Benjamin Smith

FAMILIES OF EXPLICITLY ISOGENOUS JACOBIANS OF VARIABLE-SEPARATED CURVES

We construct six infinite series of families of pairs of curves (X, Y ) of arbitrarily high genus, defined over number fields, together with an explicit isogeny J X → J Y splitting multiplication by 2, 3, or 4. The families are derived from Cassou-Noguès and Couveignes' explicit classification of pairs (f, g) of polynomials such that f (x 1 ) -g(x 2 ) is reducible.

Correspondences

We begin with a brief review of the theory of correspondences. (See [2, §11.5] and [22, §16] for further detail.)

Let X and Y be (projective, irreducible, nonsingular) curves over a field K, and let C be a curve on the surface X × Y . The natural projections from X × Y restrict to morphisms π C X : C → X and π C Y : C → Y , which in turn induce

Introduction

Our goal in this article is to give algebraic constructions of explicit isogenies of Jacobians of high-genus curves; we are motivated by the scarcity of examples. Isogenies of Jacobians are special in genus g > 3, in the sense that quotients of Jacobians are generally not Jacobians. More precisely, if φ : J X → J Y is an isogeny of Jacobians (that is, a geometrically surjective, finite homomorphism respecting the canonical principal polarizations), then its kernel is a maximal m-Weil isotropic subgroup of the m-torsion J X [m] for some integer m. On the other hand, if S is a subgroup of J X [m] satisfying this same property, then the quotient J X → J X /S is an isogeny of principally polarized abelian varieties, but in general J X /S is only isomorphic to a Jacobian if the genus of X is ≤ 3 (see [START_REF] Oort | Principally polarized abelian varieties of dimension two or three are Jacobian varieties[END_REF] and [START_REF] Donagi | The arithmetic-geometric mean and isogenies for curves of higher genus[END_REF]Theorem 6]).

Nevertheless, families of non-isomorphic pairs of isogenous Jacobians of highgenus curves exist: recently Mestre [START_REF] Mestre | Couples de jacobiennes isogénes de courbes hyperelliptiques de genre arbitraire[END_REF] and the author [START_REF] Smith | Families of explicit isogenies of hyperelliptic Jacobians[END_REF] have constructed families of hyperelliptic examples. Here, we extend the results of [START_REF] Smith | Families of explicit isogenies of hyperelliptic Jacobians[END_REF] to derive new families of isogenies of non-hyperelliptic Jacobians in arbitrarily high genus. Theorem 1.1 summarises our results. Definition 1.1. If φ is an isogeny with kernel isomorphic to a group G, then we say φ is a G-isogeny. 1 Definition 1.2. For all positive integers d and n, we define the integer g n (d) by g n (d) := 1 2 (n -1)(d -1) -(gcd(n, d) -1) .

Theorem 1.1. For each integer d > 1 and for each row of the following table, there exists a ν-dimensional family of explicit G-isogenies of Jacobians of curves of genus g n (d), defined over a CM-field of degree e; and if d is in S, then the generic fibre is an isogeny of absolutely simple Jacobians (here P denotes the set of primes).

n ν e G S P \ {3, 5, 7} 31 d -1 6 ((Z/8Z) ×(Z/4Z) 2 ×(Z/2Z) 2 ) g31(d)/3 P \ {3, 5, 31}

1 This is not the conventional notation, which replaces G with a tuple of its abelian invariants; but it is much more useful in higher dimensions, where such tuples are typically very long.

Proof. Follows from Propositions 8.1 through 14.1.

The proof of Theorem 1.1 is organised as follows: In §3, we associate a family of pairs of curves (X , Y) to each integer d > 1 and each pair of polynomials (Q X , Q Y ) such that Q X (x 1 ) -Q Y (x 2 ) has a nontrivial factorization. We also give a correspondence C on X × Y inducing an explicit homomorphism φ C : J X → J Y . In §4 and §5 we develop methods to determine the number of moduli and the kernel structure of φ C . We recall the classification of Cassou-Noguès and Couveignes [START_REF] Cassou-Nogues | Factorisations explicities de g(y) -h(z)[END_REF] in §6, and then apply our constructions to their polynomials in § §8-13. Finally, in §14 we list some values of d where J X and J Y are known to be absolutely simple.

Connections to prior work. The chief contribution of this work is the construction of non-hyperelliptic families: to our knowledge, all of the families of isogenies of Jacobians in genus g > 3 in the literature are of hyperelliptic Jacobians. The non-hyperelliptic families (those with d > 2) are all new. Technically, the main improvement over [START_REF] Smith | Families of explicit isogenies of hyperelliptic Jacobians[END_REF] is a more sophisticated approach to computing the action on differentials: this allows us to treat all d > 1 simultaneously, and to determine the isogeny kernel structures when d > 2 (the approach in [START_REF] Smith | Families of explicit isogenies of hyperelliptic Jacobians[END_REF] uses an explicit description of the 2-torsion specific to hyperelliptic curves). The hyperelliptic families (those with d = 2) have all appeared in earlier works: The families φ 2,7 , φ 2,11 , φ 2,13 , φ 2,15 , φ 2,21 , and φ 2,31 are isomorphic to the 'linear construction' families in [START_REF] Smith | Families of explicit isogenies of hyperelliptic Jacobians[END_REF].

The subfamily of φ 2,7 with s 2 = 0 and the fibre of φ 2,11 at s 2 = 0 appear in Kux's thesis [START_REF] Kux | Construction of algebraic correspondences between hyperelliptic function fields using Deuring's theory[END_REF]. The endomorphisms of Proposition 7.1 with d = 2 are isomorphic to those described by Tautz, Top, and Verberkmoes [START_REF] Tautz | Explicit hyperelliptic curves with real multiplication and permutation polynomials[END_REF].

Notation. Throughout, K denotes a field of characteristic 0 and ζ n denotes a primitive n th root of unity in Q ⊂ K. Automorphisms of K/Q act on polynomials over K by acting on their coefficients.

Files. Six files accompany this article (degree-n.m, for n in {7, 11, 13, 15, 21, 31}), containing the coefficients of the polynomials and matrices that appear in § §8-13.

(These objects are too big to be useful in printed form: for example, the matrix M 30 (A 31 ) in the proof of Proposition 13.1 is a 30× 30 matrix over a sextic number field, with 436 nonzero entries.) Each file is a program in the Magma language [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF][START_REF] Bosma | Handbook of Magma Functions[END_REF], but they should be easily adaptable for use in other computational algebra systems; in any case, the reader need not be familiar with Magma to make use of the data. If the files are not attached to this copy of the article, then they may be found from the author's webpage.

pullback and pushforward homomorphisms on divisor classes: in particular, we have homomorphisms

(π C X ) * : Pic(X) → Pic(C) and (π C Y ) * : Pic(C) → Pic(Y ). Both (π C
X ) * and (π C Y ) * map degree-0 classes to degree-0 classes, and so induce homomorphisms of Jacobians (and, a fortiori, of principally polarized abelian varieties). Composing, we get a homomorphism of Jacobians

φ C := (π C Y ) * • (π C X ) * : J X -→ J Y ; we say C induces φ C .
We emphasize that φ C is completely explicit, given equations for C: we can evaluate φ C (P ) for any P in J X by choosing a representative divisor from the corresponding class in Pic 0 (X), pulling it back to C with (π C X ) * , and pushing the result forward onto Y with (π C Y ) * . Extending Z-linearly so that φ C1+C2 = φ C1 + φ C2 , we may take C to be an arbitrary divisor on X × Y . We call divisors on X × Y correspondences.

The map C → φ C ; defines a homomorphism Div(X × Y ) → Hom(J X , J Y ); its kernel is generated by the principal divisors and the fibres of π X and π Y . The map is surjective: every homomorphism φ : J X → J Y is induced by some correspondence Γ φ on X×Y (we may take

Γ φ = (φ • α X × α Y ) * µ * (Θ Y ), where α X : X ֒→ J X and α Y : Y ֒→ J Y are the canonical inclusions, Θ Y is the theta divisor on J Y , and µ : J Y × J Y → J Y is the subtraction map). We therefore have an isomorphism Pic(X × Y ) ∼ = Pic(X) ⊕ Pic(Y ) ⊕ Hom(J X , J Y ).
Exchanging the rôles of X and Y in the above, we obtain the image of φ C under the Rosati involution:

φ † C = (π C X ) * • (π C Y ) * : J Y -→ J X . (Recall that φ † C := λ -1 X • φC • λ Y ,
where φC : J Y → J X is the dual homomorphism and λ X : J X ∼ → J X and λ Y : J Y ∼ → J Y are the canonical principal polarizations.) Composition of homomorphisms corresponds to fibred products of correspondences: if X, Y , and Z are curves, and C and D are correspondences on X×Y and

Y × Z respectively, then C × Y D is a correspondence on X × Z and φ D • φ C = φ (C×YD) .
Let Ω(X) and Ω(Y ) denote the g n (d)-dimensional K-vector spaces of regular differentials on X and Y , respectively. The homomorphism φ C : J X → J Y induces a homomorphism of differentials

D(φ C ) : Ω(X) -→ Ω(Y )
(see [START_REF] Shimura | Abelian varieties with complex multiplication and modular functions[END_REF] for details). The image of a regular differential ω on X under D(φ C ) is

D(φ C )(ω) = Tr Ω(C) Ω(Y ) (ω),
where the inclusion Ω(X) ֒→ Ω(C) and the trace Ω(C) → Ω(Y ) are induced by the natural inclusions of K(X) and K(Y ) in K(C). The map φ C → D(φ C ) extends to a faithful representation

D(•) : Hom(J X , J Y ) → Hom(Ω(X), Ω(Y ))
(the faithfulness depends on the fact that K has characteristic 0). We view differentials as row vectors, and homomorphisms as matrices acting by multiplication on the right. Composition of homomorphisms corresponds to matrix multiplication:

D(φ 2 • φ 1 ) = D(φ 1 )D(φ 2 )
for all φ 1 : J X → J Y and φ 2 : J Y → J Z . In particular, if Y = X then D(•) is a representation of rings; in general, D(•) is a representation of left End(J X )-and right End(J Y )-modules.

Example 2.1. Suppose that X is a curve with affine plane model X : F (x, y) = 0, and let x 1 , y 1 and x 2 , y 2 denote the coordinate functions on the first and second factors of X × X, respectively. Our first example of a nontrivial correspondence is the diagonal

∆ X := V (y 1 -y 2 , x 1 -x 2 ) ⊂ X × X, which induces the identity map: φ ∆X = [1] JX . More generally, if ψ is an automor- phism of X, then (Id × ψ)(X) is a correspondence on X × X inducing ψ.
Example 2.2. Let X and Y be curves with affine plane models X : F X (x 1 , y 1 ) = 0 and Y : F Y (x 2 , y 2 ) = 0. For any polynomial A(x 1 , y 1 , x 2 , y 2 ), the correspondence C = V (A) is rationally equivalent to a sum of fibres of π X and π Y , and so induces the trivial homomorphism: on the level of degree-0 divisor classes,

φ C P ∈X(K) n P (P ) = div P ∈X(K) A(x 1 (P ), y 1 (P ), x 2 , y 2 ) nP = 0.
In particular, correspondences inducing nonzero homomorphisms must be cut out by more than one defining equation (cf. Example 2.1).

Variable-separated curves and correspondences

Now let X and Y be variable-separated plane curves over K: that is, we suppose that X and Y have affine plane models

X : P X (y 1 ) = Q X (x 1 ) and Y : P Y (y 2 ) = Q Y (x 2 ),
where P X , Q X , P Y , and Q Y are polynomials over K. (This includes elliptic, hyperelliptic, and superelliptic X and Y .) We restrict our attention to the case where P X , P Y , Q X and Q Y are indecomposable: that is, they cannot be written as compositions of polynomials of degree at least two (cf. Remark 6.1).

