On the Prym variety of genus 3 covers of genus 1 curves

Given a generic degree-2 cover of a genus 1 curve D by a non hyperelliptic genus 3 curve C over a field k of characteristic different from 2, we produce an explicit genus 2 curve X such that Jac(C) is isogenous to the product of Jac(D) and Jac(X). This construction can be seen as a degenerate case of a result by Nils Bruin.


Introduction
Let C be a (smooth, projective, absolutely irreducible) curve of genus g C 2 over a field k. Let us assume that C admits a degree-n map π to a curve D/k of genus g D with 0 < g D < g C . We then know that Jac(C) is isogenous to Jac(D) × A and one would like to get as much information as possible on A from π. For instance, if C is given by explicit equations, is it possible to find a curve X/k given by explicit equations such that A is isogenous over k to Jac(X)?
There is an extensive literature dedicated to the case g C = 2 for which D and X are genus 1 curves, both for its applications to cryptography (see [14], [10] and [4]) and for its interests in pure mathematics [15]. The general theory is well understood (see for instance [20]) and explicit formulas have been worked out when n = 2 (going back to the work of Jacobi on abelian integrals, see the references in [2, p.395] or [19]), n = 3 (see [18], [22] or the appendix of [7]), n = 4 (see [5] and [9]), n = 5 (see [25]) and more generally when n 11 (see [23]).
From now on, let n = 2. When C is given by a hyperelliptic equation, we can describe the map π as the quotient by the involution (x, y) → (−x, y). Then C : y 2 = f (x 2 ) where f is a polynomial of degree g C + 1 and hence D : y 2 = f (x) is a curve of genus g D = g C +1 2 − 1. One can moreover take advantage of the existence of the hyperelliptic involution to construct the curve X : y 2 = xf (x) which is the quotient of C by (x, y) → (−x, −y). It is easy to see by the action on the differentials that Jac(C) ∼ Jac(D) × Jac(X).
Apart from the hyperelliptic cases described above, for a generic curve C, a curve X has been made explicit only in few cases. One is when D is hyperelliptic and C is a degree-2 cover either étale or ramified at two points, see respectively [26, p.346] and [24]. In [8], one can find the only case where D is neither elliptic nor hyperelliptic, namely g C = 5 and g D = 3. This work relies on beautiful geometric constructions sketched in [1, VI exercise F]. We will use this construction to get our result.
The present article deals with the case where C is a non-hyperelliptic curve of genus 3 with a degree-2 map to a genus 1 curve D over a field k of characteristic different from 2. It finds its motivation in the article [28] where cohomological techniques are derived to compute the Weil polynomial of C. Although the techniques developed there remain interesting on their own, one byproduct of the present article is that the computation of the Weil polynomial of C can be reduced to computations in genus 1 and 2.
Note that any degree-2 cover of a genus 1 curve by a non-hyperelliptic genus 3 curve C can be written in the form y 4 −h(x, z)y 2 +r(x, z) = 0 with r a degree 4 polynomial, since we can assume that the involution of C which defines the cover is given by (x : y : z) → (x : −y : z). Our main result gives an equation of X when r splits as a product of degree 2 factors. Theorem 1.1. Let C be a smooth, non-hyperelliptic genus 3 curve defined by are homogeneous degree-2 polynomials over a field k of characteristic different from 2. The involution (x : y : z) → (x : −y : z) induces a degree-2 cover π to the genus 1 curve Then Jac(C) ∼ Jac(D) × Jac(X) with X y 2 Remark 1.2. The geometric hypothesis (i.e. which is still true overk) that det(A) 0 is satisfied if C has automorphism group isomorphic to Z/2Z. Indeed since C is non-singular overk, we can assume that f = xz, g = x 2 + g 1 xz + z 2 (f g has no double root). If A is non-invertible, we have for instance that h = λf + µg but then the morphism (x : y : z) → (z : y : x) is also an involution. In that case, the Jacobian is going to decompose further in a product of three genus 1 curves which are easily determined. Remark 1.3. The arithmetic hypothesis that r = f g over k is however necessary in our proof and we do not know how to remove it. Note that it can however always be obtained over a finite extension of k.
After we wrote this article, it was brought to our attention by Victor Enolski and Yuri Fedorov that such a case was already handled by Kötter in 1892 (see [21] and also [13,]) in an analytic setting. The proof we propose here is completely different as it is algebraic and works over any field (of characteristic not 2). The shape of the equation we get for X is also much simpler and in particular does not involve any field extension of k.
The general strategy is somehow new in this context although it will probably appear very classical to algebraic geometers. The idea is to move from the smooth cover π : C → D to a singular oneπ :C →D of arithmetic genus 5 and 3, withC birational to C. The reason to do so is the following. As n = 2, one can look at A as the Prym variety P (C/D), i.e. the connected component of the identity of ker(π * : Jac(C) → Jac(D)). Now, the Prym variety of a degree-2 cover π naturally inherits a principal polarization from Jac(C) when π is étale (see for instance [26,Cor.2]). Our original smooth cover π is not étale but the coverπ will only be ramified over the singular points ofD. In this case (more precisely in the case of allowable covers, see below), Beauville showed that the generalized Prym variety ofπ is an abelian surface P (C/D), isogenous to P (C/D), and naturally principally polarized. The abelian variety P (C/D) is therefore the Jacobian of a (possibly reducible) genus 2 curve X. It then remains to find X explicitly. In order to do so, we look at the coverC →D as the degeneration of a family of unramified degree-2 covers C → D of smooth genus 3 curves by smooth degree 5 curves. As we said, for the smooth generic fiber, Bruin [8] gives explicitly the hyperelliptic curve X whose Jacobian is the Prym of the cover. Using a particular deformation, we show that we can specialize the equation of X to find X. 2. From a singular cover to the original cover

