Rational points on abelian varieties over function fields and Prym varieties

Abstract

In this paper, using a generalization of the notion of Prym variety for covers of projective varieties, we prove a structure theorem for the Mordell–Weil group of abelian varieties over function fields that are twists of abelian varieties by Galois covers of smooth projective varieties. In particular, the results we obtain contribute to the construction of Jacobians of high rank.

Twists of Albanese varieties over function fields with large ranks

“…Furthermore, [8] gives a description of the Prym variety of the m-times self product ∏ i f of the G-cover f ∶ X → Y with itself, which generalizes the results of [10] and [4] for cyclic and double covers. Such a product yields a G-Galois field extension (by considering the function fields of ∏ i X and ∏ i Y ) and hence a twist of the abelian varieties defined over the function field of these varieties.…”

“…For the classical case of Prym varieties of double coverings of curves we refer to [1]. The following result has been proven in [8], Prop 2.2 and gives an equivalent description of the Prym variety and shows furthermore that it coincides with the Prym variety of covers of curves (see [1], [6] and [9]) up to isogeny.…”

“…Let X and X be the twists of X and X by the extensions L K and L K, respectively. Then, as in Example 2.6 of [8], one can check that the twist X is given by the following set of affine equations (3.2.2)…”

“…The group G is also the Galois group of the Galois field extension L K. By the results of [2,4], the twist of A by the extension L K is equivalent to the twist by the 1-cocyle a = (a g ) ∈ Z 1 (G, Aut(A)) given by a g = g, where in the notation a g , g is viewed as a group element and on the right side as an automorphism of A corresponding to g ∈ G (i.e., we identify g with its image g ∈ G ↪ Aut(A)). By the results of [8] there is an isomorphism for A a (K) in terms of the Prym variety.…”

“…Let f ∶ X → Y be a G-Galois covering. To such a covering we have associated in [8] an abelian varietiy P (X Y ), the so-called Prym variety, that generalizes the notion of Prym variety in the case of coverings of curves. The group G is also the Galois group of the Galois field extension L K. By the results of [2,4], the twist of A by the extension L K is equivalent to the twist by the 1-cocyle a = (a g ) ∈ Z 1 (G, Aut(A)) given by a g = g, where in the notation a g , g is viewed as a group element and on the right side as an automorphism of A corresponding to g ∈ G (i.e., we identify g with its image g ∈ G ↪ Aut(A)).…”