Our aim is to give examples of correspondences inducing nontrivial homomorphisms. If C = V (A) for some polynomial A, then φ C = 0 (cf. Example 2.2); so we need to find divisors on X×Y defined by at least two equations. We investigate the simplest nontrivial case, where each involves only two variables:

C = V (A(x 1 , x 2 ), B(y 1 , y 2 )) ⊂ X × Y.
We immediately reduce to the case where P X = P Y and B(y 1 , y 2 ) = y 1y 2 : Let Z be the curve defined by Z : P X (v) = Q Y (u), and define correspondences

C 1 = V (A(x 1 , u), y 1 -v) and C 2 = V (u -x 2 , B(v, y 2 )) on X×Z and Z×Y , respectively. Then C = C 1 × Z C 2 , so φ C = φ C2 • φ C1 .
Replacing Y with Z and C with C 1 (or X with Z and C with C 2 ), we reduce to the study of curves and correspondences defined by

X : P (y 1 ) = Q X (x 1 ), Y : P (y 2 ) = Q Y (x 2 ), C = V (y 1 -y 2 , A(x 1 , x 2 )) .
For C to be one-dimensional, we must have

A(x 1 , x 2 )|(Q X (x 1 )-Q Y (x 2
)); we will see in §6 that the existence of such a nontrivial factor is special. It is noted in [9, §2.1] that if Q X and Q Y are indecomposable, then the existence of a nontrivial A implies that Q X and Q Y have the same degree

n := deg Q X = deg Q Y ;
and further that there exists some integer r such that

r = deg x1 (A(x 1 , x 2 )) = deg x2 (A(x 1 , x 2 )) = deg tot (A(x 1 , x 2 )), so we may write (1) A(x 1 , x 2 ) = r i=0 c i (x 2 )x r-i 1 with deg c i ≤ i for all 0 ≤ i ≤ r.
We have no restrictions on P , so we let it be (almost) generic2 : for each integer d > 1 we let s 2 , . . . , s d be free parameters, and define P d to be the polynomial

P d (y) := y d + s 2 y d-2 + • • • + s d-1 y + s d .
Note that P d is indecomposable. Henceforward, therefore, we consider families of curves X and Y and correspondences C in the form (2)

X : P d (y 1 ) = Q X (x 1 ), Y : P d (y 2 ) = Q Y (x 2 ), C = V (y 1 -y 2 , A(x 1 , x 2 )) ⊂ X × Y,
with Q X and Q Y indecomposable of degree n, and A as in Eq. ( 2).

The families are parametrized by s 2 , . . . , s d , together with any parameters in the coefficients of Q X and Q Y . The special case d = 2, which produces hyperelliptic families, is the linear construction of [START_REF] Smith | Families of explicit isogenies of hyperelliptic Jacobians[END_REF] (with s = -s 2 ). The Newton polygon of X (and Y) is

N (d, n) = {(λ 1 , λ 2 ) ∈ R 2 ≥0 : dλ 1 + nλ 2 ≤ dn}.
The families X and Y have (generically) nonsingular projective models in the weighted projective plane P(d, n, 1), which is the projective toric surface associated to N (d, n) (we see in [START_REF] Reid | Graded rings and varieties in weighted projective space[END_REF] that P(d, n, 1) = P(d/m, n/m, 1), where m = gcd(d, n)).

We let P(d, n) denote the set of integer interior points of the Newton polygon:

P(d, n) = {(λ 1 , λ 2 ) ∈ Z 2 >0 : dλ 1 + nλ 2 < dn}.
The geometric genus of X (and of Y) is equal to #P(d, n), and it is easily verified that if g n (d) is the function of Definition 1.2, then

g X = g Y = #P(d, n) = g n (d).
Remark 3.1. Most known nontrivial examples of explicit isogenies of Jacobians, including the isogenies of Richelot [START_REF] Bost | Moyenne arithmético-géometrique et périodes des courbes de genre 1 et 2[END_REF], Mestre [START_REF] Mestre | Couples de jacobiennes isogénes de courbes hyperelliptiques de genre arbitraire[END_REF], and Vélu [START_REF] Vélu | Isogénies entre courbes elliptiques[END_REF] and the endomorphisms of Brumer [START_REF] Brumer | The rank of J 0 (N )[END_REF] and Hashimoto [START_REF] Hashimoto | On Brumer's family of RM-curves of genus two[END_REF], are not induced by correspondences in the form of Eq. (2). However, the explicit real multiplications of Mestre [START_REF] Mestre | Familles de courbes hyperelliptiques à multiplications réelles[END_REF] and Tautz, Top, and Verberkmoes [START_REF] Tautz | Explicit hyperelliptic curves with real multiplication and permutation polynomials[END_REF] are in the form of Eq. (2). Remark 3.2. Our construction generalizes readily to the case where P d , Q X , and Q Y are rational functions instead of polynomials. While this yields many more families, it also complicates the algorithmic aspects of our constructions below.

Isomorphisms and Moduli

We want to compute the number of moduli of X : that is, the dimension of the image of X in the moduli space M gn(d) of curves of genus g n (d) over K. By Torelli's theorem, this is also the dimension of the image of the family φ C in the appropriate moduli space of homomorphisms of principally polarized abelian varieties.

We will adapt the methods of Koelman's thesis [START_REF] Koelman | The number of moduli of families of curves on toric surfaces[END_REF] to compute the number of moduli. Up to automorphism, we can determine the form of the polynomials defining any isomorphism between curves in X by considering column structures and column vectors on the projective toric surface associated to N (d, n), where X has a convenient nonsingular embedding (see [START_REF] Bruns | Polytopal linear groups[END_REF], [START_REF] Castryck | On nondegeneracy of curves[END_REF], and [START_REF] Koelman | The number of moduli of families of curves on toric surfaces[END_REF] for details).

More specifically, for d > 2, we embed X in P(d, n, 1). The K-isomorphisms between distinct curves in X must then take the form

(3) (x, y) -→ (ax + b, ey)
for some a, b, e in K with a and e nonzero. When d = 2, it is more convenient to embed X in P(1, g n (d) + 1, 1); the K-isomorphisms must then take the form

(4) (x, y) -→ ((ax + b)/(cx + d), ey/(cx + d) (gn(d)+1) )
for a, b, c, d, and e in K with e and adbc nonzero.

Lemma 4.1. Let d > 1 be an integer, K a subfield of C, and

f (x) = n i=0 f i x n-i a polynomial over K or K(t)
, where t is a free parameter, such that g n (d) > 1 and

(i) f 0 = 1, (ii) f 1 = 0, (iii) f 2 = 0, and (iv) f 3 = κf 2 for some κ ∈ K.
Let X be the family defined by X :

P d (y) = f (x). Then (1) if f i is in K(t) \ K for some 2 ≤ i < n, then X has d moduli; (2) otherwise, X has d -1 moduli.
Proof. Let U be the open subfamily of X where s 2 , . . . , s d are all nonzero. It suffices to show that the intersection of U with the isomorphism class of any curve in U is finite. First, observe that U has no nontrivial constant subfamilies: the parameters s 1 , . . . , s d (and t in Case ( 1)) appear in distinct coefficients of the defining equation of U. It is enough, therefore, to show that there are only finitely many possible defining equations for isomorphisms from a fixed curve in U to other curves in U.

Every such isomorphism has the form of Eq. ( 3) (or Eq. ( 4) for d = 2). But the defining equation of the codomain curve must satisfy (i) through (iv), which determine e and ax + b (or (ax + b)/(cx + d)) up to a finite number of choices.

The representation on differentials

Let X , Y, and C be as in Eq. ( 2). We want to make the representation D(φ C ) of §2 completely explicit, with a view to determining the structure of ker φ C . It suffices to consider the generic fibres X, Y , and C of X , Y, and C respectively. In this section, K denotes the field of definition of X, Y , and C.

First, we partition P(d, n) into disjoint "vertical" slices:

(5)

P(d, n) = b d,n i=1 {(i, j) : 1 ≤ j ≤ p d,n (i)}, where b d,n := max{i : (i, j) ∈ P(d, n)} = ⌈(1 -1/n)d⌉ -1 and p d,n (i) := ⌈(1 -i/d)n⌉ -1 for 1 ≤ i ≤ b d,n .
We fix a basis for the spaces of regular differentials on X and Y :

Ω(X) = ω i,j : (i, j) ∈ P(d, n) and Ω(Y ) = ω ′ i,j : (i, j) ∈ P(d, n) , where ω i,j := y i-1 1 P ′ d (y 1 ) d(x j 1 ) and ω ′ i,j := y i-1 2 P ′ d (y 2 ) d(x j 2 ).
This fixes isomorphisms of Ω(X) and Ω(Y ) with K gn(d) ; we view regular differentials on X and Y as row g n (d)-vectors over K. If we define subspaces

Ω(X) i := ω i,j : 1 ≤ j ≤ p d,n (i) and Ω(Y ) i := ω ′ i,j : 1 ≤ j ≤ p d,n (i) for 1 ≤ i ≤ b d,n
, then the partition of Eq. ( 5) induces direct sum decompositions ( 6)

Ω(X) = b d,n i=1 Ω(X) i and Ω(Y ) = b d,n i=1 Ω(Y ) i .
Since

y 1 = y 2 in K(C), the image of ω i,j under D(φ C ) is D(φ C )(ω i,j ) = Tr Ω(C) Ω(Y ) y i-1 1 d(x j 1 ) P ′ d (y 1 ) = y i-1 2 d Tr K(C) K(Y ) (x j 1 ) P ′ d (y 2 ) = y i-1 2 P ′ d (y 2 ) dt j ,
where

t j := Tr K(x2)[x1]/(A(x1,x2)) K(x2) (x j 1 )
. By definition, t j is the j th power-sum symmetric polynomial in the roots of A viewed as a polynomial in x 1 over K(x 2 ); but for k > 0, the k th elementary symmetric polynomial in these same roots is equal to (-1) k c k /c 0 , where c k and c 0 are as in Eq. ( 1). We can therefore compute the t j using the Newton-Girard recurrences

t 1 = - c 1 c 0 , t 2 = - 2c 2 + t 1 c 1 c 0 , • • • , t j = - jc j + j-1 k=1 c j t j-k c 0 .
Equation ( 1) implies deg t j ≤ deg c j ≤ j, so expanding t j in terms of x 2 we write

t j = j k=0 µ j,k x k 2 .
In terms of differentials we have

dt j = d( j k=0 µ j,k x k 2 ) = j k=1 µ j,k d(x k 2 ), so D(φ C )(ω i,j ) = j k=1 µ j,k ω ′ i,k .
In particular, D(φ C ) respects the decomposition of Eq. ( 6): that is,

(7) D(φ C )(Ω(X) i ) ⊂ Ω(Y ) i
for all 1 ≤ i ≤ b d,n . For each 0 < k < n, we define a matrix

M k (A) :=        µ 1,1 0 0 • • • 0 µ 2,1 µ 2,2 0 • • • 0 µ 3,1 µ 3,2 µ 3,3 • • • 0 . . . . . . . . . . . . µ k,1 µ k,2 µ k,3 • • • µ k,k        representing D(φ C )| Ω(X) k : Ω(X) k → Ω(Y ) k .
Combining Equations ( 6) and ( 7), we have

(8) D(φ C ) = b d,n i=1 M p d,n (i) (A).
The i th summand in Eq. ( 8) is (by definition) the upper-left p d,n (i)× p d,n (i) submatrix of M n-1 (A), because p d,n (i) ≤ n -1 for all i. Hence, we need only compute M n-1 (A) to determine D(φ C ) for arbitrary d. Input: An integer n ≥ 2, and a polynomial A(x 1 , x 2 ) over K in the form of Eq. (1