From a singular cover to the original cover
Let the notation and hypotheses be as in Theorem 1.1 and let us consider the curve in P 4 defined bỹ C : Multiplying the second equation by x 2 2 and substituting the first and last equation we get our initial curve Hence we get an isomorphism between C andC on the locus x 2 0. Now let us rewriteC using the matrix A. We havẽ As A is invertible, we get:C ) be the left-hand side vector, we get that C : is a degree 2 coverπ of the curveD : q 2 2 = q 1 q 3 . In order to see that the covers π andπ are birationally equivalent, let us consider the automorphism (x 2 : u : v) → (−x 2 : u : v) of C which induces the cover π : C → D. OnC, this automorphism induces the automorphism (x 1 : and hence the automorphism isι which induces the coverπ :C →D. In particular D is birationally equivalent toD over k. To conclude, let us study the singular points ofC. Since C is non-singular, they can appear only when x 2 = 0. But from the first form of the equations ofC, we see that f (u, v) = g(u, v) = 0. Since C is non-singular, f and g have no common root, hence u = v = 0. The possible singular points are then only the points p 0 = (0 : 0 : 1 : 0 : 0) and p ∞ = (1 : 0 : 0 : 0 : 0). The tangent cone at the point p 0 has equations x 1 = x 2 = f (u, v) = 0 while the tangent cone at the point p ∞ has equations x 2 = x 3 = g(u, v) = 0 (see for instance [27,III,§ 3] or [11, p.485]). Since f and g have no double root, it follows that these points are nodal singularities. Therefore we see that : ( * ) The fixed points of the involutionι are exactly the singular points, which are nodal, andι preserves the branches at these points.
This condition is phrased in Beauville In Donagi and Livné [12], covers satisfying conditions ( * ) and ( ** ) are called allowable (also admissible by other authors) and we keep this terminology. Finally we sum up the preceding discussion.

Proposition 2.1. The coverπ :C →D is birationally equivalent to the cover π : C → D and is allowable.
Let us associate to the coverC →D the connected component containing 0 of the kernel of the norm mapπ * : Jac(C) → Jac(D) between generalized Jacobians. We denote it P (C/D) and call it the generalized Prym variety. Since the coverC →D is allowable, the results [3, Prop.3.5] and [12,Lem.1] show that P (C/D) is an abelian variety which is isogenous to the classical Prym variety P (C/D) of the ramified cover C → D. Moreover the kernel of the isogeny is contained in the group generated by degree 0 divisors supported on the singular points ofC and is therefore defined over k. We hence get: The Jacobian of C is isogenous over k to D × P (C/D).
As mentioned in the introduction, what we have gained by moving from P (C/D) to P (C/D) is that the latter is principally polarized [3,Th.3.7]. From Weil (see the version in [17,Thm. 3.1]), one knows that every principally polarized abelian surface is either the Jacobian of a curve, the product of two elliptic curves (with the product polarization) or the restriction of scalars of an elliptic curve over a quadratic extension of k. Under our hypotheses, we are now going to recover P (C/D) as the Jacobian of a genus 2 curve X.

The singular cover as a special fiber
In order to find explicitly the curve X, we will 'embed' our singular coverC →D into a flat family whose generic fibre is non-singular and use the beautiful result of [8] which we recall here. 1, 2, 3) are quadratic forms, and Y :

Proposition 3.1. Let Y → Z be an unramified double cover of a genus 5 curve over a non-hyperelliptic curve of genus 3 over a field k of characteristic different from 2. Then such a cover can be written
Moreover the Prym variety Prym(Y /Z) (as a principally polarized abelian surface over k) is the Jacobian of the hyperelliptic curve H which is the projective closure of the curve with equation y 2 = − det(Q 1 + 2xQ 2 + x 2 Q 3 ), where we identify Q i with the symmetric 3 × 3 matrices ((∂ 2 Q i /∂x j ∂x k )) 1 j,k 3 .

