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Arithmetic of abelian varieties in Artin-Schreier extensions


Authors: Rachel Pries and Douglas Ulmer
Journal: Trans. Amer. Math. Soc. 368 (2016), 8553-8595
MSC (2010): Primary 11G10, 11G40, 14G05; Secondary 11G05, 11G30, 14H25, 14J20, 14K15
Published electronically: January 27, 2016
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Abstract: We study abelian varieties defined over function fields of curves in positive characteristic $ p$, focusing on their arithmetic in 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.


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Additional Information

Rachel Pries
Affiliation: Department of Mathematics, Colorado State University, Fort Collins, Colorado 80523
Email: pries@math.colostate.edu

Douglas Ulmer
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
Email: ulmer@math.gatech.edu

DOI: https://doi.org/10.1090/tran6641
Received by editor(s): June 10, 2013
Received by editor(s) in revised form: October 15, 2014
Published electronically: January 27, 2016
Article copyright: © Copyright 2016 American Mathematical Society