Samuel Le Fourn
email: samuel.le_fourn@ens-lyon.fr

Nonvanishing of central values of L-functions of newforms in S 2 (Γ 0 (dp 2 )) twisted by quadratic characters

Keywords: Nonvanishing of L-functions of modular forms, Petersson trace formula, rank zero quotients

Nonvanishing of Central Values of L-functions of Newforms in S_2(Γ_0(dp

2 )) Twisted by Quadratic Characters

Introduction

Let d ∈ {2, 3, 5, 7, 13} (that is, a prime number such that the genus of X 0 (d) is zero) and K/Q be a real extension of degree at most two, with discriminant D assumed prime to d and associated Dirichlet character χ.

The main result of this paper is the following (with the usual notations, recalled in section 2).

Theorem 1. For every prime number p not dividing dD,

f ∈Bp a 1 (f )L(f ⊗ χ, 1) = 2π + O √ D(log(D) + 1) 3 log(p) 2 p 2 ,
where B p is a Petersson-orthonormal basis of the subspace of S 2 (Γ 0 (dp 2 )) new spanned by the modular forms xed by the Atkin-Lehner involution w p 2 , and the implied constant is absolute.

Remark 2.

(a) For d = 1 and K imaginary, an analogous result (with sign (-1) for w p 2 and 4π instead of 2π) is given in [START_REF] Ellenberg | Galois Representations Attached to Q-Curves and the Generalized Fermat Equation A 4 + B 2 = C p[END_REF] (Lemma 3.10 and the associated discussion therein), itself based on the estimates given in [START_REF]On the Error Term in Duke's Estimate for the Average Special Value of L-Functions[END_REF], Theorem 1 and Corollary 2). We gave by similar methods a completely explicit bound for that case in [START_REF]Surjectivity of Galois representations associated with quadratic Q-curves[END_REF] (Proposition 5.13 of the Appendix and its proof), but for simplicity we focus here on giving good exponents in D and p, without explicit constants.

(b) The method here is the same in principle as in [START_REF] Ellenberg | Galois Representations Attached to Q-Curves and the Generalized Fermat Equation A 4 + B 2 = C p[END_REF], but the level dp 2 instead of p 2 involves more preliminary computations to deal with the contribution of the oldforms, which will be done in Lemma 7. This is also why d is assumed in {2, 3, 5, 7, 13} : for now, it is the only situation where we can take out the p 2 -old part, because S 2 (Γ 0 (d)) = 0 (see Remark 8 for further details).

(c) We assume that D and dp 2 are coprime to have a simple expression for (f ⊗ χ) |w dp 2 D 2 in terms of f |w dp 2 (Lemma 6 (a)). If it is not the case, f ⊗ χ is a modular form of smaller level ( [START_REF] Iwaniec | Analytic number theory[END_REF], Proposition 14. 19), but the action of the (smaller level) Atkin-Lehner involution seems less natural to describe.

As an essential tool for the proof of Theorem 1, we devised a new Petersson trace formula which essentially gives a closed expression for the same weighted sum as classical Petersson trace formula on S 2 (Γ 0 (N )), but for the sum only on the eigenforms having prescribed eigenvalues (1 or -1) for the possible Atkin-Lehner involutions on Γ 0 (N ). The general version is given in Proposition 14, but for now, we only give the case of one prescribed eigenvalue in Proposition 3 below.

Proposition 3 (Restricted Petersson trace formula).

Let m, n, N be three positive integers, an integer Q|N such that Q > 1 with (Q, N/Q) = 1, and ε = ±1. Let B be an eigenbasis of S 2 (Γ 0 (N )). Then, we have

1 2π √ mn f ∈B f |w Q =ε•f a m (f )a n (f ) f 2 = δ mn -2π c>0 N |c S(m, n; c) c J 1 4π √ mn c
(1)

-2πε c>0 (N/Q)|c (c,Q)=1 S(m, nQ -1 ; c) c √ Q J 1 4π √ mn c √ Q ,
where nQ -1 in the Kloosterman sum means n times the inverse of Q modulo c (see section 5 for the denitions of Kloosterman sums and the Bessel function J 1 ).

Notice there are analogues of Proposition 3 already given in the litterature, but this version seems to be the most general and the easiest to use (the comparison is made in Remark 10).

The arithmetic motivation of Theorem 1 is the following.

Corollary 4. For d ∈ {2, 3, 5, 7, 13} and p = d a prime number, let J(d, p) := J 0 (dp 2 ) p-new /(w p 2 -1)J 0 (dp 2 ) p-new .

(a) For p 1, there exists a nonzero quotient abelian variety of J(d, p) with only nitely many Q-rational points (such a quotient is called a rank zero quotient ).

(b) Let K be a real quadratic eld with discriminant D prime to d and Dirichlet character χ. Let J(d, p, χ) be the twist of J(d, p) by the extension K/Q relatively to the automorphism [χ(d)]w d , so that the points of J(d, p, χ)(Q) correspond to the points P of J(d, p)(K) such that

P σ = χ(d)w d • P,
where σ is the nontrivial automorphism of K. Then, for p D 1/4 log(D) 3 , there is a rank zero quotient of J(d, p, χ).

(c) The same results hold when replacing J(d, p) by the jacobian J (d, p) of the modular curve X ns 0 (d; p) parametrising the triples (E, C d , α p ), where C d is a cyclic subgroup of E of order d, and α p a normaliser of nonsplit Cartan structure on E. of the modular curve. To be precise, the twist J (d, p, χ) of J (d, p) by the extension K/Q relatively to the automorphism [χ(d)]w d also has a rank zero quotient for p D 1/4 log(D) 3 .

The proof of Corollary 4 (which is quite straightforward for the reader familiar with the techniques), has been displaced to the end of this paper to keep the focus on the analytic side for now.

The following remark compares it with others results in the litterature.

Remark 5.

(a) The part (a) of Corollary 4 has been already proven in [START_REF] Darmon | Winding quotients and some variants of Fermat's Last Theorem[END_REF] through modular symbols for p ≥ 7. Their result is obviously stronger, but our proof shows it can be found through analytic tools at least for large p (which can be made explicit if necessary). The real interest of Corollary 4 lies in part (b) and (c), and a possible application (which would require further work) to nd a linear lower bound for the dimension of the winding quotient of J(d, p), following the methods of [START_REF] Iwaniec | The Non-vanishing of Central Values of Automorphic L-Functions and Landau-Siegel Zeros[END_REF].

