [25, Proposition 3.1, p. 103].

Ring class elds

To prove Proposition 3.2, we will have to work with a generalization of the Hilbert class eld. Let D < 0 be any integer ≡ 0, 1 mod 4 that is not a square, and let O D be the quadratic order of discriminant D, i.e., We call this eld G because it coincides with the genus eld of K in the case that n is a prime number congruent to -1 modulo 4.

O D = Z[(D + √ D)/2]. Let K = Q( √ D) be
Finally, we dene a quadratic extension of G as follows. Dene ν ∈ Z[

√ 2] ⊂ G by setting ν = u + v √ 2.
(3.9)

Then let

L = L u,v = G( √ εν)
,

where ε = 1 + √ 2 as before.
If n is a prime number congruent to -1 modulo 8 and u and v are chosen as in the statement of Proposition 3.1, we will see that L coincides with the 4-Hilbert class eld H 4 of K.

Remark. The elds K and G are determined simply by n. In other words, had we started with another choice of integers u and v giving rise to the same n, the denitions of K and G would not change. However, the eld L may depend on the specic choice of u and v. Since we xed u and v in the beginning of the section, this should not cause any confusion.

We now introduce some notation and prove some properties of the exten-

sions K ⊂ G ⊂ L. Let ν = u -v √ 2 be the conjugate of ν in Q( √ 2)
. We now state a few consequences of the assumption that gcd(u, v) = 1. It will be useful to consider the following eld diagram.

L = G( √ εν) G = K( √ 2) A = Q( √ 2, √ εν) K = Q( √ -2m) Q( √ 2) Q Lemma 3.3.
The extension L/K is cyclic of degree 4, and the extension L/Q is dihedral of order 8.

Proof. We have

Norm G/K (εν) = Norm Q( √ 2)/Q (εν) = -νν = -n. As -n = 2 • 1 2 √ -2n 2 ∈ 2 • (K × ) 2 , the rst claim follows from Lemma 2.2, part (3). Now let A = Q( √ 2, √ εν). As -n / ∈ (Q × ) 2 ∪ 2 • (Q × ) 2 , part (1) of Lemma 2.2 implies that L = A( √ -n) is the normal closure of A/Q and Gal(L/Q) ∼ = D 8 .
Let t denote the prime of K lying above 2.

Lemma 3.4. L/K is unramied at every prime other than possibly at t.

Proof. Recall that ν = δ 2 µ, so L = Q( √ -2m, √ 2
A = Q( √ 2, √ εν). Thus, as p splits in Q( √ 2
), its ramication index in L/Q is at most 2. But p already ramies in K/Q, and hence every prime p of K lying above p must be unramied in L/K. By Lemma 3.4, the only prime that can divide the conductor f of L/K is the prime t. The following lemma gives the precise power of t dividing f. Lemma 3.5. Let f denote the conductor of L/K. Then:

1. If v ≡ 1 mod 4, then L/K is unramied and f = 1.

If

v ≡ -1 mod 4, then f = t 2 = (2). 3. If v ≡ 0 mod 2, then f = t 4 = (4).
Proof. Since t is the only prime that can divide f, we only need to study the extensions locally at the primes above 2. Let T be a prime of G lying above t and T a prime of L lying above T. Let K t , G T , and L T denote the completions of K, G, and L with respect to the primes t, T, and T , respectively.

If v is odd, then n ≡ -1 mod 8, and so

K t = Q 2 ( √ -2n) = Q 2 ( √ 2) and G T = K t ( √ 2) = K t . Thus the extension G T /K t is trivial and L T = Q 2 ( √ 2, √ εν). The extension Q 2 ( √ 2, √ εν)/Q 2 ( √ 2) is unramied if and only if εν is a square modulo t 4 ; here t = ( √ 2) is the maximal ideal in Z 2 [ √ 2]. If v ≡ 1 mod 4, then εν = (u + 2v) + (u + v) √ 2 ≡ 1 mod t 4 if u ≡ -1 mod 4, ε 2 mod t 4 if u ≡ 1 mod 4,
and hence L T /K t is unramied. This proves part (1) of the lemma. Similarly, if v ≡ 1 mod 4, then εν ≡ 3 or 1 + 2 √ 2 mod t 4 .

In this case εν is not a square modulo t 4 , and so L T /K t is ramied. The ramication is wild, and thus f must be divisible by t 2 . As εν ≡ 1 mod t 2 , the extension L T /K t can be generated by a root of the polynomial

X 2 + √ 2X + 1 -εν 2 = 1 2 √ 2X + 1 2 -εν ,
whose discriminant is 2 mod t 4 . Hence f = t 2 and part (2) of the lemma is proved.

Finally, suppose v ≡ 0 mod 2, so that n ≡ 1 mod 8.

Then K t = Q 2 ( √ -2n) = Q 2 ( √ -2) and G T = K t ( √ 2) = Q 2 (ζ 8 ). The quadratic extension G T /K t is ramied of conductor t 2 , where t = ( √ -2) is the maximal ideal in Z 2 [ √ -2]. Let s = 1 + ζ 8 be a generator of the maximal ideal s in Z 2 [ζ 8 ]. Note that s 2 = √ 2 • ζ 8 ε, so εν ≡ 1 mod s 2 .
Hence the extension L T /K t can be generated by a root of the polynomial

X 2 + s 3 ζ 6 8 ε -2 X + 1 -εν s 2 = 1 s 2 (sX + 1) 2 -εν , whose discriminant is s 6 mod s 7 . Hence the discriminant of L T /G T is s 6 .
To nish, we use the conductor-discriminant formula, i.e.,

Disc(L

T /K t ) = Disc(G T /K t )f(L T /K t ) 2 .
The discriminant formula for the tower of elds Proof. Combine Lemmas 3.2, 3.3, and 3.5.

K t ⊂ G T ⊂ L T gives Disc(L T /K t ) = Disc(G T /K t ) 2 Norm G T /Kt (Disc(L T /G T )), so that f(L T /K t ) 2 = Disc(G T /K t )Norm G T /Kt (Disc(L T /G T )). Substituting Disc(G T /K t ) = t 2 and Disc(L T /G T ) = s 6 into the formula above implies that f(L T /K t ) = t 4 ,

A computation of Artin symbols

This section contains the heart of the proof of both Proposition 3.1 and Proposition 3.2.

The integers u and v appearing in (3.8) are not unique. Given a representation n = u 2 -2v 2 , another representation can be obtained by multiplying u + v √ 2 by 3 + 2 √ 2. This transforms (u, v) into (3u + 4v, 2u + 3v).

We will show how the quantity

v u χ(u),
where χ is a Dirichlet character from Proposition 3.2, naturally arises in the computation of a certain Artin symbol. This computation is somewhat delicate because the Artin symbol will take a value in a cyclic group of order 4, and such a group has a non-trivial automorphism.

Remark. In [START_REF] Halter-Koch | An Artin character and representations of primes by binary quadratic forms[END_REF], Halter-Koch, Kaplan, and Williams compute Artin symbols in similar cyclic eld extensions L/K of degree 4. Their results, however, involve computations of Artin symbols of ideals of K of order 2 in the class group of K, and hence only give information about the 8-rank in certain quadratic elds.

Let f ∈ {1, 4}. The case f = 1 will be used to prove Proposition 3.1, while the case f = 4 will be used to prove Proposition 3.2. Let τ = f √ -2n, so that Z[τ ] is the order of K of discriminant -8nf 2 . We dene a homomorphism ψ u,v : Z[τ ] → Z/uZ by sending τ → 2vf mod u. This homomorphism is well-dened since

τ 2 = -2nf 2 = -2(u 2 -2v 2 )f 2 ≡ (2vf ) 2 mod u. Let u = ker ψ u,v . (3.10)
It is the ideal of Z[τ ] generated by u and 2vf -τ , i.e., u = (u, 2vf -τ ).

In case n = p ≡ -1 mod 8 and f = 1, the ideal class of u turns out to have order 4, as we will see later. We remark that 2vf ≡ τ mod u. (3.11) We also note that Norm(u) = u. The key idea is to relate this Artin symbol to the Artin symbol associated to a dierent but related cyclic degree-4 extension of K.

Let γ = (2 + √ 2)v ∈ Z[ √ 2]. (3.13)
Then again by Lemma 2.2, the extension G( √ γ)/K is cyclic of degree 4. The element γ was chosen so that εν ≡ γ mod u, (3.14) and at the same time so that the extension Q( √ γ)/Q mimics the cyclic degree-

K = Q( √ -2n) G = K( √ 2) K( √ β ) K( √ β) G( √ ενγ) G( √ γ) F = G( √ εν, √ γ) G( √ εν)
Here β and β are elements of K that are conjugate over Q.

Let ενγ ∈ Q( √ 2) be the conjugate of ενγ over Q. Since 2ενγ ± 2ενγ 2 = 4v((4u + 6v) ± √ -2n) = 4v f ((4u + 6v)f ± τ ) ,
we can take

β = v((4u + 6v)f -τ ) and β = v((4u + 6v)f + τ ).
The inclusion Gal(F/K( √ β)) ⊂ Gal(F/K) and projections Gal(F/K) Gal(G( √ εν)/K) and Gal(F/K) Gal(G( √ γ)/K) induce canonical isomorphisms

ψ 1 : Gal(F/K( β)) ∼ -→ Gal(G( √ εν)/K) and ψ 2 : Gal(F/K( β)) ∼ -→ Gal(G( √ γ)/K). √ v = γ. There is a canonical isomorphism Gal(Q(ζ 16 √ v)/Q) ∼ = (Z/16Z) × ∼ = -1 mod 16 × 3 mod 16
given by sending

ζ 16 √ v → ζ k 16 √ v → (k mod 16). Then Q( √ γ) is the subeld of Q(ζ 16 √ v) xed by -1.
For each prime p coprime to 2v, we have p

Q(ζ 16 √ v)/Q = p v p mod 16,
so that if we identify

ψ 3 : 3 mod 16 ∼ -→ µ 4 = i ⊂ C ×
Note that the congruence p ≡ -1 mod 8 immediately implies that both u and v are odd. Without loss of generality, we may assume that u is positive and v ≡ 1 mod 4.

(3.15)

Since the 2-part of Cl(-8p) is cyclic, rk 16 Cl(-8p) = 1 if and only if Cl(-8p) has an element of order 16. To get started, we rst produce an element of order 4 in Cl(-8p) that we can write explicitly in terms of u and v.

A class of order 4

We now produce an ideal generating a class of order 4 in the class group Cl(-8p) when p is a prime ≡ -1 mod 8. This is the main ingredient in [START_REF] Leonard | On the divisibility of the class numbers of Q( √ -p) and Q( √ -2p) by 16[END_REF].

When n = p and f = 1, the ideal t = t f dened in Section 3.1.4 is the prime ideal lying above 2. If t = (x + y √ -2p) for some x, y ∈ Z, then

x 2 + 2py 2 = Norm(t) = 2,
which is impossible. Hence the class of t in Cl(-8p) has order 2.

Now let u be dened as in (3.10) with u and v as above and f = 1. Lemma 3.9 shows that u 2 and t are in the same ideal class in Cl(-8p). Hence we have proved the following result.