): that is, A(x 1 , x 2 ) = r i=0 c i (x 2 )x r-i 1 with deg c i ≤ i for all i. Output: The matrix M n-1 (A). 1: Let c i := 0 for r < i < n. 2: For i in (1, . . . , n -1) do 2a: Set t i := -ic i + i-1 j=1 c j t i-j /c 0 . 2b: For j in (1, . . . , n -1), let µ i,j ∈ K be the coefficient of x j 2 in t i . 3: Return the matrix (µ i,j ). The representation of the Rosati dual φ † C is D(φ † C ) = b d,n i=1 M p d,n (i) (A(x 2 , x 1 )
). We make the following definition for notational convenience. Definition 5.1. We define an involution τ on K[x 1 , x 2 ] by

τ (A(x 1 , x 2 )) := A(x 2 , x 1 ). Lemma 5.2. With the notation above, if M n-1 (A)M n-1 (τ (A)) = mI n-1 for some integer m, then φ † C φ C = [m] JX (that is, φ C splits multiplication-by-m on J X ). Further, if m is squarefree, then φ C is a (Z/mZ) gn(d) -isogeny. Proof. We have D(φ † C φ C ) = D(φ C )D(φ † C
), so Eq. ( 8) implies

D(φ † C φ C ) = b d,n i=1 M p d,n (i) (A)M p d,n (i) (τ (A)) . As we noted above, M k (A) is the upper-left k×k submatrix of M n-1 (A) for all k. Both M n-1 (A) and M n-1 (τ (A)) are lower-triangular, so M k (A)M k (τ (A)) is the upper-left k×k submatrix of M n-1 (A)M n-1 (τ (A)), which is mI n-1 by hypothesis. Hence M i (A)M i (τ (A)) = mI i for all 1 ≤ i ≤ b d,n
, and therefore

D(φ † C φ C ) = b d,n i=1 mI i = mI gn(d) .
The faithfulness of D(•) implies φ † C φ C = [m] JX , proving the first assertion. The kernel of φ C must be a maximal subgroup of J X [m] with respect to the property of being isotropic for the m-Weil pairing; when m is squarefree, the second assertion follows from this together with the nondegeneracy of the Weil pairing. Lemma 5.2 determines the kernel structure of isogenies splitting multiplication by a squarefree integer. In § §11-13, we will derive isogenies splitting multiplication by 4 and 8; we will need another method to determine their kernel structures. It is helpful to specialize to an isogeny defined over a number field, and then to view the specialized isogeny as an isogeny of complex abelian varieties.

Suppose that K is a number field. Fix an embedding of K into C, and let σ denote complex conjugation; enlarging K if necessary, we assume K σ = K. Viewing J X and J Y as complex tori, there exist coordinates on C gn(d) and lattices Λ X and Λ Y in C gn (d) such that the analytic representation S(φ C ) : C gn(d) → C gn(d) and the rational representation R(φ C ) : Λ X → Λ Y are given by the matrices ( 9)

S(φ C ) = D(φ C ) and R(φ C ) = D(φ C ) 0 0 D(φ C ) σ .
We will compute the structure of ker(φ C ) using the relation

(10) ker(φ C ) ∼ = coker(R(φ C )) ∼ = Λ Y /R(φ C )(Λ X ).
The first step is a restriction of scalars from K to Q, since we do not know a priori how elements of K should act on our unknown lattices Λ X and Λ Y . Suppose that R(φ C ) is defined over the ring O K of integers of K (it is sufficient that A be a polynomial over O K ). Fixing a Z-basis γ 1 , . . . , γ e of O K , we have a faithful representation ρ : O K → Mat e×e (Z) (made explicit in Algorithm 5.3), which extends to a homomorphism

ρ * : Mat 2gn(d)×2gn(d) (O K ) -→ Mat 2egn(d)×2egn(d) (Z)
mapping a matrix (a i,j ) to the block matrix (ρ(a i,j )). We then have

(11) (Λ Y /R(φ C )(Λ X )) e ∼ = Z 2egn(d) /ρ * (R(φ C ))(Z 2egn(d) ),
so we can compute the isomorphism type of (ker φ C ) e by computing the elementary divisors of ρ * (R(φ C )). Combining Equations ( 8) and ( 9), and applying ρ * , we have

(12) ρ * (R(φ C )) = b d,n i=1 ρ * M p d,n (i) (A) ⊕ M p d,n (i) (A) σ .
For each 1 ≤ k ≤ n -1, we define

G(A, k) := Z 2ek / ρ * M k (A) ⊕ M k (A) σ (Z 2ek ) ;
then combining Equations ( 10), [START_REF] Chai | A note on the existence of absolutely simple Jacobians[END_REF], and ( 12), we have

(13) (ker(φ C )) e ∼ = b d,n i=1 G(A, p d,n (i)).
We can use this relation to deduce the structure of ker(φ C ).

Algorithm 5.3. Computes the sequence (G(A, k)) n-1 k=1 . Input: A polynomial A ∈ O K [x 1 , x 2 ]
, and an integer n.

Output: The sequence of groups

G(A, k) for 1 ≤ k ≤ n -1. 1: Compute M n-1 (A) using Algorithm 5.1. 2: Set e := [K : Q],
and compute a Z-basis γ 1 , . . . , γ e of O K . 3: For each 1 ≤ i ≤ e, let Γ (i) be the e × e integer matrix such that

γ i γ j = e k=1 Γ (i) jk γ k for all 1 ≤ j ≤ e,
and let ρ : O K → Mat e×e (Z) be the map e i=1 a i γ i → e i=1 a i Γ (i) . 4:

For each 1 ≤ k ≤ n -1,
4a: Let M be the 2ek × 2ek block matrix Remark 5.1. In our examples, the generic fibres X, Y , and C are defined over K(s 2 , . . . , s d ) or K(s 2 , . . . , s d , t), where K is a number field. But if Q X and Q Y are defined over K then so is A, so we can apply Algorithm 5.3 and use Eq. ( 13) to deduce the structure of ker φ C without choosing any particular specialization.

M := (ρ(M n-1 (A) i,j )) k i,j=1 ⊕ ρ(M n-1 (A) σ i,j ) k i,

Pairs of polynomials

To produce nontrivial examples in the form of Eq. ( 2), we need a source of pairs of polynomials (Q

X , Q Y ) such that Q X (x 1 ) -Q Y (x 2
) is reducible. For indecomposable Q X and Q Y over C, these pairs have been explicitly classified by Cassou-Noguès and Couveignes [START_REF] Cassou-Nogues | Factorisations explicities de g(y) -h(z)[END_REF]. The pairs are deeply interesting in their own right: For further background, we refer to the work of Cassels [START_REF] Cassels | Factorization of polynomials in several variables[END_REF], Davenport, Lewis, and Schinzel [START_REF] Davenport | Equations of the form f (x) = g(y)[END_REF][START_REF] Davenport | Two problems concerning polynomials[END_REF], Feit [START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF][START_REF] Feit | On symmetric balanced incomplete block designs with doubly transitive automorphism groups[END_REF][START_REF] Feit | Some consequences of the classification of finite simple groups[END_REF], and Fried [START_REF] Fried | On a conjecture of Schur[END_REF][START_REF] Fried | The field of definition of function fields and a problem in the reducibility of polynomials in two variables[END_REF][START_REF] Fried | Exposition on an arithmetic-group theoretic connection via Riemann's existence theorem[END_REF]. An excellent account of the context and importance of these results can be found on Fried's website [START_REF] Fried | [END_REF]. The plane curves cut out by the factors themselves are also interesting; Avanzi's thesis [START_REF] Avanzi | A study on polynomials in separated variables with low genus factors[END_REF] provides a good introduction to this topic. Definition 6.1. We say that polynomials f 1 and f 2 over K are linear translates if f 1 (x) = f 2 (ax+b) for some a, b in K with a nonzero. We say pairs of polynomials (f 1 , g 1 ) and (f 2 , g 2 ) are equivalent if there exists some a, b in K with a nonzero such that f 1 and af 2 + b are linear translates and g 1 and ag 2 + b are linear translates.

The "equivalence" of Definition 6.1 is indeed an equivalence relation on pairs of polynomials. From the point of view of constructing homomorphisms, equivalent pairs of polynomials give rise to isomorphic homomorphisms of Jacobians. Proposition 6.1. Let X , Y, and C be as in Eq. [START_REF] Birkenhake | Complex abelian varieties (2e). Grundlehren der mathematischen Wissenschaften[END_REF]. Suppose that

(Q Z , Q W ) is equivalent to (Q X , Q Y ) (so Q Z (x) = aQ X (a 1 x + b 1 ) + b and Q W (x) = aQ Y (a 2 x + b 2 ) + b for some a, b, a 1 , b 1 , a 2 ,
and b 2 in K with a, a 1 , and a 2 nonzero.)

(1) If A(x 1 , x 2 ) is a K-irreducible factor of Q X (x 1 )-Q Y (x 2 ), then A ′ (x 1 , x 2 ) = A(a 1 x 1 + b 1 , a 2 x 2 + b 2 ) is a K-irreducible factor of Q Z (x 1 ) -Q W (x 2 ). (2) (X , Y) is K-isomorphic to (W : P d (y 1 ) = Q W (x 1 ), Z : P d (y 2 ) = Q Z (x 2 )),
and

φ C is K-isomorphic to φ D where D = V (y 1 -y 2 , A ′ (x 1 , x 2 )) ⊂ W × Z. Proof. Part (1) is a straightforward symbolic exercise. For Part (2), let α := a -1/d . The family (Z, W) is K-isomorphic to (X , Y) via (s 2 , . . . , s d ) -→ (α 2 s 2 , . . . , α d-1 s d-1 , α d s d -b/a), (x i , y i ) -→ (a i x i + b i , αy i ).
This induces a K-isomorphism between C and D, so

φ C ∼ = φ D . The classification of pairs of indecomposable polynomials (Q X , Q Y ) over C such that Q X (x 1 ) -Q Y (x 2
) has a nontrivial factor splits naturally into two parts, according to whether Q X and Q Y are linear translates or not. Observe that if Q X and Q Y are linear translates, then by Proposition 6.1(1) we reduce to the case Q Y = Q X . We always have a factor x 1x 2 of Q X (x 1 ) -Q X (x 2 ); this corresponds to the fact that the endomorphism ring of J X always contains Z (cf. Example 2.1). Theorem 6.2 (Fried [START_REF] Fried | On a conjecture of Schur[END_REF]). Let Q X be an indecomposable polynomial of degree at least

3 over C. Then (Q X (x 1 ) -Q X (x 2 ))/(x 1 -x 2 ) is K-reducible if and only if (Q X , Q X ) is equivalent to either
(1) the pair (x n , x n ) for some odd prime n, or (2) the pair (D n (x, 1), D n (x, 1)) for some odd prime n, where D n (x, 1) is the n th Dickson polynomial of the first kind with parameter 1 (see Remark 6.3).

