Rational points on abelian varieties over function fields and Prym varieties

In this paper, using a generalization of the notion of Prym variety for covers of quasi-projective varieties, we prove a structure theorem for the Mordell-Weil group of the abelian varieties over function fields that are twists of Abelian varieties by Galois covers of irreducible quasi-projective varieties. In particular, the resutls we obtain contribute in the construction of Jacobians (of covers of the projective line) of high rank.


introduction
Let A be an abelian variety over a field k of characteristic ≥ 0. The Mordell-Weil theorem asserts that the set of k-rational points A(k) on A is a finitely generated abelian group. The rank of this abelian group is referred to as the rank of the abelian variety A with notation rank(A(k)). One interesting problem in arithmetic geometry is to find abelian varieties with arbitrary large ranks.
The present paper is a generalization of the works of Hazama and Salami, respectively in [4] and [8] to arbitrary Galois coverings. More precisely, let n ∈ N be a natural number such that the characteristic of the field k in the previous paragraph does not divide n. Suppose there is an embedding G ↪ Aut(A) with G a finite group with G = n. Further, let X, Y be irreducible quasi-projective varieties over k with function fields F, K respectively. We first define the notion of Prym variety associated to a G-Galois covering f ∶ X → Y . We will show that this is a natural generalization of the notion of Prym variety in the case of coverings of curves. The group G is well-known to be also the Galois group of the Galois field extension F K. By the results of [2] and [3], the twist of A by the extension F K is equivalent to a twist by the 1-cocyle a = (a g ) ∈ Z 1 (G, Aut(A)) given by a g = g, where g is viewed both as a group element and as an automorphism of A corresponding to g ∈ G (in other words we identify g with its image g ∈ G ↪ Aut(A)). We prove a theorem which gives an isomorphism for A a (K) in terms of the Prym variety introduced in section 2, see Theorem 2.4. Then we consider the n-times product When G is abelian, we prove a result which describes the Prym variety of the self product of G-covers of varieties in terms of the Prym variety of the cover. We apply this to the case of an abelian cover C → P 1 . More precisely we prove that rank(Jac(C) a (K)) can be made arbitrarily large by taking n large enough.

the Prym variety of a Galois covering
Let V be an irreducible quasi-projective variety over a field k. The Albanese variety of V is by definition the initial object for the morphisms from V to abelian varieties. In particular, if V is moreover non-singular over a field of characteristic zero (the classical case) it is the abelian variety, Now let X, Y be irreducible quasi-projective varieties with f ∶ X → Y a Galois covering with the Galois group G. Throughout this paper, we consider the situation where there is a finite group G with G = n that has a G-linearized action on X such that Y ∶= X G and f ∶ X → Y is the quotient map. The universal property of the albanese variety implies that there is an induced action of G on Alb(X). We denote by (Alb(X) G ) 0 the largest abelian subvariety of Alb(X) fixed (pointwise) under this action. Equivalently, (Alb(X) G ) 0 is the connected component containing the identity of the subvariety Alb(X) G of fixed points of Alb(X) under the action of G. In characteristic zero the induced action of G on Alb(X) can be described through its action on H 0 (X, Ω 1 X ). With respect to this action, we write Let g ∈ G be an element of the group G. We will denote also by g the automorphism of Alb(X) induced by g. Using this notation and following that of [7], Prop 3.1 for the case of curves, we write Nm G ∶ Alb(X) → Alb(X) for the norm endomorphism of Alb(X) given by Nm G ∶= ∑ g∈G g. Now we are ready to define the notion of Prym variety for the covering f .

RATIONAL POINTS ON ABELIAN VARIETIES OVER FUNCTION FIELDS AND PRYM VARIETIES
3 Definition 2.1. Let f ∶ X → Y be G-Galois covgering of irreducible quasi-projective varieties over a field k. The Prym variety P (X Y ) associated with the covering f is defined to be In particular P (X Y ) is the complementary abelian subvariety in Alb(X) of (the abelian For the classical case of Prym varieties of double coverings of curves we refer to [1]. The following result gives an equivalent description of the Prym variety and shows furthermore that in coincides with the Prym variety of covers of curves (see [1], [5], [6] and [7]) up to isogeny.
Proposition 2.2. With the above notation, P (X Y ) = Alb(X) Im Nm G . Furthermore it is isogeneous to the abelian variety (ker Nm G) 0 . In particular, if the varieties are defined over a field of charac- Proof. The induced linear action on the tangent space of Alb(X) at the origin gives that . In particular, this intersection is finite so that Alb(X) is isogenous to the product (Alb(X) G ) 0 ×(ker Nm G) 0 .
On the other hand the above isogeny decomposition for Alb(X) implies that dim Im Nm G = dim Alb(X) − dim (ker Nm G) 0 = dim (Alb(X) G ) 0 . Hence Im Nm G = (Alb(X) G ) 0 . This completes the proof of the claims.