Lemma 3.10. Let u be the ideal of Z[ √ -2p] dened as above. Then the ideal class of u has order 4 in Cl(-8p). Remark. Perhaps an easier, although more old-fashioned, way to prove Lemma 3.10 is via the theory of binary quadratic forms, as was done in [START_REF] Leonard | On the divisibility of the class numbers of Q( √ -p) and Q( √ -2p) by 16[END_REF]. Let [a, b, c] denote the SL 2 (Z)-equivalence class of the form ax 2 + bxy + cy 2 . The key observation is that [u, -4v, 2u] has discriminant 16v 2 -8u 2 = -8p. To compose this class with itself, one can use the special case of the composition law for concordant forms, which yields the class [u, -4v, 2u] 2 = [u 2 , -4v, 2] = [2, 0, p].

The classes [u, -4v, 2u] and [2, 0, p] correspond to the ideal classes of u and t, respectively.

Generating the 4-Hilbert class eld

Let p be a prime congruent to -1 mod 8 and let K = Q( √ -8p). The 2-Hilbert class eld, also called the genus eld of K, is known to be H 2 = K( √ 2). Lemma 3.10 implies that rk 4 Cl(-8p) = 1, and our aim is to generate the 4-Hilbert class eld H 4 over H 2 by adjoining an element that we can write explicitly in terms of u and v.

Dene π ∈ Z[ √ 2
] by setting π = ν with ν as in (3.9), i.e.,

π = u + v √ 2.
The following proposition achieves our aim.

Sums over primes

Above, we dened the governing symbol [p] for a prime p ≡ -1 mod 16 in terms of particular integer solutions u and v to the equation p = u 2 -2v 2 . The main lemma that we will use to prove Theorem 3.1, i.e., that these governing symbols oscillate, is a proposition due to Friedlander, Iwaniec, Mazur and Rubin [START_REF] Friedlander | The spin of prime ideals[END_REF].

We now state this proposition in our context.

A result of Friedlander, Iwaniec, Mazur, and Rubin

Recall that an element w = u + v

√ 2 ∈ Z[ √ 2
Norm(n) := u 2 -2v 2 ,
where u + v √ 2 is a totally positive generator of n.

We now dene an analogue of the von Mangoldt function Λ for the ring Z[ Hence Λ is supported on powers of prime ideals.

Given a sequence of complex numbers {a n } n indexed by non-zero ideals in Z[

√ 2], a good estimate for the sum of a n over prime ideals p of norm Norm(p) ≤ X can usually be derived from a good estimate of the smoother weighted sum S(X) :=

Norm(n)≤X a n Λ(n).
The idea in [START_REF] Friedlander | The spin of prime ideals[END_REF] (and even earlier in [START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]), is to bound S(X) by combinations of linear and bilinear sums in a n . Given a non-zero ideal d of Z[

a n := [w] + [ε 2 w] + [ε 4 w] + [ε 6 w], (3.28)
where w 0 is any totally positive generator of n.

A convenient fact is that if p ≡ -1 mod 16 is a prime, then exactly one of the four elements

ε 2k w = u k + v k √ 2 (0 ≤ k ≤ 3) satises u k ≡
:= [w] φ,ψ + [ε 2 w] φ,ψ + [ε 4 w] φ,ψ + [ε 6 w] φ,ψ . (3.30)
This is still well-dened, i.e. independent of the choice of w 0, by Proposition 3.2 and by (3.24). We will apply Proposition 3.4 to 8 2 sequences {a φ,ψ,n } n , one for each pair of Dirichlet characters φ, ψ, and then add together the corresponding 8 2 sums S φ,ψ (X) to obtain Theorem 3.1. The key lemma is then Lemma 3.12. If p is a prime and p is a prime ideal lying above p, then we have Hence, to prove Theorem 3.1, it now suces to prove Theorem 3.2. Let a φ,ψ,n be dened as in (3.30). For every > 0, there is a constant C > 0 depending only on such that for every X ≥ 2, we have

Norm(n)≤X a φ,ψ,n Λ(n) ≤ C X 149 150 + .

Fundamental domains

In order to obtain power-saving cancellation for linear and bilinear sums as in Proposition 3.4, we will have to choose generators of n in (3.30) carefully. The problem reduces to nding a convenient fundamental domain for the action of

ε 2 = 3 + 2 √ 2 on totally positive elements of Z[ √ 2].
In [START_REF] Friedlander | The spin of prime ideals[END_REF], the authors describe how to construct such a fundamental domain in a more general setting. We give simpler arguments tailored to our specic needs and describe a fundamental domain very explicitly. This explicit description along with the ancillary pictures is possible in large part because the degree of the extension Q( Since Norm(ε 2 ) = 1 and since the norm is multiplicative, it follows immediately that ε 2 • Ω ⊂ Ω.

√ 2)/Q is 2. Let Ω := (u, v) ∈ R 2 : u > 0, -u < √ 2v < u .
Let D be the subset of Ω dened by

D := (u, v) ∈ R 2 : u > 0, -u < 2v ≤ u (3.31)
We claim that the region D in Figure 3.1 shown above is a fundamental domain for the action of ε 2 on Ω in the following sense.

Lemma 3.13. For each element (u, v) ∈ Ω ∩ Z 2 , there exists exactly one integer k such that ε 2k • (u, v) ∈ D. For each integer k, dene integers p k and q k by the equation

(3 + 2 √ 2) 2k = p k + q k √ 2. Since p 2 k -2q 2 k = 1, it follows that q k /p k → 1/ √ 2 as k → +∞. Moreover, p -k = p k and q -k = -q k , so that q k /p k → -1/ √ 2 as k → -∞. We will also use the fact that |q k /p k | < 1/ √ 2. Now let w = (u, v) ∈ Ω ∩ Z 2 . We have m(ε 2k • w) = q k u + p k v p k u + 2q k v = q k p k + v p k (p k u + 2q k v)
.

Since u and v are integers, so is

p k u + 2q k v. If p k u + 2q k v = 0, then v u = p k 2q k > 1 √ 2 ,
which contradicts the assumption that (u, v) ∈ Ω. Hence p k u + 2q k v is a nonzero integer, so that |p k u + 2q k v| ≥ 1, and since p k → +∞ as k → +∞, we deduce that

m(ε 2k • w) → ± 1 √ 2 as k → ±∞. Moreover, we have m(ε 2 • w) -m(w) = 2u + 3v 3u + 4v - v u = 2(u 2 -2v 2 ) (3u + 4v)u . As 3 √ 2 > 4, we deduce that 3u + 4v > 3 √ 2|v| + 4v ≥ 0,
and so m(ε 2 • w) -m(w) > 0. Also, as (u + 2v) 2 ≥ 0, we deduce that

2u 2 -4v 2 ≤ 3u 2 + 4uv, so that m(ε 2 • w) -m(w) ≤ 1.
Hence multiplying w ∈ Ω by ε 2 strictly increases its slope by at most 1 and multiplying w ∈ Ω by ε -2 strictly decreases its slope by at most 1. As |m(w 1 ) -m(w 2 )| < 1 for any two elements w 1 , w 2 ∈ D, this proves that for each w ∈ Ω ∩ Z 2 , there exists an integer k such that ε 2k w ∈ D.

To show that this integer k is unique, it remains to prove that if w = (u, v) ∈ D, then ε 2 • w = (3u + 4v, 2u + 3v) / ∈ D. Suppose for sake of contradiction that ε 2 • w ∈ D. Then 2(2u + 3v) ≤ 3u + 4v, so that -u ≥ 2v, which contradicts the assumption that (u, v) ∈ D.

An immediate consequence of Lemma 3.13 is the following proposition.

Proposition 3.5. Suppose that n is a non-zero ideal of Z[

√ 2].
Then n has a unique generator in D.

Geometry of numbers in the fundamental domain: the Lipschitz principle

We now briey turn to the problem of counting lattice points and boxes inside certain compact subsets of the fundamental domain D. We state a lemma of Davenport (see [START_REF] Davenport | On a principle of Lipschitz[END_REF] and [START_REF] Davenport | Corrigendum: On a principle of Lipschitz[END_REF]).

Let R be a compact, Lebesgue measurable subset of R n . Suppose that R satises the following two conditions:

1. Any line parallel to one of the n coordinate axes intersects R in a set of points which, if not empty, consists of at most h intervals, and 2. The same is true (with m in place of n) for any of the m-dimensional regions obtained by projecting R on one of the coordinate spaces dened by equating a selection of n -m of the coordinates to zero; and this condition is satised for all m from 1 to n -1.

Lemma 3.14 (Davenport). If R satises conditions (1) and ( 2) above, then

|R ∩ Z n -Vol(R)| ≤ n-1 m=0 h n-m V m
where V m is the sum of the m-dimensional volumes of the projections of R on the various coordinate spaces obtained by equating any n -m coordinates to zero, and V 0 = 1 by convention.

We will apply Lemma 3.14 to the fundamental domain D ⊂ R 2 as well as certain variations thereof.

Let k be a positive integer, and dene Then L(D k (X)) is a compact subset of R 2 that also satises conditions (1) and (2) above, also with h = 2.

D k = D ∪ ε 2 • D • • • ∪ ε 2k • D. Let X > 0. Then the region D k (X) := {(u, v) ∈ D k : u 2 -2v 2 ≤ X}
We dene the diameter of L to be diam(L) = |a| + |b| + |c| + |d|. Hence we dene

R(d, X) := {(u, v) ∈ R(X) : (u, v) ∈ Image(L d )}
(depicted in Figure 3.2), and we rewrite the sum A d (X) as

A d (X) = (u,v)∈R(d,X) [u + v √ 2] φ,ψ . Using the fact that |[u + v √ 2] φ,ψ | ≤ 1, we obtain the trivial bound |A d (X)| ≤ (u,v)∈R(d,X) 1 = L -1 d R(X)∩Z 2 1. (3.34) Since d 1 + d 2 √ 2 ∈ D, we have the inequalities d 2 1 2 ≤ D ≤ d 2 1 ,
Proposition 3.7. Let a n = a φ,ψ,n , where a φ,ψ,n is dened as in (3.30), and let B(M, N ) be dened as in (3.26). Then for all > 0 and all M, N ≥ 2, we have In other words, if w = a + b √ 2 with a, b ∈ Z, then r(w) = a.

B(M, N ) (M + N ) 1
We say that an element w = a + b

√ 2 ∈ Z[ √ 2] is primitive if and only if gcd(a, b) = 1.
Suppose w and z are primitive. Then wz need not be primitive. Nonetheless, we have the following lemma. Then wz/d is primitive. In particular, wz is primitive whenever gcd(w, σ(z)) = 1.