Theorem 6.3 (Cassou-Noguès and Couveignes [START_REF] Cassou-Nogues | Factorisations explicities de g(y) -h(z)[END_REF]). Let (Q X , Q Y ) be indecomposable polynomials of degree at least 3 over C, and let σ denote complex conjugation. Assume the classification of finite simple groups (see Remark 6.2). If Q X and Q Y are not linear translates, then

Q X (x 1 )-Q Y (x 2 ) is reducible if and only if (Q X , Q Y ) is equivalent (possibly after exchanging Q X and Q Y ) to
(1) a pair in the one-parameter family (f 7 , f σ 7 ) defined in §8, or (2) the pair (f 11 , f σ 11 ) defined in §9, or (3) a pair in the one-parameter family (f 13 , f σ 13 ) defined in §10, or (4) a pair in the one-parameter family (f 15 , -f σ 15 ) defined in §11, or (5) the pair (f 21 , f σ 21 ) defined in §12, or (6) the pair (f 31 , f σ 31 ) defined in §13. It follows from Proposition 6.1 that we can give a complete treatment of homomorphisms induced by correspondences in the form of Eq. ( 2) by applying our constructions to the polynomials of Theorems 6.2 and 6.3. We treat x n and D n (x, 1) in §7, and the polynomials f 7 , f 11 , f 13 , f 15 , f 21 , and f 31 from Theorem 6.3 in § §8-13. Remark 6.1. The restriction to indecomposable polynomials is not too heavy, since we are primarily interested in isogenies of absolutely simple Jacobians. If Q

X (x) = Q 1 (Q 2 (x)) with deg Q 2 > 1, then we have a (deg Q 2 )-uple cover (x, y) → (Q 2 (x), y) from X to X ′ : P d (y) = Q 1 (x). If d > 2 and deg Q 1 > 1, or if d = 2 and deg Q 1 > 2,
then X ′ has positive genus and J X ′ is a nontrivial isogeny factor of J X , so J X is reducible. If d = deg Q 1 = 2, then J X is not necessarily reducible: a partial treatment of this case appears as the quadratic construction in [START_REF] Smith | Families of explicit isogenies of hyperelliptic Jacobians[END_REF]. Remark 6.2. Theorem 6.3 assumes the classification of finite simple groups [START_REF] Gorenstein | The classification of the finite simple groups[END_REF]. The classification is only required to prove the completeness of the list of pairs of polynomials (and not for the existence of the factorizations). In particular, Theorem 1.1 does not depend on the classification of finite simple groups; but one corollary of the classification is that every isogeny induced by a correspondence in the form of Eq. ( 2) is isomorphic to a composition of endomorphisms from the families in §7 and isogenies from the families in Theorem 1.1. Remark 6.3. Recall that D n (x, a) is the n th Dickson polynomial of the first kind with parameter a (see [START_REF] Lidl | Dickson polynomials. Pitman monographs and surveys in pure and applied mathematics 65[END_REF]): that is, the unique polynomial of degree n such that D n (x+a/x, a) = x n +(a/x) n . In characteristic zero D n (x, 1) = 2T n (x/2), where T n is the n th classical Chebyshev polynomial. We have D n (x, a) = a n/2 D n (a -1/2 x, 1) when a = 0, so (D n (x, a), D n (x, a)) is equivalent to (D n (x, 1), D n (x, 1)). On the other hand D n (x, 0) = x n , so Theorem 6.2(1) is essentially a specialization of Theorem 6.2(2).

Families with explicit Complex and Real Multiplication

We now put our techniques into practice. First, consider Theorem 6.2(1): Let Q X (x) = Q Y (x) = x n for some odd prime n. For each d > 1, we derive a family Z d,n : P d (y) = x n of curves of genus g n (d) with an automorphism ζ : (x, y) → (ζ n x, y) of order n. We say Z d,n is superelliptic if n ∤ d. The family has d -2 moduli: restricting the isomorphisms of §4 to Z d,n , we see that every isomorphism class in Z d,n contains a unique representative with s 2 = 1. We identify ζ its induced endomorphism of J Z d,n ; its minimal polynomial is the n th cyclotomic polynomial. Recalling that

x n 1 -x n 2 = n-1 i=0 (ζ i n x 1 -x 2 ),
we consider the correspondences

C i := V y 1 -y 2 , ζ i n x 1 -x 2 ⊂ Z d,n × Q(ζn)(s2,...,s d ) Z d,n . We have C i = (Id×ζ i )(Z d,n ) so φ Ci = ζ i Example 2.1); the C i therefore generate a subring of End(J Z d,n ) isomorphic to Z[ζ n ]. Now consider Theorem 6.2(2): Q X (x) = Q Y (x) = D n (x, 1
) for some odd prime n. For each d > 1, we derive a family

W d,n : P d (y i ) = D n (x i , 1)
of curves of genus g n (d) with d -1 moduli. In [START_REF] Lidl | Dickson polynomials. Pitman monographs and surveys in pure and applied mathematics 65[END_REF]Theorem 3.12] we see that

D n (x 1 , 1) -D n (x 2 , 1) = (x 1 -x 2 ) (n-1)/2 i=1 A n,i (x 1 , x 2 ), where A n,i (x 1 , x 2 ) := x 2 1 + x 2 2 -(ζ i n + ζ -i n )x 1 x 2 + (ζ i n -ζ -i n ) 2 .
Proposition 7.1. The endomorphisms of J W d,n induced by the correspondences x,y)), and hence

C i := V (y 1 -y 2 , A n,i (x 1 , x 2 )) ⊂ W d,n × Q(ζn)(s2,...,s d ) W d,n generate a subring of End(J W d,n ) isomorphic to Z[ζ n + ζ -1 n ].
where α 1 + α 2 = (ζ i n + ζ -i n )x and α 1 α 2 = x 2 + (ζ i n -ζ -i n ) 2 . On the other hand, π * (ζ i + ζ -i )π * ((x, y)) = 2(ζ i n β + ζ -i n β -1 , y) + 2(ζ -i n β + ζ i n β -1 , y), where β + β -1 = x. But {ζ i n β + ζ -i n β -1 , ζ -i n β + ζ i n β -1 } = {α 1 , α 2 }, so π * (ζ i + ζ -i )π * ((x, y)) = 2φ Ci ((
π * (ζ i + ζ -i )π * = [2]φ Ci .
Let m i be the minimal polynomial of ζ i n +ζ -i n ; it is irreducible, and

m i (ζ i +ζ -i ) = 0. Working in Q(φ Ci ), we have 2m i (φ Ci ) = 2m i ( 1 2 π * (ζ i + ζ -i )π * ) = π * m i (ζ i + ζ -i )π * = 0;
hence m i (φ Ci ) = 0, and the proposition follows.

Remark 7.1. The family W 2,n is isomorphic to the family C t of hyperelliptic curves of genus (n -1)/2 described by Tautz, Top, and Verberkmoes [START_REF] Tautz | Explicit hyperelliptic curves with real multiplication and permutation polynomials[END_REF]. Their families extend earlier families of Mestre [START_REF] Mestre | Familles de courbes hyperelliptiques à multiplications réelles[END_REF], replacing subgroups of the n-torsion of elliptic curves with the group of n th roots of unity in Q. Our construction of W d,n readily generalizes in the other direction to give more families of Jacobians in genus g n (d) with Real Multiplication by Z[ζ n + ζ -1 n ] (though for these families, the Dickson polynomials are replaced by certain rational functions).

8. Genus g 7 (d) families from Theorem 6.3 (1) Consider Theorem 6.3(1): Let α 7 be an element of Q satisfying

α 2 7 + α 7 + 2 = 0, so Q(α 7 ) = Q( √ - 7
). The involution σ : α 7 → 2/α 7 generates Gal(Q(α 7 )/Q). Let t be a free parameter, and let f 7 be the polynomial over Q(α 7 )[t] defined by

f 7 (x) := x 7 -7α 7 tx 5 -7α 7 tx 4 -7(2α 7 + 5)t 2 x 3 -7(4α 7 + 6)t 2 x 2 + 7((3α 7 -2)t 3 -(α 7 + 3)t 2 )x + 7α 7 t 3
(so f 7 = 7g, where g is the polynomial of [9, §5.1] with a 2 = α 7 and T = t). We have a factorization

f 7 (x 1 ) -f σ 7 (x 2 ) = A 7 (x 1 , x 2 )B 7 (x 1 , x 2 )
, where

A 7 = x 3 1 -x 3 2 -α σ 7 x 2 1 x 2 + α 7 x 1 x 2 2 + (3 -2α σ 7 )tx 1 -(3 -2α 7 )tx 2 + (α 7 -α σ 7 )t. Both
X d,7 : P d (y 1 ) = f 7 (x 1 ), Y d,7 : P d (y 2 ) = f σ 7 (x 2 ), C d,7 = V (y 1 -y 2 , A 7 (x 1 , x 2 )) ⊂ X d,7 × Q(α7)(s2,...,s d ,t) Y d,7 . The induced homomorphism φ d,7 = φ C d,7 : J X d,7 → J Y d,7 is a d-dimensional family of (Z/2Z)
       α 7 0 0 0 0 0 0 α 7 0 0 0 0 -3(2α 7 + 1)t 0 α σ 7 0 0 0 -4(α 7 + 4)t -4(α 7 + 4)t 0 α 7 0 0 35(α 7 + 2)t 2 -5(2α 7 + 1)t -5(α 7 -3)t 0 α σ 7 0 -42(α 7 -3)t 2 -21(2α 7 -3)t 2 -6(α 7 -3)t -6(2α 7 + 1)t 0 α σ 7        
.

We have

M 6 (A 7 )M 6 (τ (A 7 )) = M 6 (A 7 )M 6 (-A σ 7 ) = M 6 (A 7 )M 6 (A 7 ) σ = 2I 6 (since τ (A 7 ) = -A σ 7 ), so φ d,
C 0 = V y 1 -y 2 , x 3 1 -x 2 1 x 2 + α σ 7 x 1 x 2 2 -x 3 2 = i∈{1,2,4} V y 1 -y 2 , ζ i 7 x 1 -x 2 ⊂ Z d,7 × Q(α7)(s2,...,s d ) Z d,7 . Each V y 1 -y 2 , ζ i 7 x 1 -x 2 induces ζ i on J Z d,7 , so φ C0 = ζ + ζ 2 + ζ 4 = [α 7 ] JZ d,7 . Therefore, φ d,7 is a one-parameter deformation of [α 7 ] JZ d,7 , which splits [2] JZ d,7 .
(This gives an alternative proof of Proposition 8.1.) Remark 8.2. Given any hyperelliptic curve X of genus 3 and a maximal 2-Weil isotropic subgroup S of J X [START_REF] Birkenhake | Complex abelian varieties (2e). Grundlehren der mathematischen Wissenschaften[END_REF], there exists a (possibly reducible, and generally nonhyperelliptic) curve Y of genus 3 and an isogeny φ : J X → J Y with kernel S, which may be defined over a quadratic extension of K(S). An algorithm to compute equations for Y and φ when S is generated by differences of Weierstrass points appears in [START_REF] Smith | Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves[END_REF]. Mestre [START_REF] Mestre | Couples de jacobiennes isogénes de courbes hyperelliptiques de genre arbitraire[END_REF] gives a 4-parameter family of (Z/2Z) 3 -isogenies of hyperelliptic Jacobians; their kernels are also generated by differences of Weierstrass points. Since J X2,7 [START_REF] Birkenhake | Complex abelian varieties (2e). Grundlehren der mathematischen Wissenschaften[END_REF] is generated by differences of Weierstrass points, which correspond to roots of f 7 together with the point at infinity, we can factor f 7 (or a reduction at some well-chosen prime) over its splitting field, and then explicitly compute the restriction of φ 2,7 to J X2,7 [START_REF] Birkenhake | Complex abelian varieties (2e). Grundlehren der mathematischen Wissenschaften[END_REF] to show that its kernel is not generated by differences of Weierstrass points. Therefore, φ 2,7 is not one of the isogenies of [START_REF] Smith | Isogenies and the discrete logarithm problem in Jacobians of genus 3 hyperelliptic curves[END_REF] or [START_REF] Mestre | Couples de jacobiennes isogénes de courbes hyperelliptiques de genre arbitraire[END_REF]. 9. Genus g 11 (d) families from Theorem 6.3 (2) (so f 11 = 11g, where g is the polynomial of [9, §5.2] with a 2 = α σ 11 ). We have a factorization The field

Consider Theorem 6.3(2): Let α 11 be an element of Q satisfying α 2 11 + α 11 + 3 = 0, so Q(α 11 ) = Q( √ - 11
f 11 (x 1 ) -f σ 11 (x 2 ) = A 11 (x 1 , x 2 )B 11 (x 1 , x 2 ), where A 11 (x 1 , x 2 ) = x 5 1 -α 11 x 4 1 x 2 -x 3 1 x 2 2 + (4α 11 + 2)x 3 1 + x 2 1 x 3 2 + (α 11 + 6)x 2 1 x 2 -(2α 11 -10)x 2 1 -(α 11 + 1)x 1 x 4 2 + (α 11 -5)x 1 x 2 2 -(12α 11 + 6)x 1 x 2 + (8α 11 -7)x 1 -x 5 2 + (4α 11 + 2)x
P d (y 1 ) = f 11 (x 1 ), Y d,11 : P d (y 2 ) = f σ 11 (x 2 ), C d,11 = V (y 1 -y 2 , A 11 (x 1 , x 2 )) ⊂ X d,11 × Q(α11)(s2,...,s d ) Y d,11 . The induced homomorphism φ d,11 = φ C d,11 : J X d,11 → J Y d,11 is a (d-1)-dimensional family of (Z/3Z) g11(d) -isogenies. Proof. Both X d,
Q(α 13 ) = Q( -3 √ 13 + 1) is an imaginary quadratic extension of the real quadratic field Q(β 13 ) = Q( √ 13
). The involution σ : α 13 → β 13 /α 13 generates Gal(Q(α 13 )/Q(β 13 )). Let t be a free parameter, and let f 13 (x) = x 13 + 39((3β 13 -13)α 13 -2β 13 + 8)tx 11 + • • • be the polynomial of degree 13 over Q(α 13 )[t] defined in the file degree-13.m (we have f 13 = 13g, where g is the polynomial of [9, §5.3] with a 1 = α 13 and T = t). We have a factorization f 13 (x 1 )f σ 13 (x 2 ) = A 13 (x 1 , x 2 )B 13 (x 1 , x 2 ), where Both A 13 and B 13 are absolutely irreducible, and τ (A 13 ) = A σ 13 and τ (B 13 ) = -B σ 13 . Proposition 10.1. Let d > 1 be an integer, and consider the families defined by [START_REF] Davenport | Two problems concerning polynomials[END_REF] .