(b) The restriction d ∈ {2, 3, 5, 7, 13} comes from Theorem 1. For d / ∈ {2, 3, 5, 7, 13} prime, there is actually a natural rank zero quotient of J (d, p) (not its twist) given by the Eisenstein quotient of J 0 (d), because of (32). The existence of a rank zero quotient of J(d, p) (or its twist J(d, p, χ)) for d / ∈ {2, 3, 5, 7, 13} does not seem attainable by the methods used here. (c) The application to Mazur's method for Q-curves as designed in ([Ell05], Proposition 3.6) is doable for large enough p, K quadratic real and d ∈ {2, 3, 5, 7, 13} with d split in K i.e. χ(d) = 1, which is the limitation exposed in ([Ell05], Remark 3.7), because we obtain points P such that P σ = w d • P (modulo torsion). For d / ∈ {2, 3, 5, 7, 13} prime, the techniques do not work here, because of (b) and because it would amount to proving that the jacobian of X 0 (d)/w d has a rank zero quotient, which is not true if we admit the conjecture of Birch and Swinnerton-Dyer because for f ∈ S 2 (Γ 0 (d)) + , L(f, 1) = 0. Notice that this problem is related to the existence of quadratic Q-curves of degree d for general prime d, and a still open conjecture of Elkies [START_REF] Elkies | On Elliptic K-curves, Modular Curves and Abelian Varieties[END_REF] states that for large enough d, there are none.

We will begin with the useful notations in section 2, followed by a reminder on the approximate functional equation to estimate the L(f ⊗ χ, 1) in section 3. We then prove Lemma 7 which allows us to separate the contribution of the newforms in the moment of the a 1 (f )L(f ⊗ χ, 1) over S 2 (Γ 0 (dp 2 )) + p 2 in section 4. In section 5, we obtain the mentioned Petersson trace formula restricted to Atkin-Lehner involution spaces. Finally, we compute all the terms involved in the computation of the moment, leading to the proof of Theorem 1 in section 6 and conclude with the proof of Corollary 4 there.

Notations

Let N be a positive integer and H be the Poincaré half-plane.

• S 2 (Γ 0 (N )) is the complex vector space of cuspidal forms of weight 2 for Γ 0 (N ), and we add the superscripts {+, -, old, new} to refer respectively to the subspaces made up of the forms

f such that f |w N = f , f |w N = -f , f is old, f is new.
We will cumulate the superscripts when it is nonambiguous, for example S 2 (Γ 0 (N )) +,old is the subspace of oldforms f such that f |w N = -f .

• For f ∈ S 2 (Γ 0 (N )), one has the q-expansion

f (z) = n≥1 a n (f )e 2iπnz (z ∈ H)
and we will keep this notation a n (f ) throughout. The L-function associated to f is dened as a holomorphic series over the domain {Re(s) > 2} by

L(f, s) = n≥1 a n (f ) n s .
• For f ∈ S 2 (Γ 0 (N )) and χ a Dirichlet character, the twist f ⊗ χ is dened on H as the series

(f ⊗ χ)(z) = n≥1 χ(n)a n (f )e 2iπnz (z ∈ H)
and its L-function on {Re(s) > 2} is in the same fashion dened by the holomorphic series

L(f ⊗ χ, s) = n≥1 χ(n)a n (f ) n s ,
which extends to a holomorphic function on C (Lemma 6 (b) for details).

• For every m ∈ N, dene a m the linear form associating to any modular form f ∈ S 2 (Γ 0 (N )) the coecient a m (f ), and

L χ : f → L(f ⊗ χ, 1) • For γ = a b c d ∈ GL + 2 (R) and z ∈ H, dene γ • z := az + b cz + d , j γ (z) := cz + d.
For any holomorphic function f on H, let f |γ be the function on H dened by

f |γ (z) := det γ j γ (z) 2 f (γ • z).
We recall that this denes a right action of GL + 2 (R), satisfying the following formulas

Im(γ • z) = (det γ) Im z |j γ (z)| 2 , γ • z = a c - det γ cj γ (z) , j γγ (z) = j γ (γ • z)j γ (z). (2)
which we will frequently use without specic mention.

• The Petersson scalar product

•, • N on S 2 (Γ 0 (N )) is dened by f, g N = D f (x + iy)g(x + iy)dxdy,
where D is a choice of fundamental domain for the action of Γ 0 (N ) on H. Dened as such, the Petersson scalar product depends on the chosen congruence subgroup, for example when N divides N and f, g ∈ S 2 (Γ 0 (N )), we have

f, g N = [Γ 0 (N ) : Γ 0 (N )] f, g N . (3)
• For every positive divisor Q of N such that gcd(Q, N/Q) = 1, choose W Q a matrix of the following form:

W Q := Q y N Qt , y, t ∈ Z, det W Q = Q. ( 4
)
For every f ∈ S 2 (Γ 0 (N )), the function f |W Q does not depend on the choice of W Q , and the Atkin-Lehner involution of degree Q on S 2 (Γ 0 (N )) is the corresponding involution on this space (noted w Q to emphasize its canonical nature). For ε = ±1, the space S 2 (Γ 0 (N )) ε Q is the subspace of S 2 (Γ 0 (N )) made up with the modular forms f such that f |w Q = εf , for example S 2 (Γ 0 (N )) + = S 2 (Γ 0 (N )) + N . Note that the denition of W Q generally depends on N , so we will precise (unless the context is obvious) on which spaces we are considering them. For more details about these involutions, see [START_REF] Atkin | Hecke operators on Γ 0 (m)[END_REF]. In particular, notice that for f ∈ S 2 (Γ 0 (N )),

f |w N (z) = 1 N z 2 f i N z . (5)
• For any subspace V of S 2 (Γ 0 (N )) and any linear forms A, B on V , dene

(A, B) V N := f ∈F V A(f )B(f ), (6)
where F V is a Petersson-orthonormal basis of V . This denes a scalar product on V * independent of the choice of F V . We will in particular denote by (A, B) N the scalar product of A and B on the whole space S 2 (Γ 0 (N )), again adding natural superscripts corresponding to how V is dened as a subspace of S 2 (Γ 0 (N )). For example, Theorem 1 is exactly reformulated as

(a 1 , L χ ) + p 2 ,new dp 2 = 2π + O √ D(log(D) + 1) 3 log(p) 2 p 2 . ( 7
)
3

The approximate functional equation

We will recall here some necessary results to provide a way of evaluating L(f ⊗ χ, 1).

