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Transactions of the American Mathematical Society

Published by the American Mathematical Society, the Transactions of the American Mathematical Society (TRAN) is devoted to research articles of the highest quality in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.43.

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Arithmetic of abelian varieties in Artin-Schreier extensions
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by Rachel Pries and Douglas Ulmer PDF
Trans. Amer. Math. Soc. 368 (2016), 8553-8595 Request permission

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
  • MR Author ID: 665775
  • Email: pries@math.colostate.edu
  • Douglas Ulmer
  • Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332
  • MR Author ID: 175900
  • ORCID: 0000-0003-1529-4390
  • Email: ulmer@math.gatech.edu
  • Received by editor(s): June 10, 2013
  • Received by editor(s) in revised form: October 15, 2014
  • Published electronically: January 27, 2016
  • © Copyright 2016 American Mathematical Society
  • Journal: Trans. Amer. Math. Soc. 368 (2016), 8553-8595
  • MSC (2010): Primary 11G10, 11G40, 14G05; Secondary 11G05, 11G30, 14H25, 14J20, 14K15
  • DOI: https://doi.org/10.1090/tran6641
  • MathSciNet review: 3551581