A 13 (x 1 , x 2 ) = x 4 1 + x 4 2 + (β 13 -3)x
X d,13 : P d (y 1 ) = f 13 (x 1 ), Y d,13 : P d (y 2 ) = f σ 13 (x 2 ), C d,13 = V (y 1 -y 2 , A 13 (x 1 , x 2 )) ⊂ X d,13 × Q(α13)(s2,...,s d ,t) Y d,
The induced homomorphism φ d,13 := φ C d,13 : J X d,13 → J Y d,13 is a d-dimensional family of (Z/3Z) g13(d) -isogenies.

Proof. Both X d,13 and Y d,13 have genus g 13 (d), with d moduli by Lemma 4.1. We compute M 12 (A 13 ) (given in degree-13.m) using Algorithm 5.1; its diagonal is

(λ 1 , λ 2 , λ 1 , λ σ 1 , λ 2 , λ 2 , λ σ 2 , λ σ 2 , λ 1 , λ σ 1 , λ σ 2 , λ σ 1 )
, where λ 1 = (β 13 -4)α 13 + 2 and λ 2 = α 13 + 1 both have norm 3 in Q(β 13 ). We find M 12 (A 13 )M 12 (τ (A 13 )) = M 12 (A 13 )M 12 (A 13 ) σ = 3I 12 (since τ (A 13 ) = A σ 13 ), so the result follows from Lemma 5.2.

Remark 10.1. As in §8, we may view φ d,13 as a deformation of an endomorphism of a superelliptic Jacobian. We embed Z[α 13 ] in Z[ζ 13 ], identifying α 13 with 1+ζ via ι : (x 2 , y 2 ) → (-x 2 , y 2 ). Meanwhile, A specializes to A 0 = i∈S (ζ i 15 x 1 + x 2 ), so C d,15 specializes to

C 0 = i∈S V y 1 -y 2 , ζ i 15 x 1 + x 2 ⊂ Z d,15 × Q(α15)(s2,...,s d ) Z ′ d,
) → (x 3 i , y i ) and (x i , y i ) → (x 5 i , y i ), respectively. The endomorphism ψ = ι • φ C0 of J Z d,15 is induced by V (y 1 -y 2 , A 0 (x 1 , -x 2 )). The matrix M 14 (A 0 (x 1 , -x 2 )) is diagonal: M 14 (A 0 (x 1 , -x 2 )) = diag(α σ 15 , α σ 15 , 2, α σ 15 , -2, 2, α 15 , α σ 15 , 2, -2, α 15 , 2, α 15 , α 15
). Considering Eq. ( 8), we see that D(ψ)(ω i,j ) = 2ω i,j whenever j = 3, 6, 9, and 12 (that is, when ω i,j is the pullback of a differential on Z d,5 ), so ψ acts as [START_REF] Birkenhake | Complex abelian varieties (2e). Grundlehren der mathematischen Wissenschaften[END_REF] JZ d,15 on J Z d,5 ⊂ J Z d,15 . Similarly, D(φ C0 )(ω i,j ) = -2ω i,j for j = 5 and 10 (when ω i,j is the pullback of a differential on Z d,3 ), so ψ acts as

[-2] JZ d,15 on J Z d,3 ⊂ J Z d,15 .
Looking at the other entries on the diagonal, we see that ψ acts as multiplicationby-α 15 on the (g 15 (d) - [START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF] . This gives us a clearer description of the isomorphism in the proof of Proposition 11.1: the factors (Z/4Z) g15(d)-g3(d)-g5(d) , (Z/2Z) 2g3(d) , and (Z/2Z) 2g5(d) correspond to ker(ψ| A ), ker(φ| JZ d,3 ), and ker(φ| JZ d,5 ) respectively.

g 5 (d) -g 3 (d))-dimensional complimentary subvariety A of J Z d,3 × J Z d,5 in J Z d,
d (y 1 ) = f 21 (x 1 ), Y d,21 : P d (y 2 ) = f σ 21 (x 2 ), C d,21 = V (y 1 -y 2 , A 21 (x 1 , x 2 )) ⊂ X d,21 × Q(α21)(s2,...,s d ) Y d,21 . The induced homomorphism φ d,21 := φ C d,21 : J X d,21 → J Y d,21 is a (d -1)- dimensional family of (Z/4Z) g21(d)-g3(d) ×(Z/2Z) 2g3(d) -isogenies. Proof. Both X d,

Absolute simplicity

We want to verify that our isogenies φ : J X → J Y do not arise from products of isogenies of lower-dimensional abelian varieties. To this end, where possible, we show that the generic fibres of J X and J Y are absolutely simple. Proof. We need only prove absolute simplicity for each J X d,n (the existence of the isogeny φ d,n then implies that J Y d,n is absolutely simple). If J X d,n is reducible, then so are all of its specializations; so it suffices to exhibit an absolutely simple specialization of J X d,n . We can do this for many (d, n) by applying results of Zarhin to hyperelliptic or superelliptic specializations. For n = 7 and 13, we specialize at t = 0; then we apply [42, For some of the remaining cases, we can use the fact that X d,n is defined over a number field; by [START_REF] Chai | A note on the existence of absolutely simple Jacobians[END_REF]Lemma 6], it suffices to exhibit an absolutely simple reduction of a specialization of J X d,n modulo a prime of good reduction. We prove absolute simplicity of reductions by computing Weil polynomials (using Gaudry and Gürel's algorithm [START_REF] Gaudry | An extension of Kedlaya's point-counting algorithm to superelliptic curves[END_REF] for superelliptic curves, and the Magma system's implementation [START_REF] Harrison | Some notes on Kedlaya's algorithm for hyperelliptic curves[END_REF] of Kedlaya's algorithm [START_REF] Kedlaya | Counting points on hyperelliptic curves using Monsky-Washnitzer cohomology[END_REF] for hyperelliptic curves) and applying [START_REF] Howe | On the existence of absolutely simple abelian varieties of a given dimension over an arbitrary field[END_REF]Proposition 3]. For (d, n) = (2, 11) we specialize at s 2 = 0 and reduce at a prime over 7; for (d, n) = (2, 13) we specialize at (s 2 , t) = (1, 0) and reduce at a prime over 53; for (d, n) = (2, 15) we specialize at (s 2 , t) = (0, 1) and reduce at a prime over 17; and for (d, n) = (5, 11) we specialize at (s 2 , . . . , s 5 ) = (0, . . . , 0) and reduce at a prime over 31.

The list of values of n and d in Proposition 14.1 is not intended to be exhaustive; it simply reflects the practical and theoretical limits of the results used in the proof. We would like to prove simplicity for at least all prime d; but the Gaudry-Gürel algorithm requires n and d to be coprime, so we cannot apply it to cases such as (d, n) = [START_REF] Chai | A note on the existence of absolutely simple Jacobians[END_REF][START_REF] Chai | A note on the existence of absolutely simple Jacobians[END_REF]. Further, the reduction of a superelliptic Jacobian can only be simple if the residue field contains a primitive d th root of unity (otherwise the superelliptic automorphism does not commute with Frobenius, so the endomorphism ring is noncommutative, so the reduction is not simple). This rules out many small primes of reduction, rendering the computation much more expensive. Computing Weil polynomials for (d, n) = (7, 15), [START_REF] Bost | Moyenne arithmético-géometrique et périodes des courbes de genre 1 et 2[END_REF][START_REF] Fried | [END_REF], [START_REF] Bosma | The Magma algebra system. I. The user language[END_REF][START_REF] Lidl | Dickson polynomials. Pitman monographs and surveys in pure and applied mathematics 65[END_REF], and (5, 31) will therefore require highly optimised implementations and significant computing resources.

  4Z) g15(d)-g5(d)-g3(d) ×(Z/2Z) 2g5(d)+2g3(d) P \ {3, 5, 7} 21 d -1 2 (Z/4Z) g21(d)-g3(d) ×(Z/2Z) 2g3(d)

Algorithm 5 . 1 .

 51 Computes the maximal block M n-1 (A) of the matrix D(φ C ).

  j=1 . 4b: Compute the Hermite Normal Form of M , and let (d 1 , . . . , d 2ek ) be its elementary divisors. 4c: Set G(A, k) := 2ek i=1 (Z/d i Z). 5: Return (G(A, 1), . . . , G(A, n -1)).

  Proof. The family U d,n : P d (v) = u n + 1/u n has an involution ι : (u, v) → (1/u, v) and an automorphism ζ : (u, v) → (ζ n u, v). The double cover π : U d,n → W d,n defined by (u, v) → (u + u -1 , v) is the quotient of U d,n by ι , and π * π * = [2] JW d,n . Let (x, y) be a generic point on W d,n . On the level of divisors we have φ Ci ((x, y)) = (α 1 , y) + (α 2 , y),

7 . 8 . 1 .

 781 A 7 and B 7 are absolutely irreducible, and τ (A 7 ) = -A σ 7 and τ (B 7 ) = B σ Proposition Let d > 1 be an integer, and consider the families defined by

  ). The involution σ : α 11 → 3/α 11 generates Gal(Q(α 11 )/Q). Let f 11 be the polynomial over Q(α 11 ) defined by f 11 (x) := x 11 + 11α 11 x 9 + 22x 8 -33(α 11 + 4)x 7 + 176α 11 x 6 -33(7α 11 -5)x 5 -330(α 11 + 4)x 4 + 693(α 11 + 1)x 3 -220(5α 11 -1)x 2 -33(8α 11 + 47)x + 198α 11

  11 and Y d,11 have genus g 11 (d), and d -1 moduli by Lemma 4.1. As in Proposition 8.1, we calculate M 10 (A 11 ) (given in degree-11.m) using Algorithm 5.1; its diagonal entries are all either α 11 or α σ 11 . Using τ (A 11 ) = -A σ 11 , we find M 10 (A 11 )M 10 (τ (A 11 )) = M 10 (A 11 )M 10 (A 11 ) σ = 3I 10 , so φ d,11 is a family of (Z/3Z) g11(d) -isogenies by Lemma 5.2. 10. Genus g 13 (d) families from Theorem 6.3 (3) Consider Theorem 6.3(3): Let β 13 and α 13 be elements of Q satisfying β 2 13 -5β 13 + 3 = 0 and α 2 13 + (β 13 -2)α 13 + β 13 = 0.