Remark 3.2.
Bruin's result is stated in characteristic 0 only. But all his arguments work as well in positive characteristic different from 2.
The discriminant Disc(F) of a degree 4 homogeneous polynomial F(x 1 , x 2 , x 3 ) over a field K of characteristic different from 2 is defined as the multivariate resultant of the partial derivatives of F (see [16, p.426 If we restrict the discriminant to quartics F of the form are degree 2 homogeneous polynomials over K, we get a homogeneous polynomial H of degree 2 · 27 = 54 in the coefficients of the Q i . Note that H is nonzero since for instance the quartic has a discriminant equal to −2 40 . The equation H defines a hypersurface in The point xD representing the singular quarticD = q 2 2 − q 1 q 3 belongs to H and the pencil (Q 1 (λ), Q 2 (λ), Q 3 (λ)) = (q 1 + λq 1 , q 2 + λq 2 , q 3 + λq 3 ), λ ∈ A 1 such that Now, let S be the spectrum of the discrete valuation ring R = k[[ ]] with special fiber k and generic fiber K. Over S we can define a cover π def : C → D by C : Clearly π def is a deformation ofπ :C →D and by construction its generic fiber π K : C K → D K is smooth over K. The Prym varieties of the fibres of π def fit in a flat family which we call the Prym scheme P = P (C → D). The details of its construction are to be found in [3], Section 6, especially (6.2); we recall them briefly. The morphism π def * : Pic 0 (C) → Pic 0 (D) is surjective with smooth target hence it is flat, and its fibres are smooth. Therefore the group scheme ker(π def * ) → S is smooth and we let P → S denote the connected component of the unit section. This is the Prym scheme; it has proper fibres hence it is an abelian scheme. Using Proposition 3.1 as well as the fact that the functor Pic 0 commutes with base change, we get the following lemma. Lemma 3.3. The special fiber of P is P (C/D) and the generic fiber is P (C K /D K ) = Jac(X K ) where X K : y 2 = − det(Q 1 ( ) + 2xQ 2 ( ) + x 2 Q 3 ( )).
We are going to prove the following lemma.

Lemma 3.4.
The scheme X is smooth over S with geometrically connected fibres.
Proof. As we are considering our schemes in the weighted projective space P(1, 3, 1), the point at infinity is smooth and we only have to consider the affine part in what follows. Since the generic fiber X K is smooth, we need to consider the singular points of the special fiber X. They correspond to points where F(x) = det(q 1 + 2xq 2 + x 2 q 3 ) has a (projective) root with multiplicity greater than 1. With the notation of . We are actually going to prove that F has no multiple root. As this is a geometric problem, we can assume that we work over an algebraically closed field. In that case, since C is assumed to be non-singular, we can take f (x, z) = xz and g(x, z) = g 2 x 2 + g 1 xz + z 2 . A computation with a computer algebra system such as Magma shows that Disc(F) = 4 · g 2 · (g 2 − g 2 1 /4) 2 · Disc(h 2 − 4f g) det(A) 18 .
Since the curve D is assumed to be non-singular, we see that 4f g − h 2 has no multiple root, hence Disc(h 2 − 4f g) 0. The curve C being non-singular, one has that g 2 0 (otherwise (1 : 0 : 0) would be a singular point) and one finds that g does not have a double root, hence g 2 − g 2 1 /4 0. We hence get Disc(F) 0.
Finally, since X → S is smooth and proper, the number of connected components of the fibres is constant on S, hence equal to 1.
With this result on X , we have the right hypotheses to apply the following proposition (see for instance [6, Th. 9.5/1]). Proposition 3.5. Pic 0 (X ) is a Néron model of Jac(X K ).
We can now conclude. By Lemma 3.3, the Néron model of the Jacobian of X K is the Néron model of the generic fiber P (C K /D K ) of P . Since the special fiber of P is P (C/D) which is an abelian variety (the coverπ being allowable), we see that the Néron model of P (C K /D K ) is P . Hence the Néron model of Jac(X K ) is P and by Proposition 3.5 we have P = Pic 0 (X ). Taking the special fiber gives: P ⊗ k = P (C/D) = Pic 0 (X ) ⊗ k = Jac(X ⊗ k) = Jac(X).
We have therefore obtained a description of P (C/D) which is isogenous to P (C/D) and this concludes the proof of Theorem 1.1.