Arithmetic of abelian varieties in Artin-Schreier extensions

TL;DR

This paper investigates the arithmetic properties of abelian varieties over function fields in characteristic p within Artin-Schreier extensions, proving high order vanishing of L-functions, constructing Jacobians satisfying BSD, and analyzing Mordell-Weil groups.

Contribution

It introduces a new Artin-Schreier construction of Jacobians where BSD holds and provides explicit formulas for Mordell-Weil ranks in this setting.

Findings

High order vanishing of L-functions at the center point.

Construction of Jacobians satisfying BSD in Artin-Schreier extensions.

Explicit computation of Mordell-Weil ranks and lattices.

Abstract

We study abelian varieties defined over function fields of curves in positive characteristic $p$, focusing on their arithmetic within the system of Artin-Schreier extensions. First, we prove that the $L$-function of such an abelian variety vanishes to high order at the center point of its functional equation under a parity condition on the conductor. Second, we develop an Artin-Schreier variant of a construction of Berger. This yields a new class of Jacobians over function fields for which the Birch and Swinnerton-Dyer conjecture holds. Third, we give a formula for the rank of the Mordell-Weil groups of these Jacobians in terms of the geometry of their fibers of bad reduction and homomorphisms between Jacobians of auxiliary Artin-Schreier curves. We illustrate these theorems by computing the rank for explicit examples of Jacobians of arbitrary dimension $g$, exhibiting Jacobians with bounded rank and others with unbounded rank in the tower of Artin-Schreier extensions. Finally, we compute the Mordell-Weil lattices of an isotrivial elliptic curve and a family of non-isotrivial elliptic curves. The latter exhibits an exotic phenomenon whereby the angles between lattice vectors are related to point counts on elliptic curves over finite fields. Our methods also yield new results about supersingular factors of Jacobians of Artin-Schreier curves.