  7 and 8 respectively (also defined in degree-15.m), with τ (A 15 ) = A σ 15 and τ (B 15 ) = B σ 15 . Proposition 11.1. Let d > 1 be an integer, and consider the families defined by X d,15 : P d (y 1 ) = f 15 (x 1 ), Y d,15 :P d (y 2 ) = f σ 15 (x 2 ), C d,15 = V (y 1y 2 , A 15 (x 1 , x 2 )) ⊂ X d,15 × Q(α15)(s2,...,s d ,t) Y d,[START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF] .The induced homomorphism φ d,15 := φ C d,15 : J X d,15 → J Y d,15 is a d-dimensional family of (Z/4Z) g15(d)-g5(d)-g3(d) ×(Z/2Z) 2(g5(d)+g3(d)) -isogenies. Proof. Both X d,[START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF] and Y d,15 have genus g 15 (d), with d moduli by Lemma 4.1. We compute M 14 (A 15 ) (given in degree-15.m) using Algorithm 5.1. We findM 14 (A 15 )M 14 (τ (A 15 )) = M 14 (A 15 )M 14 (A 15 ) σ = 4I 14 (using τ (A 15 ) = A σ 15 ), so φ d,15 splits multiplication-by-4 by Lemma 5.2. After specializing t, Algorithm 5.3 givesG(A 15 , k) ∼ = (Z/4Z) 2(k-m(k)) ×(Z/2Z) 4m(k) , where m(k) = #{i : 1 ≤ i ≤ k, gcd(i, 15) = 1}, for each 1 ≤ k ≤ 14.Each of the g 15 (d) points (i, j) in P(d, 15) therefore contributes a factor of either (Z/4Z) 2 or (Z/2Z) 4 to (ker(φ d,15 )) 2 , according to whether gcd(j, 15) = 1 or not. The number of points (i, j) in P(d, 15) with gcd(j, 15) = 1 is equal to g 3 (d) + g 5 (d), so(ker φ d,15 ) 2 ∼ = (Z/4Z) 2(g15(d)-g5(d)-g3(d)) ×(Z/2Z) 4(g5(d)+g3(d)) ; the result follows. (See Remark 11.1 for more detail on the kernel structure.) Remark 11.1. As in §8 and §10, we may view φ d,15 as a deformation of an endomorphism of a superelliptic Jacobian. Let S = {0, 1, 2, 4, 5, 8, 10}; we embed Z[α 15 ] in Z[ζ 15 ], identifying α 15 with i∈S ζ i 15 . At t = 0, the family X d,15 specializes to Z d,15 : P d (y 1 ) = x 15 1 , which has an automorphism ζ : (x 1 , y 1 ) → (ζ 15 x, y), while Y d,15 specializes to Z ′ d,15 : P d (y 2 ) = -x 15 2 , which is isomorphic to Z d,15

12 .

 12 Genus g 21 (d) families from Theorem 6.3 (5) Consider Theorem 6.3(5): Let α 21 be an element of Q satisfying α 2 21α 21 + 2 = 0, so Q(α 21 ) = Q( √ -7); the involution σ : α 21 → 2/α 21 generates Gal(Q(α 21 )/Q). Let f 21 (x) = x 21 + (42α 21 + 42)x 19 + (84α 21 + 84)x 18 + (2331α 21 -861)x 17 + • • • be the polynomial of degree 21 over Q(α 21 ) defined in the file degree-21.m (such that f 21 (x) = 2 21 g(x/2), where g is the polynomial of [9, §5.5] with a 1 = α 21

  21 and Y d,21 have genus g 21 (d), with d -1 moduli by Lemma 4.1.We compute M 20 (A 21 ) (given in degree-21.m) using Algorithm 5.1. We find thatM 20 (A 21 )M 20 (τ (A 21 )) = M 20 (A 21 )M 20 (A 21 ) σ = 4I 20 (since τ (A 21 ) = -A σ 21 ), so φ 21 splits multiplication-by-4 by Lemma 5.2. Applying Algorithm 5.3, we see that G(A 21 , k) ∼ = (Z/4Z) k-⌊k/7⌋ ×(Z/2Z) 2⌊k/7⌋ for 1 ≤ k ≤ 20. Hence each point (i, j) in P(d, 21) contributes a factor of either (Z/4Z) 2 or (Z/2Z) 4 to (ker(φ d,21 )) 2 , according to whether 7 divides j or not. Therefore (ker φ d,21 ) 2 ∼ = (Z/4Z) 2(g21(d)-g3(d)) ×(Z/2Z) 4g3(d) , and the result follows.

  ) = -A σ 31 and τ (B 31 ) = B σ 31 . Proposition 13.1. Let d > 1 be an integer, and consider the families defined byX d,31 : P d (y 1 ) = f 31 (x 1 ), Y d,31 : P d (y 2 ) = f σ 31 (x 2 ), C d,31 = V (y 1y 2 , A 31 (x 1 , x 2 )) ⊂ X d,31 × Q(α31)(s2,...,s d ) Y d,31 . The induced homomorphism φ d,31 := φ C d,31 : J X d,31 → J Y d,31 is a (d -1)dimensional family of (Z/8Z) g31(d)/3 ×(Z/4Z) 2g31(d)/3 ×(Z/2Z) 2g31(d)/3 -isogenies. Proof. Both X d,31 and Y d,31 have genus g 31 (d), with d -1 moduli by Lemma 4.1. We compute M 30 (A 31 ) (given in degree-31.m) using Algorithm 5.1. We see that M 30 (A 31 )M 30 (τ (A 31 )) = M 30 (A 31 )M 30 (A 31 ) σ = 8I 30 (using τ (A 31 ) = -A σ 31 ), so φ d,31 splits multiplication-by-8 by Lemma 5.2. Algorithm 5.3 gives G(A 31 , k) ∼ = ((Z/8Z) ×(Z/4Z) 2 ×(Z/2Z) 2 ) 2k for 1 ≤ k ≤ 30, so (ker(φ d,31 )) 6 ∼ = (Z/8Z) ×(Z/4Z) 2 ×(Z/2Z) 2 2g31(d) ; the result follows.

Proposition 14 . 1 .

 141 The generic fibres of J X d,n and J Y d,n are absolutely simple for (1) n = 7 and all d ≥ 2; (2) n = 11 and all prime d = 11; (3) n = 13 and all d ≥ 2 (4) n = 15 and all prime d / ∈ {3, 5, 7}; (5) n = 21 and all prime d / ∈ {3, 5, 7}; (6) n = 31 and all prime d / ∈ {3, 5, 31}.

  Theorem 1.1] for d ≥ 5, and [44, Theorem 1.2] for d = 3 and 4. (We cannot use this approach for n = 15, because the specialization at t = 0 is always reducible: cf. Remark 11.1.) For n = 11, 15, 21, and 31 and all prime d not dividing n(n -1) we specialize at (s 2 , . . . , s d ) = (0, . . . , 0) and apply [41, Corollary 1.8]. For (d, n) = (2, 7), (2, 21), and (2, 31), we specialize at s 2 = 0 and apply [43, Theorem 2.3].

  g7(d) -isogenies. Proof. Both X d,7 and Y d,7 have genus g 7 (d), with d moduli by Lemma 4.1. Applying Algorithm 5.1 to A 7 , we find that M 6 (A 7 ) is equal to 

  [START_REF] Bruns | Polytopal linear groups[END_REF] is a family of (Z/2Z) g7(d) -isogenies by Lemma 5.2. Remark 8.1. We may view φ d,7 as a deformation of an endomorphism of the superelliptic Jacobian J Z d,7 of §7. Embed Z[α 7 ] in Z[ζ 7 ], identifying α 7 with ζ 7 + ζ 2 7 + ζ 4 7 . At t = 0, both X d,7 and Y d,7 specialize to Z d,7 , which has an automorphism ζ : (x, y) → (ζ 7 x, y) of order 7, while C d,7 specializes to

  Both A 11 and B 11 are absolutely irreducible, and τ (A 11 ) = -A σ 11 and τ (B 11 ) = B σ 11 . Proposition 9.1. Let d > 1 be an integer, and consider the families defined by X d,11 :

3 2 -(2α 11 + 12)x 2 2 + (8α 11 + 15)x 2 + 12α 11 + 6.

  2 1 x 2 2 -9(3β 13 -14)tx 1 x 2 + 12(47β 13 -202)t 2 -((β 13 -4)α 13 + 2)x 3 1 x 2 + ((β 13 -4)α 13β 13 + 3)x 1 x 3 (5β 13 -22)α 13 -9β 13 + 38)tx 1 -3((5β 13 -22)α 13 + 2β 13 -9)tx 2 .

	+ 3((17β 13 -73)α 13 -12β 13 + 50)tx 2 1 2 -3((17β 13 -73)α 13 -10β 13 + 45)tx 2 + 3(	2

  At t = 0, both X d,[START_REF] Davenport | Two problems concerning polynomials[END_REF] and Y d,[START_REF] Davenport | Two problems concerning polynomials[END_REF] specialize to the family Z d,[START_REF] Davenport | Two problems concerning polynomials[END_REF] of §7, while C d,13 specializes to -y 2 , ζ i 13 x 1x 2 ⊂ Z d,13 × Q(α13)(s2,...,s d ) Z d,[START_REF] Davenport | Two problems concerning polynomials[END_REF] .Each V y 1y 2 , ζ i 13 x 1x 2 induces the automorphism ζ i : (x, y) → (ζ i 13 x, y) of J Z d,[START_REF] Davenport | Two problems concerning polynomials[END_REF] , soφ C0 = [1] + ζ 7 + ζ 8 + ζ 11 = [λ 1 ] JZ d,13 ; hence φ d,13 is a one-parameter deformation of [λ 1 ] JZ d,13 , which splits [3] JZ d,13 . 11. Genus g 15 (d) families from Theorem 6.3 (4) the involution σ : α 15 → 4/α 15 generates Gal(Q(α 15 )/Q). Let f 15 (x) = x 15 + 15(α 15 -1)tx 13 + 15(α 15 + 7)tx 12 + • • • be the polynomial of degree 15 over Q(α 15 )[t] defined in the file degree-15.m (so f 15 = 15g, where g is the polynomial of [9, §5.4] with a 1 = α 15 and T = t). We have a factorization f 15 (x 1 ) -(-f σ 15 (x 2 )) = A 15 (x 1 , x 2 )B 15 (x 1 , x 2 ), where A 15 and B 15 are absolutely irreducible polynomials of total degree

	then λ 1 = 1 + ζ 7 13 + ζ 8 13 + ζ 11 13 . C 0 = i∈{0,7,8,11} V y 1 Consider Theorem 6.3(4): Let α 15 be an element of Q satisfying	3 13 +ζ 9 13 ;
	so Q(α 15 ) = Q( √ -15);	α 2 15 -α 15 + 4 = 0,

  [START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF] ,and φ C0 = ι i∈S ζ i = ι[α 15 ] JZ d,[START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF] . Hence φ d,15 is a one-parameter deformation of an isogeny isomorphic to the endomorphism [α 15 ] JZ d,[START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF] .We gain further insight into the structure of ker φ C0 , and hence ker φ C , by decomposing J Z d,[START_REF] Feit | Automorphisms of symmetric balanced incomplete block designs[END_REF] . We may view J Z d,5 and J Z d,3 as abelian subvarieties of J Z d,15 via the covers Z d,15 → Z d,5 and Z d,15 → Z d,3 , defined by (x i , y i

  ). We have a factorizationf 21 (x 1 )f σ 21 (x 2 ) = A 21 (x 1 , x 2 )B 21 (x 1 , x 2 ), where A 21 (x 1 , x 2 ) = x 5 + (107α 21 -162)x 2 + 136α 21 -68.Both A 21 and B 21 are absolutely irreducible, and τ (A 21 ) = -A σ 21 and τ (B 21 ) = B σ 21 . Proposition 12.1. Let d > 1 be an integer, and consider the families defined by X d,21 : P

	1 + (α 21 + 1)x 4 1 x 2 + 2α 21 x 3 1 x 2 2 + (10α 21 + 18)x 3 1 + (2α 21 -2)x 2 1 x 3 2 + (32α 21 -8)x 2 1 x 2 + (20α 21 + 4)x 2 1 + (α 21 -2)x 1 x 4 2 + (32α 21 -24)x 1 x 2 2 + (32α 21 -16)x 1 x 2 + (107α 21 + 55)x 1 -x 5 2 + (10α 21 -28)x 3 2 + (20α 21 -24)x 2 2

13 .

