  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.