Lemma 6. Let χ be a quadratic character of conductor D and f ∈ S 2 (Γ 0 (N )) with N prime to

D. (a)

The twisted modular form f ⊗ χ belongs to S 2 (Γ 0 (D 2 N )) and

(f ⊗ χ) |w D 2 N = χ(-N )f |w N ⊗ χ. (8)
(b) The holomorphic series L(f ⊗ χ, •) extends to an holomorphic function on C and for every x > 0, one has

L(f ⊗ χ, 1) = +∞ n=1 χ(n)a n (f ) n e -2πn x -χ(-N ) +∞ n=1 χ(n)a n (f |w N ) n e -2πnx D 2 N . ( 9
) (c) In particular, if f |w N = χ(-N ) • f , we have L(f ⊗ χ, 1) = 0.
Proof.

(a) This is a classical result, which can for example be found in ([Bum96], § I.5).

(b) Let M = D 2 N . We will reprove below that for every g ∈ S 2 (Γ 0 (M )), the L-function of g extends to C and

L(g, 1) = +∞ n=1 a n (g) n e -2πn x - +∞ n=1 a n (g|w M ) n e -2πnx M , (10)
so that (b) is a direct consequence of (a) and (10). On Re(s) > 2, let us dene the completed L-function of g by

Λ(g, s) := M s/2 Γ(s) (2π) s L(g, s).
As usual, by absolute convergence, we can write

Λ(g, s) = M s/2 +∞ n=1 a n (g) +∞ 0 e -t t 2πn s dt t = M s/2 +∞ n=1 a n (g) +∞ 0
e -2πny y s dy y

= M s/2 +∞ 0 +∞ n=1
a n (g)e -2πny y s dy y

= +∞ 0 g(iy)(M 1/2 y) s dy y .
We choose x > 0 and split the integral between [1/x, +∞[ and ]0, 1/x]. We obtain

Λ(g, s) = +∞ 1/x g(iy)(M 1/2 y) s dy y + 1/x 0 g(iy)(M 1/2 y) s dy y = +∞ 1/x g(iy)(M 1/2 y) s dy y + +∞ x/M g i M t 1 M 1/2 t s dt t t = 1 M y = +∞ 1/x g(iy)(M 1/2 y) s dy y + +∞ x/M M (it) 2 g |w M (it) 1 M 1/2 t s dt t using (5). We obtain the integral expression Λ(g, s) = M s/2 +∞ 1/x g(iy)y s dy y -M 1-s/2 +∞ x/M g |w M (iy)y 2-s dy y .
This immediately proves that Λ(g) extends to an entire function satisfying the functional equation

Λ(g, 2 -s) = -Λ(g |w M , s), (11)
hence L(g, •) extends to an entire function on C. For the central value s = 1, we have

Λ(g, 1) = √ M +∞ n=1 a n (g) +∞ 1/x e -2πny dy - √ M +∞ n=1 a n (g |w M ) +∞ x/M e -2πny dy = √ M +∞ n=1 a n (g) 2πn e -2πn x - +∞ n=1 a n (g |w M ) 2πn e -2πnx M ,
which proves (10) as L(g, 1) = 2πΛ(f, 1)/ √ M . (c) This is a straightforward consequence of (b) by applying (9) to x = D √ N , for which the two integrals on the right cancel each other.

4

A key lemma to isolate the contribution of the newforms Lemma 7. Let d ∈ {2, 3, 5, 7, 13} and χ be an even Dirichlet character with conductor D prime to d. For every prime number p not dividing dD, we have

(a 1 , L χ ) + p 2 ,new dp 2 = (a 1 , L χ ) + p 2 dp 2 - 1 p -1 (a 1 , L χ ) χ(p)p dp . ( 12
)
• For the choice of signp 2 , either the sign of eigenvalue for w d is -χ(d), then the sign of the twisted L-function is (-1), giving an automatic vanishing (Lemma 6 (c)), either it is χ(d), and then the proof below does not work : indeed, we could not evaluate exactly the contribution of the d-old space, as the formula analogous to (19

) for f ∈ S 2 (Γ 0 (p 2 )), g ∈ S 2 (Γ 0 (p 2 )) is f |A1 , g |A d dp 2 = f |T d , g p 2 ,
and the eigenvalues of T d on S 2 (Γ 0 (p 2 )) are not simply ±1. Actually, as one knows that the eigenvalues of T d are of modulus bounded by 2 √ d and that every a 1 (f )L χ (f ) is nonnegative when f is an eigenform (see [START_REF] Guo | On the positivity of the central critical values of automorphic L-functions for GL(2)[END_REF]), one can easily compute that the contribution of the d-old forms is bounded in absolute value by a term of the shape

O( 1 d )(a 1 , L χ ) - p 2 .
It is not needed in the present case, therefore we do not give more details.

• The number d is assumed in {2, 3, 5, 7, 13} to ensure we can evaluate the contribution of the p-old space in S 2 (Γ 0 (dp)), which is automatically trivial in this case. If d is a larger prime, as the analogue of the formula (19

) for f ∈ S 2 (Γ 0 (d), g ∈ S 2 (Γ 0 (d) is f |A1 , g |Ap dp = f |Tp , g d ,
we cannot obtain an exact formula such as (12) for the same reason as above, but we could again bound the contribution of these old forms by some term of the shape

O( 1 p )(a 1 , L χ ) d .
Proof. By denition of the newforms and oldforms, we have the orthogonal decomposition

S 2 (Γ 0 (dp 2 )) + p 2 = S 2 (Γ 0 (dp 2 )) + p 2 ,new ⊕ S 2 (Γ 0 (dp 2 )) + p 2 ,old hence (a 1 , L χ ) + p 2 ,new dp 2 = (a 1 , L χ ) + p 2 dp 2 -(a 1 , L χ ) + p 2 ,old dp 2 , ( 13
)
so we have to relate this scalar product on the oldpart to the right term in the Lemma.