Proof.

w = a + b √ 2 ∈ Z[ √ 2], dene m(w) := r(w) Norm(w) = a a 2 -2b 2 . (3.42)
Note that w is primitive if and only if m(w) = 0. In this case, the law of quadratic reciprocity implies that a a 2 -2b 2 = (-1)

a-1 2 • a 2 -2b 2 -1 2 -2 a ,
and so m(w) ∈ {±1} depends only on the residue class of w modulo 8. We have Lemma 3.16. Let w ∈ Z[ √ 2] be odd, totally positive, and primitive, and let

z 1 , z 2 ∈ Z[ √ 2] be totally positive. Then γ(w, z 1 z 2 ) = γ(w, z 1 )γ(w, z 2 )m(w). (3.43) Proof. Write w = a + b √ 2, z 1 = c 1 + d 1 √ 2, and z 2 = c 2 + d 2 √ 2. Then γ(w, z 1 )γ(w, z 2 ) = a 2 c 1 c 2 + 2ab(c 1 d 2 + c 2 d 1 ) + 4b 2 d 1 d 2 a 2 -2b 2 .
Using the facts that 4b 2 ≡ 2a 2 mod a 2 -2b 2 and that z 1 z 2 = (c

1 c 2 + 2d 1 d 2 ) + (c 1 d 2 + c 2 d 1 ) √ 2, we deduce that γ(w, z 1 )γ(w, z 2 ) = a 2 (c 1 c 2 + 2d 1 d 2 ) + 2ab(c 1 d 2 + c 2 d 1 ) a 2 -2b 2 = a a 2 -2b 2 a(c 1 c 2 + 2d 1 d 2 ) + 2b(c 1 d 2 + c 2 d 1 ) a 2 -2b 2 = m(w)γ(w, z 1 z 2 ).
The symbol γ(w, z) also satises a reciprocity law. Lemma 3.17. Let w and z be odd, totally positive, and primitive elements of Finally, we remark that γ(w, z 1 ) = γ(w, z 2 ) whenever z 1 ≡ z 2 mod Norm(w).

Twisted multiplicativity of governing symbols

Recall

that if u + v √ 2 is a totally positive odd element of Z[ √ 2], we dene the governing symbol [u + v √ 2] to be [u + v √ 2] = v u . Thus [u + v √ 2] = 0 whenever u + v √ 2 is not primitive.
A key feature of the governing symbol

[wz] = [w][z],
does not hold for all totally positive w and z. Instead, the equation above becomes essentially valid when twisted by γ(w, z). We now state our result more precisely.

We now introduce notation that will simplify the subsequent arguments. Suppose that f 1 and f 2 are functions Z r → C. For x ∈ Z r , we write f 1 ∼ f 2 (or more conveniently f 1 (x) ∼ f 2 (x)) if there exists a function δ : Z r → {±1} such that δ factors though (Z/16Z) r , i.e. the value of δ(x) depends only on the congruence classes of the coordinates of x modulo 16, and such that

f 1 (x) = δ(x)f 2 (x)
for all x ∈ Z r . For instance, [u+v Proof. When wz is not primitive, then [wz] = 0 and γ(w, z) = 0, and so the result follows. Hence we may assume that wz is primitive.

√ 2] φ,ψ ∼ [u+v √
First note that wz = (ac + 2bd) + (ad + bc) √ 2.

We set ρ = (a, d) and dene a 1 and d 1 by the equalities a = ρa 1 and d = ρd 1 , respectively. Then

[wz] = ad + bc ac + 2bd = ad + bc ρ ad + bc a 1 c + 2bd 1 , and since ρ divides ad, the above simplies to

[wz] = bc ρ ad + bc a 1 c + 2bd 1 .

Now, since w is primitive, a 1 is relatively prime to b and hence also to a 1 c+2bd 1 .

Hence we may write c ≡ -2bd 1 /a 1 (mod a 1 c + 2bd 1 ), so that the second factor in the expression above becomes

ad + bc a 1 c + 2bd 1 = ad -2b 2 d 1 /a 1 a 1 c + 2bd 1 = a 1 d 1 a 1 c + 2bd 1 ρ 2 -2b 2 /a 2 1 a 1 c + 2bd 1 . As a 2 -2b 2 = a 2 1 (ρ 2 -2b 2 /a 2 1 ), we deduce that [wz] ∼ bc ρ a 1 d 1 a 1 c + 2bd 1 a 2 -2b 2 a 1 c + 2bd 1 .
We write the last factor in the expression above as

a 2 -2b 2 a 1 c + 2bd 1 = a 2 -2b 2 ρ a 2 -2b 2 ac + 2bd ,
and use the fact that

a 2 -2b 2 ρ = -2b 2 ρ = -2 ρ to conclude that [wz] ∼ -2bc ρ a 1 d 1 a 1 c + 2bd 1 a 2 -2b 2 ac + 2bd
.

The law of quadratic reciprocity implies that

a 2 -2b 2 ac + 2bd ∼ ac + 2bd a 2 -2b 2 , so that [wz] ∼ -2bc ρ a 1 d 1 a 1 c + 2bd 1 γ(w, z).
We again use the law of quadratic reciprocity to treat the middle term above.

We get

a 1 a 1 c + 2bd 1 = (-1) ν1(a,b,c,d,ρ) 2 a 1 bd 1 a 1 , where ν 1 (a, b, c, d, ρ) ≡ a 1 -1 2 • r 1 -1 2 mod 2 and r 1 = a 1 c + 2bd 1 .
Similarly, we write d 1 as

d 1 = 2 e d 2 ,
where d 2 is odd, and compute that

d 1 a 1 c + 2bd 1 = (-1) ν2(a,b,c,d,ρ) d 1 a 1 c , where now ν 2 (a, b, c, d, ρ) ≡ e r 2 1 -1 8 + d 2 -1 2 • r 1 -1 2 + d 2 -1 2 • a 1 c -1 2 +e a 2 1 c 2 -1 8 mod 2.
We thus have

[wz] ∼ (-1) ν1+ν2 2 a 1 -2bc ρ b a 1 d 1 c γ(w, z), which simplies to [wz] ∼ (-1) ν1+ν2+ν3 -1 ρ b a d c γ(w, z),
where

ν 3 = ν 3 (c, ρ) ≡ ρ -1 2 • c -1 2 mod 2.
It remains to show that (-1) ν1+ν2+ν3 -1 ρ depends only on the residue classes of a, b, c, d modulo 16. First note that whether e = 0, e = 1, or e ≥ 2 depends only on the residue class of d modulo 4 (and hence also modulo 16). Hence we can split into cases e = 0, e = 1, and e ≥ 2.

Note that if e ≥ 2 or e = 1 and b ≡ 0 mod 2, then r 1 ≡ a 1 c mod 8. Using this observation and the denitions of ν 1 , ν 2 , and ν 3 , we nd that First suppose e ≥ 2. Then r 1 ≡ a 1 c mod 8 and ν 2 ≡ 0 mod 2. Suppose rst that c ≡ 1 mod 4. Then ν 3 ≡ 0 mod 2 as well. Moreover, a 1 ≡ r 1 mod 4, so that

ν 1 ≡ a 1 -1 2 • a 1 -1 2 ≡ a 1 -1 2 mod 2.
It is enough to estimate (3.45) for each 0 ≤ k ≤ 3. First, suppose u + v √ 2 0

is primitive and odd. Then by Proposition 3.8, we have

[ε 2k (u + v √ 2)] ∼ [u + v √ 2][ε 2k ]γ(ε 2k , u + v √ 2) ∼ [u + v √ 2].
We write w = a + b Recall that γ(w, z 1 ) = γ(w, z 2 ) whenever z 1 ≡ z 2 mod Norm(w). Set q := Norm(w 1 w 2 ).

Under this assumption, we claim that gcd(d, a 2 -2b 2 ) = gcd(q, g) = gcd(d, g) = 1.

(3.51) These equalities will be useful in subsequent manipulations of Jacobi symbols.

We now prove the claim.

First, suppose that there is a prime p dividing d and a 2 -2b 2 . Let ξ = gcd(w 1 , σ(w 2 )), so that d = ξσ(ξ). Suppose p divides ξ or σ(ξ). In the former case, this would mean that p divides w 1 , while in the latter case it would mean that p divides w 2 . Both of these cases contradict the assumption that w 1 and w 2 are primitive. Hence p cannot divide either ξ or σ(ξ). Since z ≡ z 0 mod 16 and z 0 is an odd congruence class modulo 16, we see that a 2 -2b 2 is odd, and so also that p is odd. If p is inert in Z[ √ 2], then since p divides d, p must divide either ξ or σ(ξ), and this is a contradiction. Hence we may assume that p splits in Z[ √ 2], i.e. p = πσ(π) for some prime π in Z[ √ 2]. Again, as p divides neither ξ nor σ(ξ), we can assume without loss of generality that π divides ξ and σ(π) divides σ(ξ). This means that π divides w 1 and σ(π) divides w 2 . Now, since p (and hence π) divides a 2 -2b 2 = zσ(z), we nd that π divides z or σ(z). In the former case, σ(π) divides σ(z), which means that σ(π) divides gcd(w 2 , σ(z)), and this contradicts assumption (3.50). In the latter case, π divides gcd(w 1 , σ(z)), which again contradicts (3.50). Hence we have shown that (d, a 2 -2b 2 ) = 1. Now suppose that there is a prime p dividing q and g. As g is a rational integer, σ(g) = g, and so, as z = gz , we see that p divides σ(z). Since w 1 and w 2 are odd, p must be odd. If p divides w 1 or w 2 , then p divides gcd(w 1 w 2 , σ(z)), which contradicts assumption (3.50). Hence p cannot divide either w 1 or w 2 . If p is inert in Z[ √ 2], then, as q = w 1 w 2 σ(w 1 )σ(w 2 ), p divides at least one of w 1 , w 2 , σ(w 1 ), and σ(w 2 ). In fact, as p = σ(p), we see that p must divide either w 1 or w 2 , which is a contradiction. Hence we may assume that p splits in Z[ √ 2], i.e. p = πσ(π) for some prime π in Z[ √ 2]. Again, as p divides neither w 1 nor w 2 , we can assume without loss of generality that π divides w 1 . But then π divides gcd(w 1 , σ(z)), which contradicts assumption (3.50). Hence we have shown that gcd(q, g) = 1.

Finally, as d divides q, we immediately deduce that gcd(d, g) = 1. This nishes the proof of (3.51).

By denition of γ(•, •), as g is a rational integer, we have γ(w i , z) = g Norm(w i ) γ(w i , z )

for i = 1, 2. Hence γ(w 1 , z)γ(w 2 , z) = g q γ(w 1 , z )γ(w 2 , z ).

91

Now we can apply the reciprocity law from Lemma 3.17 twice to obtain γ(w 1 , z)γ(w 2 , z) = g q (γ(z , w 1 )m(w 1 z )) (γ(z , w 2 )m(w 2 z )) .

Recall that m(α) depends only on the residue class of α modulo 8. Using the fact that w 1 ≡ w 2 mod 16, we deduce that w 1 z ≡ w 2 z mod 16, and so m(w 1 z ) = m(w 2 z ). The assumption gcd(w 1 w 2 , σ(z)) = 1 ensures that w i z is primitive (see Lemma 3.15), and so that m(w i z ) ∈ {±1} for i = 1, 2. Hence m(w 1 z )m(w 2 z ) = m(w 1 z ) 2 = 1 and the expression above simplies to γ(w 1 , z)γ(w 2 , z) = g q γ(z , w 1 )γ(z , w 2 ). Again, z is primitive, so m(z ) ∈ {±1}. The above simplies to γ(w 1 , z)γ(w 2 , z) = g q γ(z , d)γ(z , w 1 w 2 /d).

We again use the reciprocity law from Lemma 3.17 on γ(z , w 1 w 2 /d) to obtain γ(w 1 , z)γ(w 2 , z) = g q γ(z , d)γ(w 1 w 2 /d, z )m(z w 1 w 2 /d).