 13 Genus g 31 (d) families from Theorem 6.3 (6) Consider Theorem 6.3(6): Let α 31 and β 31 be elements of Q satisfying 7β 31 + 4)α 31 + β 31 = 0. Note that Q(α 31 ) is a sextic CM field, and Q(β 31 ) is its totally real cubic subfield. The involution σ : α 31 → β 31 /α 31 generates Gal(Q(α 31 /Q(β 31 )). Let f 31 (x) = x 31 -31( 1 4 (β 2 31 -5β 31 -10)α 31 -(β 2 31 -7β 31 + 12))x 29 -31( 1 2 (β 2 31 -5β 31 -10)α 31 -(2β 2 31 -14β 31 + 24))x 28 + • • • be the polynomial of degree 31 over Q(α 31 ) defined in the file degree-31.m (such that f 31 (x) = 2 31 g(x/2), where g is the polynomial of [9, §5.6] with a 1 = α 31 ). We have a factorization f 31 (x 1 )f σ 31 (x 2 ) = A 31 (x 1 , x 2 )B 31 (x 1 , x 2 ), where A 31 and B 31 are absolutely irreducible polynomials of total degree 15 and 16, respectively, with τ (A 31

	β 3 31 -13β 2 31 + 46β 31 -32 = 0 and α 2 31 -1/2(β 2 31 -

We could define P d to be the generic monic polynomial of degree d, but we can always change variables to remove its trace term in characteristic zero, and this will be convenient in the sequel.

Acknowledgements. We thank John Voight, for his suggestions at the Explicit Methods in Number Theory workshop at the FWO in Oberwolfach, 2009; the workshop organisers and the FWO itself, for the fruitful environment in which this work was begun; Wouter Castryck, for his patient help and for pointing out Koelman's thesis; and Frederik Vercauteren, for sharing his implementation of the Gaudry-Gürel point counting algorithm.

INRIA Saclay-Île-de-France / Laboratoire d'Informatique de l' École polytechnique (LIX), 91128 Palaiseau Cedex, France E-mail address: smith@lix.polytechnique.fr URL: http://www.lix.polytechnique.fr/Labo/Ben.Smith/


Nils Le Gal¹ , ²

Vincent Lagneau¹
email: vincent.lagneau@mines-paristech.fr

Arnaud Charmoille²
email: arnaud.charmoille@ineris.fr

Mechanisms of gas migration in flooding post-mining context

Keywords: gas migration, flooding, methane, post-mining

The French national institute for industrial environment and risks (ineriS), in a support mission to state services and as a partner of the european project FLOMineT, is studying gas migration during and after mine flooding. in situ flow measurements and laboratory records acquired with an experimental device are being used to study gas transfer and to characterize the influence of hydrostatic pressure on methane release from coal. in parallel, several models are in development using the HYTec code to describe mine methane migration. The combined use of numerical and experimental approaches will make it possible to improve risk prevention concerning surface gas emission above the old mines.

Introduction

residual mine voids, during exploitation and after closing, have significant environmental impacts. numerical models are being developed to characterize the impact of old mines at the watershed scale. The French national institute for industrial environment and risks (ineriS) is studying the mechanisms of gas transfer and migration in flooded mines, in partnership with MineS ParisTech and as a partner of the european FLOMineT (FLOoding management for underground coal Mines considering regional mining neTworks) program. The mines we are interested in are the coal mines of the Lorraine basin (in northeastern France). Among the diverse underground gas products, this study is restricted to methane, carbon dioxide and radon. in this paper, we will focus on methane, cH₄.

This study has two main objectives: to characterize underground mine gas migration and to determine the influence of flooding on mobilization of sorbed gases. Several gases are naturally produced by rocks, so mine voids can contain different gases, for example methane from coal beds (e.g. doyle 2001;[START_REF] Ar | Hydrogeologic factors affecting gas content distribution in coal beds[END_REF][START_REF] Besnard K | Bilan des principales connaissances sur ls mécanismes géochimiques et biochimiques à l'origine de la création d'atmosphères vicieuses en contexte minier[END_REF]. According to [START_REF] Ar | Hydrogeologic factors affecting gas content distribution in coal beds[END_REF], hydrologic parameters significantly influence gas distribution in coal beds. during flooding, two phases are passing through mine voids and porous media: a gaseous phase and water. despite the dissolution of a part of the gaseous phase, two-phase flow occurs in old mines, pushing the gas with water and/or bubble flow. due to competition between cH₄ and water molecules for sorption sites [START_REF] Ji | Sorption of methane in moist coal[END_REF][START_REF] Doyle Br | Hazardous gases Underground[END_REF]crosdale et al. 2008;charrière,2009), mine water can be enriched in cH₄ and carry it through galleries, fractures, porous media, possibly up to the surface. The water can degas where pressure decreases, either naturally or due to pumping. The released gas can accumulate under gastight structures, like the caps of obstructed wells.

Methods

Aiming to improve our knowledge of gas migration during and after flooding, two complementary tools are being used: an experimental device and a numerical approach. The experimental device makes it possible to reproduce the effects of coal flooding; the results are used in numerical models which simulate the migration of cH₄ in old mines.

Laboratory experiment (CASPER)

An experimental device named cASPer (sorption capacity under high hydrostatic pressure during rock flooding) has been developed at ineriS to evaluate gas sorption and gas release under variable hydrostatic pressure. it reproduces the depth conditions in water-saturated context in an autoclave cell. At the equilibrium, we measured the released quantity of cH₄. The main objective of this experimental device is to determine the necessary hydrostatic pressure level to stop gas release from coal. This topic has already been introduced by [START_REF] Pokryszka | gas Migration from closed coal Mines to the Surface -risk Assessment Methodology and Prevention Means[END_REF] and Pokryszka and krauser (2007). knowledge of this parameter is essential to estimate the possibility of gas migration from flooded mines to the surface.

Modelling with HYTEC

A reactive transport model developed by MineS ParisTech: HYTec (van der [START_REF] Van Der Lee | Module-oriented modeling of reactive transport with HYTec[END_REF] simulates migration of aqueous species, and has recently been improved to take into account gas migration. numerical models are developed in order to reproduce and characterize methane migration during and after mine flooding. Several parameters are considered: the saturation of the media, the porosity, the permeability, the diffusivity, the dispersion. The methane sorption phenomenon on coal has been defined for our particular case with the geochemical module cHeSS. considering the solubility of cH₄ at a given temperature and pressure, and the properties of coal, we were be able to determine sorption constants under high hydrostatic pressure.

Results

Laboratory experiment

A first run was performed with a reduced apparatus (still incomplete monitoring environment). The obtained qualitative results were consistent with the solubilities calculated using the duan and Mao ( 2006) model. The experimental results were compared with predictions based on our hypothesis concerning the dissolution of cH₄ after desorption from the coal surface.

The cASPer device is currently being improved. Once the device is complete and the gas chromatograph calibrated, and the first tests with the rOLSi TM are performed, further quantitative results will be available. The cASPer device will allow for the monitoring of the evolution of dissolved cH₄ and validate that equilibrium is reached. Additionally, the kinetic of desorption under high hydrostatic pressure will be characterized.

Simulation of CH₄ migration and flooding

The first step of modelling was to determine adsorption constant of the experiment. Figure 1 shows the cH₄ adsorption equilibrium curve on coal surface in flooded conditions. Once this constant is established at given temperature and pressure, the coal influence can be taken into account within migration models. The next step was to reproduce cH₄ migration through different structures like galleries, wells and fractures including sorption on coal. For lack of strong sorption data yet, we will focus on migration mechanisms at the gallery scale.

considering a cH₄-enriched water that enters a flooded gallery surrounded by a damaged zone connected to a fracture, the migration of aqueous cH₄ has been simulated (fig. 2): in this scenario, the model highlights the effect of a fracture on the migration of a cH₄-enriched plume.

galleries and open fractures are preferential pathways for water flow. As a consequence, they enhance the migration of dissolved cH₄. Figure 2, which shows two different models results, illustrates the effect of a fracture on aqueous cH₄ migration. in the case of a gallery surrounded by

Figure 1 CH₄(aq) adsorption equilibrium curve on coal at 25 °C and 25 bar (experimental conditions)

a fresh rock, the cH₄ plume spreads over 7 m on the horizontal axis within 5 years simulation. When the gallery is connected to a fracture, the cH₄ plume front reaches 14 m on the horizontal axis in the same time interval. When the water rises to the surface, surface emission of methane (or other gases) can be increased. indeed, it appears that a fracture connected to a gallery can enhance the migration of methane to the surface, as a function of its own geometry and hydrodynamic properties. The damaged zone surrounding the gallery can also act like a drain.

The presence of coal in the medium induces cH₄ adsorption. Figure 3 shows the evolution of the distribution of cH₄ in the rock at 2 m from the damaged zone, just below the fracture. As the cH₄ penetrates the porous medium, the sorption sites on the coal surface are progressively saturated.

The third part of modelling aims to reproduce mine flooding. These models are still in development, and will not be further discussed in this paper.

Sydney, NS

IMWA 2010 "Mine Water and Innovative Thinking"

Wolkersdorfer & Freund (Editors) 49 The experimental device gave an initial result that was consistent with the solubility model of duan and [START_REF] Duan | A thermodynamic model for calculating methane solubility, density and gas phase composition of methane-bearing aqueous fluids from 273 to 523 k and from 1 to 2000 bar[END_REF]. it also confirmed the hypothesis of competition of water and methane molecules on sorption sites. The results are used in numerical models to investigate the parameters of cH₄ sorption on coal. This work will make it possible to precisely quantify the cH₄ release from flooded coal and improve our knowledge of sorption kinetic under high hydrostatic pressure. The numerical models showed the influence of various structures like galleries, damaged zones, and fractures. Permeable structures enhance the transfer of aqueous cH₄. Our models also account for cH₄ adsorption on coal during the migration of the enriched plume. We are also able to reproduce mine flooding and we are improving the models to better simulate the reality of the Lorraine basin, reproducing gallery filling and medium saturation. Also two-phase flow is currently incorporated to HYTec: making it possible to describe gas diffusion in porous medium. combined flow of gas and water in mine voids will thus be integrated in our models.

Figure 2 Figure 3

 23 Figure 2Migration of aqueous CH₄ through fresh porous medium (left) and fractured rock (right), with a head gradient equal to 0.01 after 5 years simulation. The red point corresponds to the evolution curves of figure3

Wolkersdorfer & Freund (Editors)


G Aouad

L De Windt

D Damidot

3D simulation of the leaching of cement-based materials in order to compare different leaching tests

Leaching of cement-based materials is a complex process that depends on both the material intrinsic properties and the leaching test. The microstructure and cement-type of the material are typical examples of the former whereas the later consist of solution pH and composition, solution renewal rate and the liquid/solid volume ratio. As most of the tests are not normalized, this leads to a diversity of results in terms of alteration layer thicknesses and leaching kinetics. As a consequence, it is often difficult to compare and transpose experimental data from one experience to the other. 1D reactive transport modeling has proven to be adequate to simulate the coupling between diffusion, dissolution / precipitation and sorption processes that take place during leaching. The aim of this study is to demonstrate the usefulness of 3D reactive transport modeling to accurately simulate the features of the leaching device features, which is essential to compare results obtained by different leaching tests. Such simulations also enable to assess the mutual interaction between samples if the device contains several of them. Additionally, these simulations represent an interesting numerical tool in order to design new or complementary leaching tests.