Following the notations of [START_REF] Atkin | Hecke operators on Γ 0 (m)[END_REF], let us dene, for every positive integer δ, A δ = δ 0 0 1 and for every positive integers M and N such that M |N and δ divides N/M , the operator

A δ : S 2 (Γ 0 (M )) → S 2 (Γ 0 (N )) f → f |A δ .
Looking at the q-expansions, we immediately see that for every δ ≥ 1,

f |A δ ⊗χ = χ(δ) (f ⊗ χ) |A δ hence L χ (f |A δ ) = +∞ 0 (f |A δ ⊗ χ)(iu)du = δ • χ(δ) +∞ 0 (f ⊗ χ)(iδu)du = χ(δ)L χ (f ). (14)
In particular,

L χ (f |A1 ) = L χ and L χ (f |A δ ) = 0 if L χ (f ) = 0. ( 15
)
By denition, the old part of S 2 (Γ 0 (dp 2 )) is the subspace spanned by the f |A1 , f |Ap with f ∈ S 2 (Γ 0 (dp)) (it is the p-old space) and by the

f |A1 , f |A d , f ∈ S 2 (Γ 0 (p 2 )) (it is the d-old space).
Let us begin with the d-old space: as d and p are coprime, by Lemma 26 of [START_REF] Atkin | Hecke operators on Γ 0 (m)[END_REF], for all f ∈ S 2 (Γ 0 (p 2 )),

f |A1 |w p 2 = f |w p 2 |A1 and f |A d |w p 2 = f |w p 2 |A d . (16)
In particular, f |A1 and f |A d have the same eigenvalue for w p 2 in S 2 (Γ 0 (dp 2 )) as f in S 2 (Γ 0 (p 2 )), which proves that S 2 (Γ 0 (dp 2 )) + p 2 ,d-old is generated by the f |A1 , f |A d where f ∈ S 2 (Γ 0 (p 2 )) + . The Lemma 6 (c) tells us in this case that L χ (f ) = 0 because χ(-p 2 ) = 1, hence L χ is zero on the d-old space by (15) and

(a 1 , L χ ) + p 2 ,old dp 2 = (a 1 , L χ ) + p 2 ,p-old dp 2
.

We will now compute the contribution of the p-old space. Our hypothesis on d ensures that S 2 (Γ 0 (dp)) = S 2 (Γ 0 (dp)) p-new because S 2 (Γ 0 (d)) = 0. Let f and g be two (p-new) eigenforms on S 2 (Γ 0 (dp)). By denition of the Petersson scalar product, we immediately obtain

f |A1 , g |A1 dp 2 = [Γ 0 (dp) : Γ 0 (dp 2 )] f, g dp = p f, g dp , (17)
and

f |Ap , g |Ap dp 2 = p 2 D f (px + ipy)g(px + ipy)dxdy = pD f (x + iy)g(x + iy)dxdy,
where D is a fundamental domain for Γ 0 (dp 2 ). It readily implies that pD is a fundamental domain for the subgroup Γ of matrices of Γ 0 (d) which are diagonal modulo p, and this subgroup is of index p in Γ 0 (dp) so we obtain

f |Ap , g |Ap dp 2 = p f, g dp . (18)
Next, using again a linear change of variables, we obtain

f |A1 , g |Ap dp 2 = p D f (x + iy)g(p(x + iy))dxdy = 1 p pD f ((x + iy)/p)g(x + iy)dxdy = f |A -1 p , g Γ
with the same Γ as above, but with Lemma 12 and notations (2.2) and (3.1) of [START_REF] Atkin | Hecke operators on Γ 0 (m)[END_REF], as 1 j 0 1 is a system of coset representatuives of Γ 0 (dp)\Γ for 0 ≤ j ≤ p -1 and f and f |A -1 p are both modular forms for Γ, we get

f |A1 , g |Ap dp 2 = f |Up , g dp (19)
and as f is a p-new eigenform, it is an eigenform for U p and w p (Theorem 3 of [START_REF] Atkin | Hecke operators on Γ 0 (m)[END_REF]) and the eigenvalues are opposite. Dening ε f ∈ {±1} such that

f |wp = ε f • f, (20)
we nally obtain

f |A1 , g |Ap dp 2 = -ε f f, g dp . (21)
Now, let B be an orthonormal eigenbasis of S 2 (Γ 0 (dp)) = S 2 (Γ 0 (dp)) p-new (for the Hecke operators T q , q = d, p and U p ). When f runs through B, the vector spaces Vect(f |A1 , f |Ap ) are pairwise orthogonal because of the formulas (17), ( 18) and ( 21). This allows us to build from B an orthonormal basis of S 2 (Γ 0 (dp 2 )) + p 2 ,p-old in the following way. From Lemma 26 of [AL70], we know that for f ∈ B, with the notation (20),

(f |wp ) |Ap = (f |A1 ) |w p 2 and (f |Ap ) |w p 2 = (f |wp ) |A1 = ε f f |A1 .
An orthogonal basis of S 2 (Γ 0 (dp 2 )) + p 2 ,p-old is then made up with the

f |A1 + f |A1 |w p 2 = f |A1 + ε f f |Ap , f ∈ B.
For f ∈ B, we know from the formulas (17), ( 18) and (21) that:

f |A1 + ε f f |Ap , f |A1 + ε f f |Ap dp 2 = (2p -2ε 2 f ) f, f dp = 2(p -1).
To summarize, an orthonormal basis of S 2 (Γ 0 (dp 2 )) + p 2 ,p-old is obtained by taking the elements f of B and considering the (f |A1 + ε f f |Ap )/ 2(p -1). Finally, by (15),

a 1 (f |A1 + ε f f |Ap )L χ (f |A1 + ε f f |Ap ) = a 1 (f )(L χ (f ) + ε f χ(p)L χ (f )),
in particular the left term is zero when ε f = -χ(p). Summing this over all f ∈ B such that ε f = χ(p) and after orthornormalisation, we get

(a 1 , L χ ) + p 2 ,p-old dp 2 = 1 p -1 (a 1 , L χ ) χ(p)p dp
, which proves the Lemma.

We now need to calculate both terms on the right of (12), and to do this we will use a version of Petersson trace formula in the next section.

5

The semi-orthogonality relation with respect to Atkin-Lehner involutions Let us begin with the necessary denitions for the trace formulas.

Denition 9 (Kloosterman sums and Bessel function).

For all positive integers m, n, c, the Kloosterman sum associated to m, n, c is dened by

S(m, n; c) = k∈(Z/cZ) * e 2iπ(mk+nk -1 )/c .
The Kloosterman sums satisfy the Weil bounds ([IK04], Corollary 11.12):

|S(m, n; c)| ≤ (m, n, c) 1/2 τ (c) √ c, (22)
with (m, n, c) the gcd of m,n and c and τ (c) the number of positive divisors of c. The Bessel function of the rst kind and order 1 is the entire function J 1 dened by

J 1 (z) = +∞ n=0 (-1) n n!(n + 1)! z 2 2n+1 .
It has the following integral representation ([Wat22], 6.21, Formula 8)

J 1 (z) = z 4iπ x+i∞ x-i∞ e w-z 2 4w w 2 dw (23)
for all z ∈ C and all x > 0.