As before, since g is a rational integer, γ(w 1 w 2 /d, z) = g q/d 2 γ(w 1 w 2 /d, z ).

By equation (3.51), the Jacobi symbols g q and g q/d 2 are non-zero. Hence g q g q/d 2 = g q 2 /d 2 = 1, .

Note that (r, s) depends only on the congruence classes of r and s modulo 4. Since g is odd, g 2 ≡ 1 mod 4, and so (d, a 2 -2b 2 ) = (d, a 2 -2b 2 ). In particular, as z ≡ z 0 mod 16, (d, a 2 -2b 2 ) depends only on the congruence classes of d and z 0 modulo 4, and not on z. Proceeding in the same way as for (3.55), we get m(z w 1 w 2 /d) = (x , x 2 -2y 2 ) -2

x .

(3.59)

As x = gx , we have -1

x = -1 x -1 g ,
that is, (x -1)/2 ≡ (x -1)/2 + (g -1)/2 mod 2. Hence (x , x 2 -2y 2 ) = (x, x 2 -2y 2 ) -1 x 2 -2y 2 g-1 2

.

(3.60)

Again, as x = gx , we also have Mama, Tata, Ano, i Mlao, hvala vam na va²oj podr²ci. Volim vas!

[ 2 ,

 2 Theorem 8, p. 231]). In light of this, instead of studying a particular quadratic ring, one might study the average behavior of certain arithmetic invariants attached to quadratic rings in families parametrized by special types of discriminants. The subject dealing with these types of problems is called arithmetic statistics.

  (1.5) by showing that rk 8 Cl(8p) + = rk 8 Cl(-8p) = 1 and rk 16 Cl(-8p) = 0 occurs for a positive density of primes p ≡ 1 mod 4. Similar improvements on the lower bound in (1.5) could be achieved by proving density results about the 16-rank of Cl(-4p) for primes p ≡ 1 mod 4. Limitations in certain analytic tools prevented us from proving density results about the 16-rank for either of the families {Q( √ -p)} p≡1 mod 4 and {Q( √ -2p)} p≡1 mod 4 . Instead, we proved results about the 16-rank for a subfamily of {Q( √ -p)} p≡1 mod 4 and for the family {Q( √ -2p)} p≡-1 mod 4 .

4 . 8 .

 48 lim X→∞ #{p ≤ X : p = a 2 + c 4 with c even and rk 16 Cl(-4p) = 1} #{p ≤ X : p = a 2 + c 4 with c even} = 1 The second result concerns the family {Q( √ -2p)} p≡-1 mod 4 . Theorem B. We have lim X→∞ #{p ≤ X : p ≡ -1 mod 4 and rk 16 Cl(-8p) = 1} #{p ≤ X : p ≡ -1 mod 4} = 1 Theorem A and Theorem B are the rst non-trivial density results about the 16-rank in families of quadratic number elds. Both of these theorems follow from new criteria for the 16-rank and estimates of sums of type (1

( 2 . 20 )

 220 Let g be the multiplicative function supported on cubefree integers dened in[19, Equation 3.16, p.961] as follows: let χ 4 denote the character of conductor 4; for p ≥ 3 set g(p)p = 1 + χ 4 (p) 1 -1 p and g(p 2 )p 2 = 1 + (1 + χ 4 (p)) 1 -1 p ;

  19, Section 3, p.955-962]. Dene A d (x) := n≤x n≡0 mod d a n and A (x) := A 1 (x).The goal is to check that the error terms r d (x) dened byr d (x) := A d (x) -g (d)A (x) (2.24)are small on average. To do this, we will prove an analogue of [19, Lemma 3.1, p.956], with M d (x) (representing the main term and dened in[19, p.955]) replaced byM d (x) = 1 dq 1 0<a 2 +b 2 ≤x Z (b)ρ(b; d) if (d, q) = 1and M d (x) = 0 otherwise; here ρ(b; d) is dened as in [19, p.955], i.e. it is the number of solutions α mod d to α 2 + b 2 ≡ 0 mod d.

( 2 .|α n | 2 ,

 22 29) is ≤ n0 mod q1 d≤D (d,q1)=1 ν mod d ν 2 +1≡0 mod d m≤(N -n0)/q1 α m,n0 e ν m d . so that by (2.28) we get d≤D (d,q1)=1

3 4 -

 4 8ε . All of the rene- ments from [19, Section 4, p.962-966] remain valid for our modied sequence (a n ). We briey recall these renements. First note that it is enough to prove the analogue of [19, Proposition 4.1, p.963]:

[ 19 ,

 19 p.967], so that (2.41) becomes B * (M, N ) = (ww,zz)=1 Imwz≡a0 mod q1 α w β z Z (Rewz).

( 2 . 42 )

 242 Similarly as in[19, (5.7), p.967], we split the sum B * (M, N ) into O(q4

4 (

 4 , p.968] to get B (M, N ) = B * (M, N ) + O M log N ) 3 where B (M, N ) := Imwz≡a0 mod q1 α w β z Z (Rewz).

  .50) by the same reasoning as in [19, Lemma 8.4, p.980].

Lemma 2 .

 2 10 now follows from this analogue of [19, Proposition 10.2, p.986]:

  is the main reason that Conjecture 3 has remained open for k ≥ 4 for such a long time. Theorem B gives a positive answer to Conjecture 3.1 for k = 4 without appealing to a governing eld. Instead, we use a criterion for the 16-rank of Cl(-8p) that is conducive to analytic techniques. In [30, Theorem 3, p.205], Leonard and Williams stated the following criterion. A prime p ≡ -1 mod 16 can be written as p = u 2 -2v 2 (3.1) where u and v are integers, u > 0, and u ≡ 1 mod 16.

  symbol. The rst few primes satisfying the above criterion are 127, 223, 479, 719, . . . . Note that integers u > 0 and v satisfying (3.1) and (3.2) are not unique. Nonetheless, the criterion (3.3) is valid for any choice of integers u > 0 and v satisfying (3.1) and (3.2). If u and v are such integers, then criterion (3.3) states that

√ 2 ]. 2 ]

 22 (3.1) and(3.2). Theorem 3.1 is an equidistribution result reminiscent of[START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF] Theorem 2, p.948]. In[START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF], Friedlander and Iwaniec associate a binary symbol (i.e., a quantity taking values in {±1}) to each non-zero ideal in Z[i] and show that its value is equidistributed over prime ideals in Z[i] ordered by the norm. Theorem 3.1 is a very similar type of result for the ring Z[ √ 2], although we encounter substantial new diculties coming from the more complicated unit group in Z[ In essence, an odd ideal in Z[ √ does not have a canonical generator, and we resort to averaging over four carefully chosen generators to dene an analogous binary symbol. Proving that the resulting symbol is well-dened already requires signicant new ideas. Section 3.1 contains the class eld theoretic construction of the governing symbol [p] = v u χ(u) for the 16-rank in the family {Q( √ -8p)} p≡-1(4) (see

  the eld of fractions of O D . Then K is an imaginary quadratic number eld of discriminant Disc(K) satisfying the equality D = f 2 Disc(K) for some positive integer f , called the conductor of O D . Let Cl(D) denote the class group of O D . Then there is a unique abelian extension R D /K called the

Lemma 3 . 6 .

 36 which completes the proof of part (3) of the lemma. L is contained in the ring class eld R D of the imaginary quadratic order O D of discriminant D = 16 • -8m.

√

  εν be a square root of εν. Then, by Lemma 2.2, the extension G( √ εν)/K is cyclic of degree 4. We are interested in computing the Artin symbol u G( √ εν)/K .

√ 2 ]

 2 . For a non-zero ideal n of Z[ √ 2], we set Λ(n) = log(Norm(p)) if n = p k for some prime ideal p and integer k ≥ 1 0 otherwise.

√ 2 )

 2 two sequences of complex numbers {α m } and {β n }, each indexed by non-zero ideals in Z[ √ 2], we dene the bilinear sum B(M, N ) := Norm(m)≤M Norm(n)≤N α m β n a mn . [p] for both of the prime ideals (u + v lying above p. Proposition 3.2 states that [w] = [ε 8 w] for any w ∈ Z[ √ 2], so we might naively dene

√ 2 . 2 ]√ 2 ]

 222 Then the lattice points (u, v) ∈ Ω ∩ Z 2 precisely enumerate the totally positive elements w = u + v The ring Z[ √ acts on itself by multiplication, and this induces an action Z[ × Ω → Ω given by (a, b) • (u, v) := (au + 2bv, bu + av).

Figure 3 . 1 :

 31 Figure 3.1: The region Ω and the fundamental domain D

is a compact subset of R 2 2 .

 22 and satises conditions (1) and (2) above with h = 2. Moreover, one can check that there exist positive real numbers a k and k such that Vol(D k (X)) = a k X (3.32) and Vol(∂(D k (X))) = k X 1 Now let L : R 2 → R 2 be an invertible linear transformation of the form ad -bc = 0.

Figure 3 . 2 :

 32 Figure 3.2: The region R(X) and the lattice points R(d, X)

  Before we begin the proof of Proposition 3.7, we rst dene a quantity γ(w, z) that oscillates in both arguments w, z ∈ Z[ √ 2].

3. 5 . 1

 51 The symbol γ(w, z) Let σ denote the non-trivial automorphism of Q( √ 2). Dene the rational part of an element w ∈ Z[ σ(w)) .

Lemma 3 . 15 .

 315 Suppose w and z are primitive. Let d = Norm(gcd(w, σ(z)).

  Then γ(w, z)γ(z, w) = m(wz).

(3. 44 )

 44 In particular, if γ(w, z) = 0 whenever gcd(w, σ(z)) = 1. Proof. We have γ(w, z)γ(z, w)

2 ]

 2 φ ,ψ for any four Dirichlet characters φ, ψ, φ , ψ . The following proposition is analogous to [19, Lemma 20.1, p. 1021]. Proposition 3.8. Let w = a + b √ 2 and z = c + d √ 2 be two primitive, totally positive, odd elements of Z[ √ 2]. Then [wz] ∼ [w][z]γ(w, z).

2 mod 2

 22 if e = 0 and b ≡ 1 mod 2 1 mod 2 if e = 1 and b ≡ 1 mod 2 0 mod 2 otherwise.

√ 2

 2 and z = c + d √ 2 and split (3.45) into 8 2 • 16 2 sums by xing congruence classes of a, b, c, and d modulo 16 (where the congruence classes of a and c are invertible). Then it suces to estimate each sum ± w∈D(M ) w≡w0 mod 16 z∈D(N ) z≡z0 mod 16 α w β z [wz]. Unless both w and z are primitive, wz is not primitive, and hence [wz] = 0 . Using Proposition 3.8 again and replacing α w by α w [w] and β z by β z [z], we are left to estimate sums of the type Q * (M, N ) := * w∈D(M ) w≡w0 mod 16 * z∈D(N ) z≡z0 mod 16 α w β z γ(w, z), (3.46) where * restricts the summation to primitive elements of Z[ √ 2]. The cancellation in the bilinear sum (3.46) comes from the double oscillation of the term γ(w, z) in the formula above. We also dene the closely related sum Q(M, N ) := * w∈D(M ) w≡w0 mod 16 z∈D(N ) z≡z0 mod 16α w β z γ(w, z),(3.47) and note that Q * (M, N ) is a special case of Q(M, N ) where the complex numbers β z are supported on primitive elements z.The Cauchy-Schwarz inequality implies that|Q(M, N )| 2 ≤ 1 , z)γ(w 2 , z).Since β z is bounded in modulus by N , Lemma 3.14 applied to L = Id gives z∈D(N ) z≡z0(16) |β z | 2 N Vol(D(N )) + N O(Vol(∂(D(N ))) + 1) N 1+ . (3.48)

Lemma 3 .