Introduction

The degradation of cement-based materials depends on many factors. Some of them do not depend on the experimental device as the material composition, temperature, pH and solution composition [START_REF] Adenot | Durabilité du béton : caractérisation et modélisation des processus physiques et chimiques de dégradation du ciment[END_REF][START_REF] Faucon | Long-term behaviour of cement pastes used for nuclear waste disposal: review of physico-chemical mechanisms of water degradation[END_REF][START_REF] Planel | Long-term performance of cement paste during combined calcium leaching-sulfate attack: kinetics and size effect[END_REF][START_REF] Moranville | Physicochemical equilibria of cement-based materials in aggressive environments-experiment and modelling[END_REF][START_REF] Vichot | La durabilité des bétons-Bases scientifiques pour la formulation de bétons durables dans leur environnement. 2ème édition[END_REF]. Other parameters such as the solution renewal rate and the liquid/solid volume ratio are strictly related to the experimental device. The variability of parameters leads to a diversity of results in terms of portlandite dissolution front, flux of Ca ions and thus, globally, in terms of leaching kinetics. Let's take the example of the leaching of Portland cement pastes. Using a leaching solution of pH 11.5, Revertegat et al. 1992 [START_REF] Revertegat | Effect of pH on the durability of cement pastes[END_REF] measured a degraded thickness of 1.2 mm after 36 months while [START_REF] Adenot | Influence du pH sur la dégradation chimique de pâte de CPA et CLC. Note technique du CEA[END_REF] [START_REF] Adenot | Influence du pH sur la dégradation chimique de pâte de CPA et CLC. Note technique du CEA[END_REF] measured a thickness of 1.91 mm after 8.5 months. Moreover, a thickness of 1.5 mm was observed by Moranville et al 2004 [4] after 114 days of alteration for a leaching solution of pH 7.

As most of the tests are not normalized, it is difficult to compare different studies, even when they only differ by a single parameter such as temperature, duration of the experiment, pH or hydrodynamic conditions. Considerable progress has been made in the field of reactive transport models and computers are more and more powerful. This offers the possibility of simulating geochemical systems with very complex geometry and hydrodynamic context. Therefore, reactive transport modeling can be used to compare data from literature. Provided we can correctly simulate a leaching experience with its associated device, it is possible to simulate the same experience for another device and eventually compare the results.

After a short description of the mechanisms involved in the degradation of cement paste, we present a reactive transport calculation in 3D-cylindrical geometry in a view of comparing leaching results obtained with the same brine and cement pastes but leached according to two different devices.

L eaching of cement-based materials

Many studies have been conducted in order to understand the main mechanisms involved in the leaching of cement paste with pure water [START_REF] Adenot | Durabilité du béton : caractérisation et modélisation des processus physiques et chimiques de dégradation du ciment[END_REF][START_REF] Faucon | Long-term behaviour of cement pastes used for nuclear waste disposal: review of physico-chemical mechanisms of water degradation[END_REF][START_REF] Planel | Long-term performance of cement paste during combined calcium leaching-sulfate attack: kinetics and size effect[END_REF][START_REF] Moranville | Physicochemical equilibria of cement-based materials in aggressive environments-experiment and modelling[END_REF]. Leaching is a combined diffusion-dissolution/precipitation process. The concentration gradients between the pore solution and the leaching solution (generally a pure water) cause diffusion of calcium ions from the pore solution to the surrounding free ion water. The reduction in concentration of calcium ions in the pore solution forces the dissolution of calcium hydroxide (CH) since the pore solution is in equilibrium with the hydrates of the cement pastes. This results in increased material porosity, and, consequently, in diffusion enhancement.

Impact of the solution chemical composition

The chemical composition of the leaching solution is of major importance due to its effect on diffusion, as well as dissolution/precipitation reactions. The greater is the concentration gradient between the solution and the pore solution, the greater is the diffusion and the hydrate dissolution. Therefore, pure or deionized water is one of the strongest decalcifying agents of cement-based materials. In contrast, the presence of carbonate ions in the solution decreases the leaching kinetics of the cement matrix [START_REF] Bescop | Dégradation des matériaux à base de liants hydrauliques en milieu saturé -Application au comportement à long terme les colis « béton[END_REF]. For example, the calcium flux generated by a leaching solution containing 2.5x10 -3 mol/L of NaHCO 3 is 40 times lower compared to leaching with deionized water. The degraded layer is consequently lower. After 6 months of leaching, the degraded thickness is 2 mm for pure water but 0.3 mm with a carbonated water. In addition, the mineralogy and texture of solids are modified by the precipitation of calcium carbonate in the later case: the solid/liquid interface consists mainly of a dense calcite layer, which reduces the porosity and, therefore, the diffusivity of the material.

The pH of the leaching solution also greatly influences the degradation kinetics [START_REF] Delagrave | Influence of chloride ions and pH level on the durability of high performance cement pastes[END_REF]. The concentration gradient between the pore and the leaching solutions leads to the diffusion of hydroxyl ions. Since the pore solution of cement paste has a pH value around 13, the lower the pH of the solution is, the greater the OH - concentration gradient and the dissolution rate are.

Impact of material microstructure

The diffusion coefficient and the leaching kinetics depend on the microstructure. Generally, the diffusivity of cement paste increases with the total porosity. As the porosity increases with the Water/Cement ratio, this means that the degradation should increase with the W/C ratio.

In conclusion, the diversity of experiences in the literature can be explained by the number of parameters influencing the cement paste degradation. Apart from standardized tests, it is scarce to find two studies using the same W/C ratio, leaching solution pH and composition, solution renewal rate or ratio liquid/solid volume (V l /V s ). While some of these parameters can be taken into account by a 1D modeling, such as temperature and pH effects, 3D modeling is required to take into account factors such as solution renewal rates and V l /V s ratios.

3D modeling used to compare leaching tests

Reactive transport calculations were done with 3D-cylindrical geometry in order to compare leaching results obtained on the same cement paste leached in two different cylindrical cells with the same brine at 80 °C.

Three monoliths were placed in each cell. The dimensions of cell 1 are: a 3.8 cm diameter and a 57 cm height (Fig. 1). The dimensions of cell 2 are: a 12 cm diameter of cell 2 and a 36 cm height. Two main differences between the two cells could influence the mechanisms of degradation; the first one is the V l /V s ratio, which is 33 in cell 1 and 215 in cell 2. The second difference is the solution renewal rate: 3 L/d and 0.5 L/d in cell 1 and 2, respectively. The composition of the solution (brine) and the cement pastes are given in Tab. 1.

Model feature

The reactive transport code HYTEC [START_REF] Van Der Lee | Module-oriented modeling of reactive transport with HYTEC[END_REF] was used to simulate the dynamic leaching tests taking into account diffusive transport of solutes and chemical reactions. Transport and chemistry are coupled through a sequential iterative algorithm. In addition, the HYTEC code is strongly coupled, e.g., the effective diffusion coefficients change when mineral precipitation or dissolution modifies the local porosity. Thus, HYTEC accounts for clogging by carbonation, or to the contrary, for porosity increase by leaching of calcium hydroxide. A modified version of the Archie's law, which is implemented in HYTEC, was used in a first attempt at modeling the feedback of chemistry on mass transport:

       ) )( ( ) ( 0 0 c c e e D D    (Eq.1)
where  is the porosity and D e is the effective diffusion coefficient. The parameter  c is a critical porosity threshold under which diffusion stops and α is the empirical Archie coefficient ranging from 1.3 for unconsolidated sand to 2.2 for consolidated rocks. A zero porosity threshold and an Archie coefficient of 2 were used in this study.

The modeling grids were made according to the configuration represented in Fig. 1, assuming a cylindrical 3D-geometry. Three monoliths were immersed in each leaching cell and a permanent leachant renewal at a given flow rate applied. The leaching solution is the brine detailed in Tab. 1 (e.g. pH of 7.7).

F igure 1: Schematic representation of the dynamic leaching cells (left) and the corresponding cylindrical grids used in the simulations. Cylindrical symmetry removed faster due to the higher flowrate.

Results

Figure 3 presents the mineralogical composition of the first monoliths, i.e. the closest to the injection point of the leachant. The altered zone is about 3 mm thick in both cells. As a matter of fact, a dissolution front around 3 mm has been experimentally identified for cell 2 in agreement with the modeling results. The portlandite (CH) is completely dissolved within the first two millimeters and partially within the 3 rd mm in the two cells. The decalcification of CSH is highlighted by the transformation of CSH (II) with high Ca/Si ratio to CSH (I) with lower report. In the degraded zone, hydrogarnet dissolves while calcium monosulfoaluminate are formed (data not shown). This phenomenon was experimentally observed in the case of cell 2 [START_REF] Neuville | Étude et modélisation de l'altération physico-chimique de matériaux de cimentation des puits pétroliers[END_REF].

The most important differences between the two cells are related to calcite precipitation and porosity clogging. Indeed, if a calcite layer is formed at the surface of both monoliths, the amount of calcite precipitated at the surface is 50 times greater in cell 1 than in cell 2. The brine flowrate is 6 times higher in cell 1, which means that the inflow of carbonate ion is also significantly larger. This could explain the greater amount of calcite precipitation calculated in cell 1. If the porosity increases in the degraded zone due to the portlandite dissolution in both cells, the formation of calcite on the surface decreases the porosity at the solid/liquid interface in the case of cell 1. The porosity drops from the starting value of 1 in the liquid to 0.45.

Conclusions

3D reactive transport modeling was used to compare two experimental devices using the same brine and the same cement pastes. The difference in flowrate and V l /V s ratio induces a significant difference in pH between the two cells. The higher flowrate of the cell 1 stabilizes the pH around 8 (against 11 in cell 2). This suggests that the first device (cell 1) is more deleterious. On the contrary, the inflow of carbonate ions is larger in cell 1 enhancing calcite precipitation and porosity decrease. Since the effectiveness of a calcite layer to protect a cementitious material has been demonstrated [START_REF] Bescop | Dégradation des matériaux à base de liants hydrauliques en milieu saturé -Application au comportement à long terme les colis « béton[END_REF][START_REF] Neuville | Étude et modélisation de l'altération physico-chimique de matériaux de cimentation des puits pétroliers[END_REF], the first device (cell 1) is less deleterious than cell 2 with this respect.

The interest of such an approach is either to reduce the number of experiments required to study various scenarios of degradation or to compare an experimental result to another one from the literature obtained in other experimental conditions.

Figure 2

 2 Figure2reports the pH data in the leachate over 1 month. Globally, modeling is in good agreement with experiments. The diffusion of hydroxyl ions tends to increase the pH of the brine, which is initially 7.7. The pH increases more significantly in cell 2, to reach a steady-state value of 11, than in cell 1 where the pH remain around 8. This is due to the imposed lower flowrate. In cell 2, the hydroxyl ions leached from cement pastes are evacuated more slowly, which results in increasing the solution pH. In cell 1, the leached hydroxyl ions are

2 F igure 2 :

 22 Evolution of pH in the leachate of cells 1 and 2 F igure 3: Concentration profiles (in kg/dm 3 ) and porosity after 30 days.

Table 1 :

 1 Brine and cement paste compositions

	B rine	mmolal	C ement paste	K g/dm 3
	Ca 2+	3.3	CH	0.33
	Cl -	378	CSH (II)	0.86
	HCO 3 -	6	Hydrogarnet	0.16
	K +	34	MonoSulfo	0.25
	Mg 2+	2	Calcite	0,03
	Na +	343	Porosity	0.28
	SO 4 2-

Proceedings of Conmod 2010, EPFL, Switzerland