The goal of this subsection is to prove Propositions 3 and 14. With our notations, the left-hand term of (1) is exactly

1 2π √ mn (a m , a n ) ε Q N .
Before the proof, let us make some remarks about Proposition 3.

Remark 10.

(a) Summing for any Q the formulas for ε = 1 and ε = -1, we recover the original Petersson trace formula ([IK04], Proposition 14.5), which generalises to every weight k ≥ 2. However, its proof for k = 2 is more involved because the Poincaré series cannot be dened as uniformly convergent series so we will focus on this case (it is also the only one we need), but it is very likely to be generalised to k > 2 as well. The trace formula above has been originally proved for Q = N in the Chapter 3 of [START_REF] Akbary | Non-vanishing of modular L-functions with large level[END_REF], but to our knowledge, not for any other Q.

(b) For Q = N prime, formula (1) can be found (in a dierent form) in ([IK04], Proposition 14.25). Notice that there is a mistake in one of the arguments of J 1 in that book, as it should actually be rewritten (with our notations) for prime level q and m, n ≥ 1 :

(a m , a n ) ε q 2π √ mn = δ mn -2π √ qδ m,nq -2π q|c S(m, n; c) c J 1 4π √ mn c (24) + 2πε √ q q|c S(m, nq; c) c J 1 4π √ mnq c .
The proof of this result is simply based on the natural system of formulas (which crucially uses Theorem 3 of [AL70] hence the hypothesis q prime)

(a m , a n ) q = (a m , a n ) + q + (a m , a n ) - q (a m , a nq ) q = -(a m , a n ) + q + (a m , a n ) - q
combined with the original Petersson trace formula. Notice there is a δ m,nq appearing here but not in (1) so even under this form, the fact that formulas (1) and ( 24) are the same is not obvious.

To see it, we use that for c ≥ 1 such that q||c, S(m, nq; c) = -S(m, nq -1 ; c/q) (e.g. by Theorem 68 of [START_REF] Hardy | An Introduction to the Theory of Numbers[END_REF]), and that for all m, n ≥ 1, we can check (separating between oldforms and newforms) that 0 = (a m , a nq ) q 2 = δ m,nq -2π

q 2 |c S(m, nq; c) c J 1 4π √ mnq c ,
and we readily obtain the equivalence using these two results.

To prove (1), we will use the Poincaré series in weight 2, whose classical properties are recalled below ([IK04], Lemma 14.2 and [Ran77], section 5.7).

Denition 11 (Poincaré series of weight 2).

For every positive integers n, N there are cuspidal forms of weight 2 for Γ 0 (N ) denoted by P n (•, N ) and called Poincaré series of weight 2 such that (a) The Poincaré series P n (•, N ) is the uniform limit on every compact subset of H when s → 0 + of the series P n (•, s, N ) dened by

P n (z, s, N ) := γ∈Γ∞\Γ0(N ) e 2iπnγ•z j γ (z) 2 |j γ (z)| 2s .
These series satisfy by uniform convergence the transformation formula

P n (γ • z, s, N ) = j γ (z) 2 |j γ (z)| 2s P n (z, s, N ). (b) For every m > 0, a m (P n (•, N )) = δ mn -2π     m n c>0 N |c S(m, n; c) c J 1 4π √ mn c     . (25) (c) For every f ∈ S 2 (Γ 0 (N )), f, P n (•, N ) N = a n (f ) 4πn .
The essential result to prove for our trace formula is the following.

Proposition 12. For every positive integers m, n, N and every divisor Q > 1 of N such that

(Q, N/Q) = 1, a m (P n (•, N ) |w Q ) = -2π m n m≥1 c>0 (N/Q)|c (Q,c)=1 S(m, nQ -1 ; c) c √ Q J 1 4π √ mn c √ Q . ( 26
)
Let us explain rst why this implies the trace formula. Dene

P + Q n (•, N ) := P n (•, N ) + P n (•, N ) |w Q . It belongs to S 2 (Γ 0 (N )) + Q , and for any f ∈ S 2 (Γ 0 (N )) + Q , as w Q is self-adjoint, we have f, P + Q n (•, N ) = f, P n (•, N ) + f, P n (•, N ) |w Q = f, P n (•, N ) + f |w Q , P n (•, N ) = 2 f, P n (•, N ) .
Hence, for F N,Q an orthonormal basis of S 2 (Γ 0 (N )) + Q , the property (c) of Poincaré series gives us

P + Q n (•, N ) = f ∈F N,Q f, P + Q n f = 2 f ∈F N,Q f, P n f = f ∈F N,Q a n (f ) 2πn f,
and the property (b) of Poincaré series together with Proposition 3 give us, by identication of Fourier coecients, for every m > 0:

(a n , a m )

+ Q N 2πn = δ mn -2π m n c>0 N |c S(m, n; c) c J 1 4π √ mn c -2π m n m≥1 c>0 (N/Q)|c (Q,c)=1 S(m, nQ -1 ; c) c √ Q J 1 4π √ mn c √ Q ,
hence the trace formula for Q and ε = 1. We can obtain the trace formula for Q and ε = -1 by the same means or by dierence with the usual Petersson trace formula, as mentioned earlier.

Remark 13. Actually, the same argument gives us the combined Petersson trace formula (which will be useful in section 6), written below.

Proposition 14 (Restricted Petersson trace formula with multiple eigenvalues).