 3 15 ensures that w 1 w 2 /d is primitive. Hence we can now use the multiplicativity formula from Lemma 3.16 twice to obtainγ(w 1 , z)γ(w 2 , z) = g q γ(z , w 1 w 2 )m(z ) = g q (γ(z , d)γ(z , w 1 w 2 /d)m(z )) m(z ).

  and the above simplies toγ(w 1 , z)γ(w 2 , z) = γ(z , d)γ(w 1 w 2 /d, z)m(z w 1 w 2 /d).

( 3 .

 3 52)By denition of γ(•, •),γ(z , d) = a d a 2 -2b 2 = m(z ) d a 2 -2b 2 .

( 3

 3 .53) By equation (3.51), we use the law of quadratic reciprocity to writed a 2 -2b 2 = (d, a 2 -2b 2 ) a 2 -2b 2 d = (d, a 2 -2b 2 ) a 2 -2b 2 d ,(3.54)where for odd integers r and s, (r, s) = (-1)

2 ,

 2 Putting together (3.52),(3.53), and (3.54), we see that to accomplish our goal(3.49), it remains to show that the value of the factor m(z )m(z w 1 w 2 /d) is independent of z. By denition of m(•), the law of quadratic reciprocity, and the fact that g 2 ≡ 1 mod 4, we havem(z ) = (a , a 2 -2b 2 ) is, (a -1)/2 ≡ (a -1)/2 + (g -1)/2 mod 2. Hence (a , a 2 -2b 2 ) = (a, a 2 -2b 2) 55), (3.56), and (3.57), and using the denition of m(•), we get m(z ) = m(z) we note that m(z) depends only on the xed congruence class z 0 modulo 16 and not on z. 93 We now dene integers e and f by the equation w 1 w 2 /d = e + f √ and dene integers x, y, x , and y by the equations zw 1 w 2 /d = x + y √ 2 = g(x + y √ 2).

1 (a 2 - 2 ,

 122 59), (3.60), and (3.61) as before, and using the denition of m(•), we get m(z w 1 w 2 /d) = m(zw 1 w 2 /d) factor m(zw 1 w 2 /d) depends on w 1 , w 2 and the xed congruence class z 0 modulo 16 but not on z. Combining (3.58) and (3.62), and using the fact that x 2-2y 2 = (a 2 -2b 2 )(e 2 -2f 2 ), we nd that m(z )m(z w 1 w 2 /d) = δ(w 1 , w 2 , z 0 ) -2b 2 ) 2 (e 2 -2f 2 )where δ(w 1 , w 2 , z 0 ) ∈ {±1} depends only on w 1 , w 2 and the residue class of z 0 modulo 16. Finally, note thate 2 -2f 2 = Norm(w 1 w 2 /d) = Norm(w 1 w 2 ) d 2and that Norm(w 1 ) ≡ Norm(w 2 ) mod 4 (since again w 1 ≡ w 2 ≡ w 0 mod 16). Hence e 2 -2f 2 ≡ Norm(w 1 ) 2 ≡ 1 mod 4, and so m(z )m(z w 1 w 2 /d) = δ(w 1 , w 2 , z 0 ).

I

  am very grateful to my advisors Étienne Fouvry and Peter Stevenhagen for their guidance during the past three years, both mathematical and otherwise.Prof. Stevenhagen shared his immense expertise in class eld theory and gave me the algebraic tools necessary to attack problems related to class groups of quadratic number elds. In our very rst mathematical discussion, he introduced me to the problems about the 16-rank that captivated my curiosity and whose solutions form the backbone of my thesis. His help was unceasing, even in the face of great tribulations.Prof. Fouvry used his great knowledge of analytic techniques to help me overcome many obstacles in the course of my research. He always made sure that I made timely progress and also that I had a platform to share my results with the wider mathematical community. Above all, his greatest contribution to my thesis are his many words of encouragement he showed great trust in me and made me believe that anything is possible. I would also like to thank Régis de la Bretèche, Kevin Destagnol, Christian Elsholtz, Jan-Hendrik Evertse, Florent Jouve, Hendrik Lenstra, Peter Koymans, Ramon Moreira Nunes, Carlo Pagano, Marco Steng, and other professors and colleagues for creating a rich mathematical environment conducive to creativity. Ila Varma deserves special thanks for telling me about the ALGANT program. Thank you Corneel, Dawid, Chiara, Giacomo, Sydney, Giulio, Rosa, Abtien, Corina, Simon, Mima, Nikola, Iuliana, Liu, Chloe, Yan, Sudi, Drew, Pauline, Dino, Pnar, Martin, Kaitlin, Du²an, Izabela, Thanos, Rossella, Andrea, Bianca and other friends for sharing many wonderful experiences and helping me grow as a person during the past ve years.

  If d is divisible by a prime number p ≡ 3 mod 4, then (P -) clearly has no solutions. Let D be the set of positive squarefree integers not divisible by a prime p ≡ 3 mod 4. Stevenhagen[START_REF] Stevenhagen | The number of real quadratic elds having units of negative norm[END_REF] made the remarkable conjecture that the set of squarefree d for which (P -) is solvable has a positive density inside the set D, given in terms of an explicit innite product (see[START_REF] Stevenhagen | The number of real quadratic elds having units of negative norm[END_REF]

It follows from

(1.3

) that comparing the ordinary and the narrow class groups of quadratic number elds corresponds exactly to solving (P -) for squarefree integers d. jecture 1.2, p. 122]). Using the criterion (1.4), Fouvry and Klüners made signicant progress on Stevenhagen's conjecture; they proved strong upper and lower bounds for the proportion of squarefree d in D for which (P -) is solvable (see

  Cl(8p) + = 1 and rk 4 Cl(8p) = 0 =⇒ x 2 -2py 2 = -1 is not solvable. The 4-rank of the ordinary class group and the 8-rank of the narrow class group are determined by variants of fourth-power residue symbols. More precisely, for a prime p ≡ 1 mod 8, let [2, p] 4 = 1 if 2 is a fourth power modulo p and let [2, p] 4 = -1 otherwise. Similarly, for a prime p ≡ 1 mod 8, let [p, 2] 4 = 1 if p ≡ 1 mod 16 and let [p, 2] 4 = -1 otherwise. Then, for a prime p ≡ 1 mod 8,

	which gives a lower bound of
	1 2	≤ lim inf	(1.7)
	Now, again by (1.6) and (1.3), we have the implication
	rk 4

1 is solvable.

It turns out that for a prime p ≡ 1 mod 4,

rk 4 Cl(8p) + = 1 ⇐⇒ p ≡ 1 mod 8, X→∞ ρ(X). rk 4 Cl(8p) = 1 ⇐⇒ [2, p] 4 = [p, 2] 4 . (1.8)

Passing to Gaussian integers and using the ebotarev Density Theorem, it is not too hard to see that the condition above is satised for one-half of primes p ≡ 1 mod 8. This gives the upper bound in (1.5). Next, to improve the lower bound in (1.7), we use the implication rk 4 Cl(8p) + = rk 4 Cl(8p) = 1 and rk 8 Cl(8p) + = 0 =⇒ x 2 -2py 2 = -1 is solvable and the criterion, valid for primes p ≡ 1 mod 8, rk 8 Cl(8p) + = 1 ⇐⇒ [2, p] 4 = [p, 2] 4 = 1.

  2 with a ≡ α mod 16 and c ≡ γ mod 4 0 otherwise, where α and γ are specied congruence classes modulo 16 and modulo 4, respectively. Proving an asymptotic formula for the sum (1.10) with a n dened

as above is a very dicult problem, and its solution by Friedlander and Iwaniec

[START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF]

in the 1990's is still considered a major achievement in analytic number theory.

For Theorem B, the relevant sequence {a n } n is much more dicult to dene for general n. At prime indices p, the sequence is given by

  dened as follows. Let p be a prime ideal of F which is unramied in E and let P be any prime ideal of E lying above p. Let Norm(p) be the cardinality of the residue eld at p. Then the Artin symbol Cl is the ordinary class group of F . Similarly, the Artin symbol induces a canonical isomorphism between the narrow class group Cl + and the Galois group Gal(H + /F ), where H + denotes the narrow Hilbert class eld of F , i.e., the maximal unramied at all nite primes abelian extension of F . -part of the narrow class group Cl(qp) + is cyclic. Let K denote the quadratic eld Q( √ qp), and let Cl + = Cl(qp) + . Suppose for the moment that rk 2 n Cl + = 1. Then (Cl + ) 2 n is a subgroup of Cl + of index 2 n . We dene the 2 n -Hilbert class eld H 2 n to be the subeld of H xed by the the image of (Cl + ) 2 n under the isomorphism(1.11). Since the 2-primary part of Cl is cyclic, it follows immediately that H 2 n is the unique unramied at all nite If we have an ideal of K, explicitly dened in terms of p, that generates the class of order 2 in Cl(qp) + , then we might be able to deduce a criterion for rk 2 n Cl(qp) + once we have found the 2 n-1 -Hilbert class eld of K, again explicitly in terms of p. Indeed, we see from (1.12) and the denition of the Artin symbol that rk 2 n Cl(qp) + = 1 if and only if the ideal generating the class of order 2 splits in the 2 n-1 -Hilbert class eld.

	The Artin map is multiplicatively The Hilbert class eld H of F is the maximal unramied abelian extension the group homomorphism • E/F : I F → Gal(E/F ) p E/F is the unique element of Gal(E/F ) such that p E/F (α) ≡ α Norm(p) mod P for all α in the maximal order of E. We then extend • E/F to I F . of F . The Artin symbol induces a canonical isomorphism of groups • H/F : Cl -→ Gal(H/F ), (1.11) where The isomorphism (1.11) shows that information about the class group of F is encoded in the Galois theory of unramied abelian extensions of F . When-ever we can construct such an extension E/F , it must hold true that E ⊂ H, and so Gal(E/F ) is canonically isomorphic to a quotient of Cl. 1.7.2 2 n -Hilbert class elds Let qp be a fundamental discriminant divisible by exactly two primes, as in the beginning of Section 1.5. Then Gauss's genus theory (see (1.2)) implies n extension of K. Moreover, (1.11) induces a canonical isomorphism of cyclic groups of order 2 n • H 2 n /K : Cl + /(Cl + ) 2 n -→ Gal(H 2 n /K). (1.12) 1.7.3 General strategy The general strategy to prove density statements about 2 n -ranks of class groups Cl(qp) with q xed and p varying is to nd a criterion, in terms of p, for the existence of the 2 n -Hilbert class eld of K = Q( √ qp). We then use the criterion to encode some information about the 2 n -rank of Cl(dp) via a complex number a p and study the sum p≤X a p . To prove something interesting about the sum above, our criterion must be suciently conducive to the available analytic techniques. In practice, we often have to extend the denition of a n to all natural numbers n in some structured way. If the function n → a n is multiplicative, we can usually apply the classical theory of L-functions to deduce interesting results about the sum over primes. Otherwise, if the function n → a n is not multiplicative, in special cases we may be able to apply more advanced sieving techniques. The main diculty in proving density results about 16-ranks of the narrow class groups Cl(qp) + with q xed and p varying is that there is no known way to generate the 8-Hilbert class eld H 8 explicitly enough in terms of p so that one could apply analytic techniques. This is also the reason that density results about 8-ranks of the ordinary class groups Cl(8p) and 16-ranks of the narrow class groups Cl(8p) + are both dicult if rk 8 Cl(8p) + = 1, then rk 8 Cl(8p) = 1 that the 2primes, cyclic, degree-2 if and only if H 8 is totally real.