Remark 2.3. For curves, the albanese variety coincides with the Jacobian variety. Hence if
X and Y are curves, the Prym variety P (X Y ) coincides with the Prym variety for covers of curves, see [5] and [6]. In this case there are two fundamental homomorphisms: the norm homomorphism Nm f ∶ Jac(X) → Jac(Y ) and the pull-back homomorphism f * ∶ Jac(Y ) → Jac(X) and it holds that: Nm G = f * ○ Nm f , so that P (X Y ) = (ker Nm f ) 0 = (ker Nm G) 0 , see [7], Prop 3.1.
Recall from [2] and [3] that the twist of A by the extension F K is equivalent to a twist by the 1-cocyle a = (a g ) ∈ Z 1 (G, Aut(A)) given by a g = g. We denote this twist by A a .
Theorem 2.4. Assume that there exists a k-rational point x 0 ∈ X(k). Then there is an isomorphism of abelian groups Proof. Note that the universal property of the Albanese variety together with the fact that a regular morphism of abelian varieties is obtained by a homomorphism followed by a translation, (see for example [5],Prop 1.2.1) shows that We assume that the Albanese map ι X ∶ X → Alb(X) satisfies ι X (x 0 ) = 0 so that it is defined over k. Recall that [3], Prop 1.1 shows that This implies in our particular case that for any g ∈ G, viewed as remarked earlier also as an automorphism of Alb(X) we have that (α, Q) ∈ A a (K) if and only if g(α ○ g, Q) = (α, Q) or equivalently (α ○ g, Q) = g −1 (α, Q). But this is the case if and only if α annihilates Im Nm G and Q ∈ A[n](k). Proposition 2.2 then shows that it must actually its lie in P (X Y ), so we obtain the claimed isomorphism in 2.4.1. Which implies that as Z-modules, it holds that rank(A a (K)) ≥ n⋅ rank(End k (U )) Given a G-Galois covering f ∶ X → Y , one can form the n-times self product ∏ -Galois covering. Suppose now that G is abelian. Then the diagonal embedding G ↪ ∏ i G ∶= G × ⋯ × G gives a subgroup of ∏ i G isomorphic to G. We denote this subgroup byG. This gives rise to an intermediate Galois covering Let us write X = ∏ n i=1 X and Y = (∏ i X) G . We are interested in the Prym variety P (X Y).
In fact we show, Proposition 2.5. With the above notation, there is an isogeny Proof. It suffices to treat only the case n = 2. The general case follows by an induction argument.
So suppose n = 2 and denote the Galois group of the cover X Y byG(≅ G). By Proposition 2.2 it suffices to show that there is a k-isogeny (ker NmG) In fact we show that there is an isomorphism between these abelian varieties. Notice that there is an isomorphism The isomorphism β is given as follows: Let j i ∶ X i → X , for i = 1, 2 be the natural inclusions.
Then β =j 1 +j 2 , wherej i denotes the induced homomorphism Alb(X i ) → Alb(X ). This isomorphism is compatible with the action ofG, namely, there is the following commutative diagram. From this, one deduces the isomorphism ker NmG ∼ → ker Nm G × ker Nm G which implies the desired isomorphism.
Consider an abelian cover C → P 1 with Galois group G. Consider the product C n = ∏ n i=1 C, i.e., the product of n copies of the same abelian cover in the above and letG be the image of G under the diagonal embedding G ↪ ∏ n i=1 G as above. Set D n = C n G . By Proposition 2.5, we have that
Note that the function field K(C) of C is generated over the function field K(P 1 ) = K(z) of P 1 by taking roots of (transcendental) elements of K(C), i.e., it is of the form K(z)(x Then the function field L n of C n is K(z)(x We define the 1-cocyle Z 1 (G, Aut(C)) by a g = g. Let Jac(C) a be the twist corresponding to this 1-cocyle. By applying 2.4.1 and 2.5.4, it follows that Jac(C) a (K) ≅ Hom k (P (C n D n ), Jac(C)) ⊕ Jac(C)[n](k) ≅ Hom k ((Jac(C)) n , Jac(C)) ⊕ Jac(C)[n](k) ≅ End k (Jac(C)) n ⊕ Jac(C)[n](k).