Let m, n, N be three xed positive integers. Let E be a group morphism from a subgroup H of the group W of Atkin-Lehner involutions on N (identied as the set of Q|N such that (Q,

N/Q) = 1 below) to {pm1}. For every Q ∈ W , let us dene S Q = 2π m n m≥1 c>0 (N/Q)|c (Q,c)=1 S(m, nQ -1 ; c) c √ Q J 1 4π √ mn c √ Q .
and for B an eigenbasis of S 2 (Γ 0 (N )),

(a m , a n ) E N := f ∈B ∀Q∈H, f |w Q =E(Q)f a m (f )a n (f ) f 2 .
Then, we have

|E| 4π √ mn (a m , a n ) E N := δ mn - Q∈H E(Q)S Q Proof. Let us dene P n (•, N ) E := Q∈H E(Q)P n (•, N ) |w Q By construction, for every Q ∈ H, one has P n (•, N ) E |w Q = E(Q)P n (•, N ) E
. Now, let B E be the subset of B made up with the eigenforms f having the good signs for the morphism E. For every f ∈ B E :

f, P n (•, N ) E = Q∈H E(Q) f, P n (•, N ) |w Q = Q∈H E(Q) f |w Q , P n (•, N ) = |E| f, P n (•, N ) ,
hence by the properties of Poincaré series,

f, P n (•, N ) E = |E|a n (f ) 4πn .
Now, B E is an orthogonal basis of S 2 (Γ 0 (N )) E , and decomposing P n (•, N ) E on the basis B E gives Proposition 14 by identication of m-th Fourier coecients on both sides, using (26).

Let us now prove the Proposition on Poincaré series.

Proof. Let us choose an Atkin-Lehner involution matrix W

Q = Q y N Qt with y, t ∈ Z and
Qt -(N/Q)y = 1. We will compute the Fourier coecients of n (•, s, N ) |W Q : here, this depends on the choice of W Q because P n (•, s, N ) is not a modular form.

For any s > 0,

P n (•, s, N ) |W Q (z) = det W Q j W Q (z) 2 P n (W Q • z, s, N ) = Q j W Q (z) 2 γ∈Γ∞\Γ0(N ) e 2iπnγW Q z j γ (W Q z) 2 |j γ (W Q z)| 2s = Q|j W Q (z)| 2s γ∈Γ∞\Γ0(N )W Q e 2iπnγz j γ (z) 2 |j γ (z)| 2s . Now, for every a b c d ∈ Γ 0 (N ), a b c d W Q = aQ + bN ay + bQt cQ + dN cy + dQt ,
so it belongs to the set of matrices a b c d with integer coecients such that N divides c , Q divides a and d , and with determinant Q. Actually, this set is exactly Γ 0 (N )W Q as we check immediately by multiplication by W -1 Q , and for

Q > 1, c is necessarily nonzero, so Γ ∞ \Γ 0 (N )W Q is in natural bijection with the set R N,Q of triples (a, c, d) of integers such that c > 0, N |c, Q|(a, d), ad = Q mod c and 0 ≤ a < c. Moreover, for γ = a b c d ∈ Γ 0 (N )W Q built from such a triple (a, c, d), γ • z = a c - Q c(cz + d) hence P n (•, s, N ) |W Q (z) = Q|j W Q (z)| 2s (a,c,d)∈R N,Q e 2iπna/c e -2iπ nQ c(cz+d) (cz + d) 2 |cz + d| 2s = Q|j W Q (z)| 2s c>0 N |c (Q,c/Q)=1 1 c 2+2s a 0≤a<c Q|a e 2iπna/c d Q|d ad≡Q[c] e -2iπnQ c 2 (z+d/c) (z + d/c) 2 |z + d/c| 2s .
For xed a and c, the set of d satisfying the property in the second sum is a congruence class modulo c, so we can choose its representative d between 0 and c -1, therefore

d Q|d ad≡Q[c] e -2iπnQ c 2 (z+d/c) (z + d/c) 2 |z + d/c| 2s = ∈Z e - 2iπnQ c 2 (z+d /c+ ) (z + d /c + ) 2 |z + d /c + | 2s = F c/ √ Q,n,s (z + d /c)
with the auxiliary function F c,n,s on H dened for c > 0, n > 0, s > 0 by

F c,n,s (z) := ∈Z f c,n,s,z ( ), with f c,n,s,z (x) := e -2iπn c 2 (x+z) (x + z) 2 |x + z| 2s .
We will now give another expression for F c,n,s allowing us to compute more precisely the terms of

P n (•, s, N ) |W Q .
As f c,n,s,z is C ∞ on R and integrable as its two rst derivatives, we can apply Poisson summation formula to rewrite

F c,n,s (z) = m∈Z +∞ -∞ f c,n,s,z (x)e -2iπmx dx.
Let us x for now η > 0, and restrict to the domain Im z ≥ η. The function f c,n,s,z then extends to an holomorphic function on | Im x| < η when we use the usual determination of the logarithm on C\R -to write, for x ∈ R,

(x + z) 2 |x + z| 2s = (x + z) 2+s (x + z) s .
The right term is clearly holomorphic in x (when z is xed) so we can extend it on the domain | Im x| < η. Notice that we still have |(x + z) s | = |x + z| s by denition of the determination of logarithm. As f c,n,s,z is holomorphic on this domain, we can shift the imaginary part of the integration axis by εη/2, with ε = -1 if m > 0 and ε = 1 otherwise, so that Re(-2iπmx) = 2πm Im(x) = -π|m|η.

We then have

+∞

-∞ f c,n,s,z (x)e -2iπmx dx = iεη/2+R f c,n,s,z (x)e -2iπmx dx ≤ iεη/2+R e -π|m|η |x + z| 2+2s dx ≤ e -π|m|η R 1 (η 2 /4 + x 2 ) 1+s dx.
By real translation in the integral, we also see that for every y ∈ R and every m ∈ Z,

R f c,n,s,z+y (x)e -2iπm(x+y) dx = R f c,n,s,z (x)e -2iπmx dx.
We can then rewrite

P n (•, s, N ) |W Q (z) Q|j W Q (z)| 2s = c>0 N |c (Q,c/Q)=1 1 c 2+2s 0≤a,d≤c Q|(a,d) ad≡Q[c] e 2iπna/c m∈Z R f c/ √ Q,n,s,z+d/c (x)e -2iπmx dx = c>0 N |c (Q,c/Q)=1 1 c 2+2s 0≤a,d≤c Q|(a,d) ad≡Q[c] e 2iπ(na+md)/c m∈Z R f c/ √ Q,n,s,z (x)e -2iπmx dx = c>0 N |c (Q,c/Q)=1 1 c 2+2s m∈Z 0≤a,d≤c Q|(a,d) ad≡Q[c] e 2iπ(na+md)/c R f c/ √ Q,n,s,z (x)e -2iπmx dx.
For a xed c, the integers a and d go through the multiples of Q between 0 and c such that ad = Q mod c. This amounts to say that a = Qa and d = Qd where a , d go through the integers between 0 and c/Q such that Qa d = 1 mod c, i. e. d is equal to Q -1 a -1 modulo c/Q. This proves the equality