  Suppose p is a prime of the form a 2 + c 4 , where a and c are integers. Let h(-4p) denote the class number of Q( Proposition 2.1. Let a 0 ∈ {1, 3, 5, 7, 9, 11, 13, 15} and c 0 ∈ {0, 2}. Then, uniformly for X ≥ 3, we have the equality

	main theorem that we prove gives a new and very explicit criterion for divisibility by 16 of class numbers of Q( √ -4p) for p of the form (2.2). a≡a0 mod 16 Theorem 2.1. a 2 +c 4 ≤X c≡c0 mod 4

  where p is a prime of the form a 2 + c 4 with c even. This is yet another piece of evidence suggesting that Conjecture 2.1 is true for k = 4. However, we also note that Conjecture 2.1 for k = 4 does not imply Corollary 2.1, as knowledge of the behavior of the class numbers An immediate byproduct of Theorem 2.1 and criterion (2.4) is the following corollary.Corollary 2.2. Suppose p is a prime of the form a 2 + c 4 , where a is odd and c is even. Let p = T + U √ p denote a fundamental unit of Q(

	4 8 log X X 3/4 8 log X The proof of Proposition 2.1 will take a signicant portion of this chap-and #{p ≤ X : h(-4p) ≡ 8 mod 16} ≥ . ter. Although the ideas required to generalize [19, Theorem 1] in this way are not particularly deep, implementing them turns out to be quite complicated simply because the proof of [19, Theorem 1] itself is very dicult. One can thus view Sections 2.4-2.6 as a summary of the proof of [19, Theorem 1] in a slightly more general context. Since primes of the form a 2 + c 4 with c even have density 0 in the set of all primes, our methods cannot be used to tackle Conjecture 2.1. Nonethe-less, each of the cases (i) -(iv) in Theorem 2.1 occurs with the same density among all primes this form, so the analogous conjecture for k = 4 deduced from the Cohen-Lenstra heuristics above holds within the thin family of imag-inary quadratic number elds Q( √ -4p) √ p).

of Q( √ -4p) over the set of all primes p does not necessarily give information about their behavior over a thin subset of all primes.

We now give a consequence of our results and a criterion for divisibility by 16 due to Williams [45]. Let p ≡ 1 mod 8, and let p be a fundamental unit of the real quadratic eld Q( √ p), written in the form p = T + U √ p, where T and U are integers. The criterion states that if 8|h, then h ≡ T + p -1 mod 16, (2.4) so that 16|h if and only if T ≡ 1 -p mod 16. (i) If a ≡ ±1 mod 16 and c ≡ 0 mod 4, then T ≡ 0 mod 16 and U ≡ ±1 mod 8. (ii) If a ≡ ±3 mod 16 and c ≡ 2 mod 4, then T ≡ 8 mod 16 and U ≡ ±5 mod 8. (iii) If a ≡ ±7 mod 16 and c ≡ 0 mod 4, then T ≡ 8 mod 16 and U ≡ ±1 mod 8. (iv) If a ≡ ±5 mod 16 and c ≡ 2 mod 4, then T ≡ 0 mod 16 and U ≡ ±5 mod 8. This can be viewed as an extension of [27, Corollary 1.2(i), p.115-116] to primes of the form p = a 2 + c 4 . Now Proposition 2.1 gives Corollary 2.3. For a prime p ≡ 1 mod 8, let p

  .5) Hence the class [t] belongs to Cl 2 n if and only if t has trivial Artin symbol in Gal(H 2 n /K). By class eld theory, this is equivalent to t splitting completely in H 2 n . Therefore

  2.1.3 Explicit constructions of H 4 and H 8 Recall from (2.6) that 4 divides h if and only if the prime t of Q( √ -4p) lying over 2 splits in H 2 , which happens if and only if (1 + i) splits in H 2 /Q(i). As H 2 is obtained from Q(i) by adjoining a square root of p, Lemma 2.1 implies that this happens if and only if p ≡ ±1 (mod m 5 ), which, for p ≡ 1 (mod 4), is true if and only if p ≡ 1 (mod 8). Thus we have recovered the criterion for divisibility by 4.

  Indeed, ππ is a square in H 2 , so both square roots of π are also contained in H 2 (

	√	π). Hence we conclude
	by Lemma 2.4 that the 4-Hilbert class eld is given by

  we nd a criterion for divisibility by 8. Recall that h is divisible by 8 if and only if t splits completely in H 4 , i.e. if and only if π is a square in Q 2 (i). By Lemma 2.1, this happens if and only if π ≡ ±1 (mod m 5 ). In terms of a and b from (2.7), this means that

8|h ⇐⇒ a + b ≡ ±1 mod 8.

  With the above choice of y, it remains to prove Lemma 2.5 with A

d (x) replaced by A d (f ). Similarly as on [19, p.958], we write

  Recall the following inequality from[19, (3.6), p.957]: for any complex numbers α n and any D, N ≥ 1, we have , and the implied constant is absolute. Lemma 2.7 can be proved in the same way as[START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF] Lemma 3.3, p.957] given the following analogue of inequality (2.28).

	d≤D	ν mod d ν 2 +1≡0 mod d	n≤N	α n e	νn d	D	1 2 (D + N )	1 2 α ,	(2.28)
	where						1
							2
			α :=		|α n | 2
					n

  3.1 to the sequence (a n ) dened in(2.19). Hypotheses (H1)-(H7) for (a n ) are veried in the same way as hypotheses (H1)-(H7) for the sequence (a n ) dened in(2.22) (see comment at the end of Section 2.3.2).

	Proposition 2.3 implies that (a n ) satises hypothesis (R) for ε = 1/8000,
	D = x	3 4 -1 1000 ,

(2.35)

which is indeed in the range (2.16), and x r = x r (ε) large enough.

Applying Proposition 2.4 with the same D as in (2.35), with P any number in the range (2.30), with A = 5+2 26 , and with η = 1 100 establishes hypothesis

(B)

  (see[19, (4.14), p.965]). It then suces to show Proposition 2.4 with β(n, C) replaced by a smoothed version

		20
	(see [19, p.1018]) and dene
	ϑ := (log x) -A
	and
	θ := (log x) -A	(2.36)
	as on [19, p.965]. We split the sum (2.34) by using a smooth partition of unity.
	Let p be a smooth function supported on an interval
	N < n ≤ (1 + θ)N

with N < N < 2N , and suppose that p is twice dierentiable with p

(j)

(θN ) -j for j = 0, 1, 2

  , p.980]. If 16 • 2 ν with ν ≥ 1 is the exact power of 2 dividing ∆, we get

  1 and γ 2 = 2ω 2 so that N (a; r) is 4 times the number of solutions (ω 1 , ω 2 ) modulo r/4 to the Hence if 16 • 2 ν with ν ≥ 2 is the exact power of 2 dividing ∆, we get

	N (a; r) r	=	N (a; r/4) r/4	-	N (a; r/16) r/16	.
	system of congruences
	ω 1 ≡ ω 2 ≡ 1 mod 2
	aω 2 1 ≡ ω 2 2 mod r/4.

  The restriction on the support of β z can be detected by multiplicative characters modulo 64q 1 , so that we can simply transform χ into a character for the group (Z[i]/64q 1 dZ[i]) × . (see[19, Lemma 16.2, p. 1012]) and this modulus is dierent from 4d by a xed constant. Similarly, for the sums over large moduli, the key sum to bound from above is the character sum χ is a multiplicative character of the group (Z[i]/ξbdZ[i]) × (where b is an integer and d is again bounded by a large power of log N ) but β z is now

	S k χ (β ) =	z	β z χ(z)	z |z|	k	,	(2.53)
			The sum (2.52) is bounded by
	studying the Hecke L-functions
	L(s, ψ) =
	where the sum ranges over the non-zero odd ideals a of Z[i] and
	ψ(a) := χ(z)	z |z|

.52)

where χ is a multiplicative character of the group (Z[i]/ξdZ[i]) × (see

[19, (16.14)

, p. 1005]). a ψ(a)(N a) -s , k

where z is the unique primary Gaussian integer which generates a. The dependence on χ of the bound given for S k χ (β) is only through the modulus of χ where

  , p. 1014] and[19, (17.12), p. 1015]). Again, the restriction on the support of β z (and hence also β z ) can be detected by multiplicative characters modulo 64q 1 , so that we can transform χ into a character for the group (Z[i]/64q 1 bdZ[i]) × . Cancellation in the sum (2.53) is now achieved due to the oscillation of the symbol

	Chapter 3
	On the 16-rank of class groups of Q( √	-8p)
	i	r-1 2	s |r|
		51

as z varies over primary Gaussian integers, but again the dependence on χ of the bound given for (2.53) is only through the modulus of χ (see

[

19, Proposition 17.2, p. 1016]) and this modulus is again dierent from 4bd by a xed constant. This shows that Lemma 2.13 follows from [19, Proposition 10.2] and hence Proposition 2.4 is proved.

  , √ εµ). As the norm of µ is m, every prime that ramies in L/Q must divide 2m. Let p be a rational prime dividing m. Suppose p factors as ππ in Z[ √ 2], and, without loss of generality, suppose π divides ν. As u and v are coprime, ν and ν are coprime in Z[

	√	2]
	and hence π does not ramify in

  4 subextension of the cyclotomic extension Q(ζ 16 )/Q. Finally, let F be the compositum of G( We have the following eld diagram.

	√	εν) and G( √ γ).