0≤a,d≤c Q|(a,d) ad≡Q[c] e 2iπ(na+md)/c = S(m, nQ -1 ; c/Q) where Q -1 is the inverse of Q modulo c/Q, so P n (•, s, N ) |W Q (z) = Q|j W Q (z)| 2s c>0 N |c (Q,c/Q)=1 S(m, nQ -1 ; c/Q) c 2+2s m∈Z R f c/ √ Q,n,s,z (x)e -2iπmx dx.
Now, using the Weil bounds (22) on Kloosterman sums:

S(m, nQ -1 ; c/Q) c 2+2s m∈Z R |f c/ √ Q,n,s,z (x)e -2iπmx |dx ≤ n 1/2 τ (c/Q) c 3/2 e -π|m|η R 1 η 2 /4 + x 2 dx
which is the general term of an absolutely convergent series, allowing us to exchange the sum and the integral in the expression of P n (•, s, N ) |W Q (z), hence

P n (•, s, N ) |W Q (z) = Q|j W Q (z)| 2s m∈Z c>0 N |c (Q,c/Q)=1 S(m, nQ -1 ; c/Q) c 2+2s R f c/ √ Q,n,s,z (x)e -2iπmx dx.
We can also take the limit s → 0 + of this equality as the previous bound of absolute convergence does not depend on s, therefore we obtain

P n (•, N ) |w Q (z) = Q m∈Z c>0 N |c (Q,c/Q)=1 S(m, nQ -1 ; c/Q) c 2 R f c/ √ Q,n,0,z (x)e -2iπmx dx = Q m∈Z       c>0 N |c (Q,c/Q)=1 S(m, nQ -1 ; c/Q) c 2 R e - 2iπn c 2 /Q(x+z) -2iπm(x+z) (x + z) 2 dx       e 2iπmz .
Let us compute this integral. Dene

G m,n,c = -2πc m n J 1 4π √ mn c , (27)
hence

P n (•, N ) |W Q (z) = -2πQ m/n m≥1 c>0 N |c (Q,c/Q)=1 S(m, nQ -1 ; c/Q) c 2 c/ QJ 1 4π √ mn c/ √ Q e 2iπmz = -2π m/n m≥1 c>0 (N/Q)|c (Q,c)=1 S(m, nQ -1 ; c) c √ Q J 1 4π √ mn c √ Q e 2iπmz
after reindexation of c by c/Q, which nishes the proof.

Final computations and proof of Corollary 4

We can now regroup all our results to obtain an exact formula for (a 1 , L χ )

+ p 2 ,new dp 2
and then estimate the error terms.

For every N ≥ 1, every divisor Q of N such that (Q, N/Q) = 1 and every x > 0, dene

A N,Q (x) = 2π +∞ n=1 χ(n) √ n e -2πn/x c>0 (N/Q)|c (c,Q)=1 S(1, nQ -1 ; c) c √ Q J 1 4π √ n c √ Q and B N,Q (x) = 2π +∞ n=1 χ(n) √ n e -2πnx/(D 2 N ) c>0 (N/Q)|c (c,Q)=1 S(1, nQ -1 ; c) c √ Q J 1 4π √ n c √ Q (so that B N,Q (x) = A N,Q (D 2 N/x)).
We recognize here terms appearing in Proposition 3, summed as indicated by the approximate functional equation of Lemma 6 (c). More precisely, by Lemma 7,

(a 1 , L χ )

+ p 2 ,new dp 2 = (a 1 , L χ ) + p 2 dp 2 - 1 p -1 (a 1 , L χ ) χ(p)p dp
, and by (9), for any x > 0 :

(a 1 , L χ )

+ p 2 dp 2 = n≥1 χ(n)(a 1 , a n ) + p 2 dp 2 n e -2πn x - n≥1 χ(n)(a 1 , a n • w dp 2 ) + p 2 dp 2 n e -2πnx dp 2 = n≥1 χ(n)(a 1 , a n ) + p 2 dp 2 n e -2πn x - n≥1 χ(n)((a 1 , a n ) + p 2 ,+ d dp 2 -(a 1 , a n ) + p 2 ,-d dp 2
) n e

-2πnx dp 2 (a 1 , L χ ) + p 2 dp 2 = 2πe -2π x -2π(A dp 2 ,1 (x) + A dp 2 ,p 2 (x)) + 2π(B dp 2 ,dp 2 (x) + B dp 2 ,d (x)) (28)
(we used here Remark 13). Here is also implicitly appearing the sign of the functional equation : for example, for (a 1 , L χ ) + p (which is 0 by Lemma 6 (c)), we would have no principal term such as 2π, and only error terms. In the same fashion, we obtain that for any x > 0, (a 1 , L χ )

χ(p)p dp = 2πe -2π x -2π(A dp,1 (x) + χ(p)A dp,p (x)) + 2πχ(p)(B dp,dp (x) + B dp,d (x)). (29)
Consequently, we only have to give good estimates for the A N,Q (x) and B N,Q (x) (simultaneously in x). The idea for those is that we will choose x of the same order of magnitude as D 2 N , so that B N,Q (x) is very small (given its exponential factors), whereas A N,Q (x) is not too large. Therefore, Theorem 1 is a direct consequence of the following lemma (notice that the only cases of Q = N appearing in (28) and (29) are for B N,N (x), hence made up to be small with our choice of x).

Lemma 15. For any N ≥ 1, any divisor Q of N such that (Q, N/Q) = 1, any x > 0 and any quadratic Dirichlet character χ of conductor D prime to N ,

|A N,Q (x)| √ D(log(D) + 1)(log(N ) + log(x)) 2 τ (N )e -2π x N + δ Q=N xe -2π x τ (D) N D 3/2 |B N,Q (x)| √ D(log(D) + 1)(log(N ) + log(x)) 2 τ (N )e -2πx D 2 N N + δ Q=N √ De -2πx D 2 N τ (D) x
Therefore, choosing x = (D 2 N ) log(D 2 N ), we obtain after simplication and use of natural bounds that for Q = N :

|A N,Q (x)| + |B N,Q (x)| √ D(log(D) + 1) 3 log(N ) 2 τ (N ) N
with an absolute implied constant. Applied to N = dp 2 , this gives us the error term of Theorem 1.

Proof. As we remarked before, B N,Q (x) = A N,Q (N/x) so it is enough to obtain a bound on A N,Q (x) for all x > 0. The double sum dening A N,Q (x) is absolutely convergent (e.g. by Weil bounds (22)), so

A N,Q (x) = 2π c>0 (N/Q)|c (c,Q)=1 A N,Q,c (x), with A N,Q,c (x) = 1 c √ Q +∞ n=1 χ(n) √ n e -2πn/x S(1, nQ -1 ; c) c √ Q J 1 4π √ n c √ Q .
With Weil bounds (22) and the bound |J 1 (t)| |t| for t real, we obtain

|A N,Q,c (x)| +∞ n=1 e -2πn/x c 2 Q τ (c) √ c τ (c) Qc 3/2 +∞ n=1 e -2πn/x |A N,Q,c (x)| xe -2π/x τ (c) Qc 3/2 . ( 30
)
On another side, if c = D, there is a natural cancellation in the terms dening A N,Q,c (x). To see this, note that if c = D, one can apply Polya-Vinogradov techniques to obtain that for every integers K, K ,

K n=K χ(n)S(1, nQ -1 ; c) ≤ 4c √ D π 2 (log(Dc) + 1.5) c √ D(log(Dc) + 1)
(this inequality is proved in Lemma 5.9 of [START_REF]Surjectivity of Galois representations associated with quadratic Q-curves[END_REF]). Dening

T (c, n) = n k=1 χ(k)S(1, kQ -1 ; c) and f T (y) = J 1 4π √ y c √ Q 4π √ y/c √ Q e -2πy/x ,
we can write by Abel transform To conclude this paper, we prove how Theorem 1 implies Corollary 4.

A N,Q,c (x) = +∞ n=1 χ(n)S(1, nQ -1 ; c) 4πf T (n) c 2 Q = 4π c 2 Q +∞ n=1 T (c, n)(f T (n) -f T (n +
Proof of Corollary 4.

(a) By Theorem 1 in case χ = 1, there is an eigenform f ∈ S 2 (Γ 0 (dp 2 )) + p 2 such that L(f, 1) is nonzero. By the famous result of Kolyvagin and Logachev ([KL90], Theorem 0.3), this implies that the abelian variety A f associated to f obtained as a quotient of J(d, p) is of algebraic rank zero.

(b) By Theorem 1, there is an eigenform f ∈ S 2 (Γ 0 (dp 2 )) + p 2 ,new such that L(f ⊗χ, 1) is nonzero. Such an eigenform necessarily satises f |w d = -χ(d)f because if f |w d = χ(d)f , L(f, 1) = 0 (Lemma 6 (c) because χ(-1) = 1 here). By Theorem 0.3 of [START_REF] Kolyvagin | Finiteness of the Shafarevich-Tate group and the group of rational points for some modular abelian varieties[END_REF] applied to f ⊗ χ, the abelian variety A f obtained as a quotient of J(d, p) then has its twist by K/Q relatively to [-1] of algebraic rank zero (because this twist is isogenous over Q to the abelian variety A f ⊗χ ), and the canonical quotient morphism J(d, p) → A f satises π • w d = -χ(d)π because f |w d = -χ(d)f , therefore this twist is a rational quotient of J(d, p, χ). Indeed, let i : J(d, p) → J(d, p, χ) be the twist isomorphism (i.e. dened over K and such that j σ = χ(d)j • w d ) and i : A f → A f ⊗ χ (i.e. dened over K such that i σ = -i, and consider π f : J(d, p, χ) → A f ⊗χ the natural quotient morphism making the diagram below commutative J(d, p)

π f / / j A f i J(d, p, χ) π f / / A f ⊗ χ .
Therefore, π f is dened over K and

π f σ = i σ • π σ f • (j -1 ) σ = -i • π f • (χ(d)w d • j -1 ) = (-χ(d)) 2 i • π f • j -1 = π f ,
which proves that A f ⊗ χ is a rational quotient of J(d, p, χ).

(c) This is a consequence of (a) and (b) coming from the fact that for d prime to p and squarefree, there is a rational isogeny

J (d, p) → J(d, p) ⊕ J 0 (d) (32)
equivariant under the action of the Hecke algebra generated by the T , prime not dividing dp, and w d such as it can be dened naturally on both sides. For d ∈ {2, 3, 5, 7, 13}, J 0 (d) = 0 hence the result. This fact is due to [START_REF] Chen | On Relations between Jacobians of Certain Modular Curves[END_REF] for d = 1 and cited in [START_REF] Darmon | Winding quotients and some variants of Fermat's Last Theorem[END_REF], and its principle has been generalised to any d by [START_REF] Smit | Sur un résultat d'Imin Chen[END_REF]. For the details, one proof can be found in [START_REF] Le | Points entiers et rationnels sur des courbes et variétés modulaires de dimension supérieure[END_REF] (Lemma I.6.2).

  As the term to integrate is holomorphic on C * , we can integrate on any horizontal line of ordinate α > 0, hence G m,n,c (z) does not depend on z, we denote it by G m,n,c . For m ≤ 0, we have |G m,n,c | = to 0 when α goes to +∞, so G m,n,c = 0 when m ≤ 0. Now, for m > 0, G m,n,c = i+R e integral representation of J 1 (Denition 9). We nally obtain

  n) -f T (n + 1)|and by denition of the f T , this sum is e -2π/x times less than the total variation of J 1 (y)/y on ]0, +∞[, which is so we obtain|A N,Q,c (x)| (log(Dc) + 1) √ D cQ e -2π/x .(31)Notice that this bound is naturally almost uniform on x, but not convergent in c, as opposed to the bound obtained previously.Comparing quickly (30) and (31), it is natural to choose (31) for c < x 2 (except if c = D) and (30) for c > x 2 . This gives (omitting for now the possible term c = D)|A N,Q (x)| /Q) log(x) + log(x) 2 e -2π/x + τ (N/Q) Q/N xe -2π/x N log(x) x √ D(log(D) + 1) N (log(N ) + log(x)) 2 + τ (N )e -2π/x log(x) N √ D(log(D) + 1) N (log(N ) + log(x)) 2 τ (N )e -2π/x .Finally, notice that the possible term c = D can appear only if Q = N because (D, N ) = 1, and we apply (30) to it hence the δ Q=N terms in the Lemma.

Remark 8. We made here an assumption on d and a choice of eigenvalue for w p 2 . Let us discuss the (similar) reasons behind these choices.

Acknowledgements

I wish to thank Olga Balkanova for her helpful answers to related problems, and the anonymous referees for their careful reading and very constructive remarks about the previous versions.