  ] is totally positive if and only if Norm(w) = u 2 -2v 2 > 0 and u > 0. We sometimes write w 0 to say that w is totally positive. is a principal ideal domain and since the norm of the fundamental unitε over Q is -1, an ideal n in Z[ √ 2]can always be generated by a totally positive element. For an ideal n of Z[ √ 2], recall that the norm of n is given by

	Since Z[ √	2]

  1 mod 16. Indeed, multiplying u + v √ 2 by ε 2 (resp. ε 4 ) transforms (u, v) into (3u + 4v, 2u + 3v) (resp. (17u + 24v, 12u + 17v)). If p ≡ -1 mod 16, then u ≡ ±1 mod 8 and v is odd. Hence u 4 ≡ u + 8 mod 16, and one can now easily check that multiplying u+v √ 2 successively by ε 2 cycles u mod 16 through the set {1, 7, 9, 15}. denition (3.28) does not quite suce for our purposes because we want to isolate those p that are congruent to -1 mod 16 and representations p = u 2 -2v 2 with u ≡ 1 mod 16. Hence we weight the formula (3.28) by Dirichlet characters modulo 16. More precisely, for each pair of Dirichlet characters φ and ψ modulo 16 and totally positive u + v

	The √	2, we set
	[u + v	√	2] φ,ψ :=	v u	φ(-u 2 + 2v 2 )ψ(u).	(3.29)
	For a non-zero ideal n in Z[ √	2] generated by a totally positive element w, we
	set
	a φ,ψ,n

  If p is inert in Z[ divides either w or z, which contradicts the assumption that w and z are primitive. Now suppose that p splits in Z[√ 2] (resp. p = 2), so that p = ξσ(ξ) (resp. p = -ξσ(ξ)) for some prime ξ ∈ Z[ √ 2]. If p k is the exact power of p dividing wz, then the assumption that w and z are primitive implies that ξ k |w and σ(ξ k )|z, which is true if and only if ξ k | gcd(w 1 , σ(w 2 )).Our choice of terminology is inspired by the Dirichlet symbol dened in a slightly dierent setting in[19, Section 19, p. 1018[19, Section 19, p. -1021]].The symbol γ(w, z) is almost multiplicative in the second argument. More precisely, for an odd, totally positive

	√ 2], p The lemma now follows by unique factorization in Z[ 2] and p|wz, then by unique prime factorization in Z[ √ √ 2].
	Given an odd, totally positive, primitive w ∈ Z[ z ∈ Z[ √ 2], we dene the generalized Dirichlet symbol γ(w, z) to be √ 2] a totally positive
	γ(w, z) :=	r(wz) Norm(w)	,	(3.41)
	where z = c + d • • is the Jacobi symbol. More concretely, if we write w = a + b √ 2, then	√	2 and
	γ(w, z) =	ac + 2bd a 2 -2b 2 .

  [•] which leads to signicant cancellation in (3.26) is that [•] is not multiplicative, i.e. the relation

√ -4p).(i) If a ≡ ±1 mod 16 and c ≡ 0 mod 4, then h(-4p) ≡ 0 mod 16.(ii) If a ≡ ±3 mod 16 and c ≡ 2 mod 4, then h(-4p) ≡ 0 mod 16.(iii) If a ≡ ±7 mod 16 and c ≡ 0 mod 4, then h(-4p) ≡ 8 mod 16.(iv) If a ≡ ±5 mod 16 and c ≡ 2 mod 4, then h(-4p) ≡ 8 mod 16.Once we prove Theorem 2.1, Theorem A follows from the following generalization of a powerful theorem of Friedlander and Iwaniec (see[START_REF] Friedlander | The polynomial X 2 + Y 4 captures its primes[END_REF] Theorem 1]):

≡ 1 2 + 1 4 mod 6 is not invertible modulo 6. This does not mean, however, that there are no primes of the form a 2 +c 4 with a ≡ 1 mod

and c ≡ 1 mod 2; one such prime is

2 + 1 4 .

Acknowledgements

CV Acknowledgements

 Proposition 3.1). Another aim of Section 3.1 is to prove an invariance result for the Jacobi symbol v u (see Proposition 3.2). In Section 3.2, we construct binary symbols that both encode behavior of the 16-rank in our family and are conducive to analytic techniques (see Equations (3.29) and (3.30)). We also reduce Theorem 3.1 to a purely analytic statement (see Theorem 3.2) that can be attacked by the machinery of Friedlander, Iwaniec, Mazur, and Rubin (see Proposition 3.4). The goal of Section 3.3 is to construct convenient fundamental domains for the multiplicative action of a fundamental unit 1 +

. In Section 3.4, we use a Polya-Vinogradov-type estimate to give bounds for linear sums of the binary symbol. In Section 3.5, we give bounds for general bilinear sums of the binary symbol, thus completing the proof of Theorem 3.1. In the nal section, we show that if a governing eld for the [START_REF] Fouvry | The parity of the period of the continued fraction of √ d[END_REF]rank in the family {Q( √ -8p)} p≡-1(4) were to exist, Theorem 3.1 would give error terms for certain prime-counting functions that are far better than any which could be obtained via the best known zero-free regions of L-functions.

Finally, we say a few words about the family {Q( √ -8p)} p≡1(4) . Given a prime p ≡ 1 mod 4, the 4-rank of Cl(-8p) is equal to 1 if and only if p ≡ 1 mod 8. Then, given a prime p ≡ 1 mod 8 and a representation of p as p = u 2 -2v 2 for integers u ≡ 1 mod 4 and v, rk 8 Cl(-8p) = 1 if and only if u p = 1 (see [30, 2.2, P.204]). Finally, rk 16 Cl(-8p) = 1 if and only if the binary symbol u p 4 is 1; see [START_REF] Leonard | On the divisibility of the class numbers of Q( √ -p) and Q( √ -2p) by 16[END_REF]Theorem 2,p.204]. Here the quantity u p 4 is equal to 1 or -1 according to whether u is a fourth power modulo p or u is a square but not a fourth power modulo p, respectively. Heuristically, we once again expect that the value of this binary symbol is equidistributed as p ranges over the prime numbers congruent to 1 modulo 8 such that u is a square modulo p.

However, although we could generalize most of the ingredients in the proof of Theorem 3.1 to this new setting, we are unable to obtain power-saving cancellation in the linear sums as in Section 3.4 without a Burgess-type estimate for short character (modulo q) sums of length q 1 8 -. As such a result on short character sums is currently well out of reach, we do not deal with the family {Q( √ -8p)} p≡1(4) .

Governing symbols

The purpose of this section is to generalize [START_REF] Leonard | On the divisibility of the class numbers of Q( √ -p) and Q( √ -2p) by 16[END_REF]Theorem 3,p.205] and to develop a framework conducive to the analytic techniques of Friedlander, Iwaniec, Mazur, and Rubin [START_REF] Friedlander | The spin of prime ideals[END_REF].

Let χ be a character (Z/16Z) × → Cl × with kernel {±1}. In other words, we have χ(±1 mod 16) = 1 and χ(±7 mod 16) = -1. Then our generalization of [START_REF] Leonard | On the divisibility of the class numbers of Q( √ -p) and Q( √ -2p) by 16[END_REF]Theorem 3,p.205] is as follows:

Proposition 3.1. Let p ≡ -1 mod 16 be a prime number. Let u and v be integers such that p = u 2 -2v 2 and such that u > 0 and v ≡ 1 mod 4. Then

The choice of u and v in the proposition above is not unique. Let

then the complete set of integer solutions (u, v) to the system (3.5) is of the

for some integer k. An interesting consequence of Proposition 3.1 is that the

is independent of the choice of u and v satisfying (3.5). This allows us to make the following denition.

For a prime p ≡ -1 mod 16, we dene the governing symbol for the 16-rank to be

where u and v are integers satisfying (3.5). The quantity [p] determines the 16-rank of the class group Cl(-8p). It is interesting to note that the 16-rank of Cl(-8p) depends on a quantitative aspect of the splitting behavior of p in Z[

√ 2] that appears to allow no description purely in terms of the qualitative splitting behavior of p in some normal extension of Q.

Leonard and Williams claim that [START_REF] Leonard | On the divisibility of the class numbers of Q( √ -p) and Q( √ -2p) by 16[END_REF]Theorem 3,p.205] can be proved by numerous manipulations of Jacobi symbols and applications of quadratic reciprocity. We instead prove Proposition 3.1 by interpreting the Jacobi symbol v u as an Artin symbol of an ideal that depends on the decomposition of a prime p as p = u 2 -2v 2 in an extension of Q( √ -8p) that depends on the same decomposition p = u 2 -2v 2 . Moreover, a by-product of our proof is the following proposition, which turns out to be essential for a successful application of the analytic tools we wish to use. (3.7)

In the case f = 1, so that D = Disc(K), the ring class eld R D coincides with the Hilbert class eld of K.

The main property of ring class elds of imaginary quadratic orders that we will use is stated in the following lemma.

Lemma 3.2. Let K be an imaginary quadratic number eld of even discriminant, and let L/K be a cyclic extension such that:

• L/Q is a dihedral extension, and

• the conductor of L/K divides (4). Then L is contained in the ring class eld R D of the imaginary quadratic order

Proof. See [START_REF] Cox | Primes of the form x 2 + ny 2[END_REF]Theorem 9.18,p. 191] and [START_REF] Cox | Primes of the form x 2 + ny 2[END_REF]Exercise 9.20,.

A special family of quadratic elds

Let u and v be coprime integers such that u is odd and positive and such that

is positive as well. Let K be the imaginary quadratic number eld dened by

Note that n ≡ ±1 mod 8, and moreover n ≡ 1 mod 8 if and only if v is even. Let m and d be the unique positive integers such that m is squarefree and

We emphasize that both m and d are odd. As gcd(u, v) = 1, every prime dividing n splits in Q( √ 2). Hence there exist δ and µ in Q( √ 2) of norm d and m, respectively, such that u + v √ 2 = δ 2 µ.

We dene a quadratic extension G/K by

Using (3.11), we nd that if p is a prime ideal dividing u, then

and so p splits in K( √ β). By Lemma 3.1, for any prime P of K( √ β) lying above a prime ideal p dividing u, we have

Multiplying over all prime ideals p dividing u, we have proved the following key lemma.

Lemma 3.7. Let u be dened as in (3.10). Then

We have

so that, by Lemma 3.7, we have

Multiplying over all primes p dividing u and using Lemma 3.7, we nally obtain the following result.

Lemma 3.8. Let ψ : Gal(G( √ εν)/K) ∼ -→ µ 4 be the isomorphism of cyclic groups of order 4 dened by

An ideal identity

We keep the same notation as in Sections 3.

given by sending τ → 2vf . The homomorphism τ f is well-dened because

Then t f = (2vf -τ, 2f 2 ). The following identity of between ideals in Z[τ ] will be useful in proofs of both Proposition 3.1 and Proposition 3.2.

Lemma 3.9. Let u be dened as in (3.10). Then

Proof. The principal ideal 2vf -τ is invertible of norm 2u 2 f 2 . Since u is odd and gcd(u, v) = 1, we deduce that u is coprime to the discriminant -8nf 2 of Z[τ ] and is thus invertible. No rational primes can divide 2vf -τ and u divides (2vf -τ ) by denition, so it must be that u 2 divides (2vf -τ ).

The ideal t f of norm 2f 2 contains (2vf -τ ) and has the same norm as the invertible ideal (2vf -τ )u -2 . Hence we must have (2vf -τ )u -2 = t f .

Proof of Proposition 3.1

We apply the results of Sections 3.1.3 and 3.1.4 in the case n = p ≡ -1 mod 8 is a prime number and f = 1. Suppose p ≡ -1 mod 8 is a prime number. Then p splits in Q( √ 2), so there exist integers u and v such that

), and let π be as above. Then the 4-Hilbert class eld of K is

Proof. Since the 2-part of the class group Cl(-8p) is cyclic, it suces to show that H 2 ( √ επ) is an unramied, cyclic, degree-4 extension of K.

We apply the lemmas of Sections 3.1.2 and 3.1.3 with n = m = p, e = 1, and u and v as above. By Lemma 3.3, the extension

Conclusion of the proof of Proposition

which proves Proposition 3.1.

Proof of Proposition 3.2

As in the statement of Proposition 3.2, let u 1 and v 1 be integers such that u 1 is odd and positive and such that u 2 1 -2v 2 1 > 0. We dene u 2 and v 2 by the equality

where, as before, ε = 1 + √ 2. Our goal is to prove the following equality of Jacobi symbols

(3.17)

By the Euclidean algorithm, we have the equality

then both sides of (3.17) are equal to 0, and hence (3.17) holds true. Proof. Let k ∈ {1, 2}. By Lemma 3.9, we have

) is as in Section 3.1.4. By (3.16), we have

Let α = (17u 1 + 24v 1 ) + 3τ.

We claim that

We rst note that

Expanding α 2 , we get

Then

By Lemma 3.6, L is contained in the ring class eld of Z[τ ]. Hence, by Lemma 3.11, the images of both u 1 and u 2 under the map (3.7) coincide, i.e.,

Applying Lemma 3.8, we get

Equation (3.16) implies that

Hence, as χ is a character modulo 16, we have χ(u 1 ) = χ(u 2 ), and so Proposition 3.2 is nally proved.

We consider bilinear sums where the complex numbers α m and β n satisfy Suppose that there exist two real numbers 0 < θ 1 , θ 2 < 1 such that: for every > 0, we have

uniformly for all non-zero ideals

and all X ≥ 2, and

uniformly for all M, N ≥ 2 and sequences of complex numbers {α m } and {β n } satisfying (3.27).

Then for all X ≥ 2 and all > 0, we have the bound

+ .

In other words, power-saving estimates for linear and bilinear sums imply power-saving estimates for sums supported on primes. Note that this result is now classical in the context of rational integers, thanks to the pioneering work of Vinogradov [START_REF] Vinogradov | The method of trigonometrical sums in the theory of numbers[END_REF].

Extending governing symbols

In light of Proposition 3.4, our current goal is to dene a sequence a n over all non-zero ideals n of Z[ √ 2] so that if p ≡ -1 mod 16 is a prime and p is a prime ideal of Z[ √ 2] lying above p, then a p coincides with the governing symbol [p] dened in (3.6). We rst dene [•] for all totally positive elements of Z[

We put

where the implied constant is absolute.

Linear sums

In this section we prove that the estimate (A) from Proposition 3.4 holds for the sequence {a φ,ψ,n } n dened in (3.30) with θ 1 = 1/6. Proposition 3.6. Let a n = a φ,ψ,n , where a φ,ψ,n is dened as in (3.30), and let A d (X) be dened as in (3.25). Then for all > 0 and all X ≥ 2, we have

Proof. Recall that

Since the sequence a n is supported on odd ideals n, we see that A d (X) = 0 unless d is odd. Hence we may assume without loss of generality that d is an

By Proposition 3.5 and denition (3.30), we have

φ,ψ is dened as in (3.29).

We now reformulate the congruence condition u + v √ 2 ≡ 0 mod d. Proposition 3.5 implies that there is an element

2 ∈ D which generates d. Then the congruence above is equivalent to saying that there exist integers e 1 and e 2 such that u + v

In other words, (u, v) is in the image of the linear transformation

where the implied constant is absolute. This estimate will be useful when D is large compared to X.

Next we split the sum A d (X) into 8 • 16 sums where the congruence classes of u and v modulo 16 are xed, say u ≡ u 0 mod 16 and v ≡ v 0 mod 16 for some congruence classes u 0 and v 0 modulo 16 with u 0 invertible modulo 16. For u and v satisfying these congruences, we have

where δ(u 0 , v 0 ) ∈ {±1} depends only on the congruence classes u 0 and v 0 modulo 16. Hence it remains to give estimates for sums of the type

Splitting the sum according to the value of u, we obtain

where

and I u is an interval (or a union of 2 disjoint intervals) of size ≤ 2R 2 (X),

where

.

We now unwind the condition (u, v) ∈ L d (Z 2 ), i.e. that (u, v) is in the image of L d . Consider the system of equations in x and y:

Now suppose u ≡ 0 mod d, and let x u , y u ∈ Z be such that

Then all solutions (x, y) ∈ Z 2 to the rst equation in (3.37) are given by

Thus we have proved that if u ≡ 0 mod d, then

.

Let e u = gcd(v u , 16D/d), write 16D/d = e u d u , v u = e u v u , and perform a change of variables v = e u v , so that

where I u = I u /e u . Since gcd(v u , d u ) = 1, we can now detect the congruence condition v ≡ v u mod d u via Dirichlet characters modulo d u . In other words,

where v u denotes the multiplicative inverse of v u modulo d u . Let χ be a Dirichlet character modulo d u . If the character

is trivial, then u = f g 2 for some f dividing d u (and therefore dividing 16D/d) and some integer g. The number of such u 4 .

In this case we use the trivial bound

where the implied constant in is absolute. Hence the contribution of such

On the other hand, if the character

and so the Polya-Vinogradov inequality gives the estimate

Combining this with (3.36), (3.38), and (3.39), we have proved the bound A d (X) D

Bilinear sums

We are left with proving the estimate (B) from Proposition 3.4, which we do with θ 2 = 1/12 in much the same way as in [19, Sections 19-21, p. 1018[19, Sections 19-21, p. -1028]].

Finally, as a = a 1 ρ,

and so ν 1 + (ρ -1)/2 ≡ (a -1)/2 mod 2. Now suppose c ≡ 3 mod 4. Then ρ and cρ are odd and dierent modulo 2, and so ν 3 + (ρ -1)/2 ≡ 1 mod 2. Moreover, r 1 ≡ 3a 1 mod 4, so that r 1 and a 1 are odd and dierent modulo 4. Hence at least one of (r 1 -1)/2 and (a 1 -1)/2 is 0 mod 2 and so ν 1 = 0.

Collecting these results, we get

Now suppose e = 1. Then splitting into cases similarly as above, we get

Finally, suppose e = 0. Then This proves the lemma.

Proof of Proposition 3.7

We are now ready to prove Proposition 3.7. Let

where again D is dened as in (3.31). We will say that u + v √ 2 ∈ D(X) to mean that (u, v) ∈ D(X). Then the bilinear sum (3.26) can be written as

Here α w = α (w) and β z = β (z) , i.e. α w (resp. β z ) depends only on the ideal generated by w (resp. z).

Hence we can split the inner sum over z into residue classes modulo 16q. More precisely, we write z 0 = z 01 +z 02 √ 2 and dene L to be the linear transformation L = 16q • Id + (z 01 , z 02 ) : R 2 → R 2 . Then Lemma 3.14 gives z∈D(N ) z≡z0( 16)

where a 0 is dened as in (3.32) Proposition 3.9. Let w 0 and z 0 be odd congruence classes modulo 16 in

be primitive, totally positive, and odd. Suppose

Let gcd(w 1 , σ(w 2 )) denote a totally positive generator for the greatest common divisor of the ideals (w 1 ) and (σ(w 2 )) in Z[

where δ ∈ {±1} possibly depends on w 1 , w 2 and the xed congruence class z 0 mod 16 but not on z.

It is possible that z is not primitive, so that we cannot directly apply the reciprocity law from Lemma 3.17 to γ(w 1 , z) and γ(w 2 , z). However, we can factor out the greatest common factor of a and b to obtain First assume that gcd(w 1 w 2 , σ(z)) = 1.

(3.50) Thus, in the case that gcd(w 1 w 2 , σ(z)) = 1 (see (3.50)), we have proved (3.49).

Suppose now that (3.50) is not satised, i.e., (w 1 w 2 , σ(z)) = 1. Then, by Lemma 3.17, either γ(w 1 , z) = 0 or γ(w 1 , z) = 0. Moreover, either (w 1 w 2 /d, σ(z)) = 1 or (w 1 w 2 /d, σ(z)) = 1. In the former case, (w 1 w 2 /d, σ(z)) divides both d and zσ(z) = a 2 -2b 2 , so that (a 2 -2b 2 , d) = 1. In the latter case, (w 1 w 2 /d, σ(z)) = 1, so γ(w 1 w 2 /d, z) = 0 again by Lemma 3.17. As 0 = 0, we have once again proved Note that there exists an integer t such that t 2 ≡ 2 mod q because q is a norm of an element in Z[

Let t be such that t ≡ 2f /e mod q/d 2 ; this is possible since, by denition, q/d 2 = e 2 -2f 2 and w 1 w 2 /d is primitive. Then, as d divides q, we may rewrite the above sum as x mod q x q/d = q y mod d y d

x mod q/d x q/d , and this implies the desired result.

Since ϕ(d)ϕ(q/d) ≤ q and q ≤ M 2 , we deduce that whenever q and d are both squares, z∈D(N ) z≡z0( 16)

By unique factorization in Z[ √ 2], the number of elements w ∈ D such that Norm(w) = n is at most τ (n), the number of divisors of n. Hence, setting m 1 = Norm(w 1 ) and m 2 = Norm(w 2 ), and using (3.27) and (3.48), we get the upper bound

Using the estimate τ (n) n , we obtain Lemma 3.18. Let Q(M, N ) be dened as in (3.47). Then

(M N ) .

We now apply Hölder's inequality with k even to (3.47) to get

By Lemma 3.18, we have

Using this estimate with k = 6 along with the upper bound (proved similarly as (3.48))

w∈D(N ) w≡w0( 16) This completes the proof of Theorem 3.2 and hence also Theorem 3.1.

Counting primes

In this section we give evidence that a governing eld for the 16-rank of the family {Q( √ -8p)} p≡3(4) does not exist. To explain why, we rst dene a prime counting function. Suppose M/Q is a normal extension. Let S be a subset of Gal(M/Q) which is a union of conjugacy classes. We dene π(M, S, X) := #{p ≤ X : the Artin class of p in Gal(M/Q) is a subset of S} Given any normal extension M/Q of degree d and a subset S of Gal(M/Q) stable under conjugation, the ebotarev Density Theorem using the best known zero-free regions of L-functions gives [39, Théorème 2, p. 132], for some constant c > 0, π(M, S, X) = #S #Gal(M/Q) Li(X) + O(#SX exp(-cd -1/2 log 1/2 X)).

Hence given any two subsets S 1 and S 2 of Gal(M/Q) which are stable under conjugation and of the same size, π(M, S 1 , X) -π(M, S 2 , X) #SX exp(-cd -1/2 log 1/2 X)

is the best known bound. Note that this bound is weaker than X 1-δ for any δ > 0. For instance, it is not known if # {p ≤ X prime : p ≡ 1 mod 4}-# {p ≤ X prime : p ≡ -1 mod 4} X 0.9999 .

However, we have the following result.

Theorem 3.3. Suppose that there exists a governing eld M for the 16-rank of the family {Q( √ -8p)} p≡3(4) . Then there exist disjoint subsets S 1 and S 2 of Gal(M/Q) which are stable under conjugation and of equal size such that π(M, S 1 , X) -π(M, S 2 , X) ε X 149 150 + 97 Proof. We simply let S 1 be the union of Artin classes c p for primes p satisfying rk 16 Cl(-8p) = 1 and S 2 be the union of Artin classes c p for primes p satisfying rk 8 Cl(-8p) = 1 but rk 16 Cl(-8p) = 0. The result now immediately follows from Theorem 3.1.

However, with our current methods of complex analysis applied to Lfunctions, we are not able to produce an error term of the form O(x 1-δ M ) for any δ M > 0. This leads us to believe that a governing eld M for the 16-rank of the family {Q( √ -8p)} p≡3( 4) is unlikely to exist.