Arithmetic of abelian varieties in artin-schreier extensions

Rachel Pries, Douglas Ulmer

Research output: Contribution to journalArticle

1 Citation (Scopus)

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.

Original languageEnglish (US)
Pages (from-to)8553-8595
Number of pages43
JournalTransactions of the American Mathematical Society
Volume368
Issue number12
DOIs
StatePublished - Jan 1 2016
Externally publishedYes

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Abelian Variety
Elliptic Curves
Function Fields
Towers
Curve
Vector Lattice
Geometry
Fibers
Positive Characteristic
L-function
Conductor
Homomorphisms
Parity
Functional equation
Galois field
Vanish
Count
Fiber
Higher Order
Angle

ASJC Scopus subject areas

  • Mathematics(all)
  • Applied Mathematics

Cite this

Arithmetic of abelian varieties in artin-schreier extensions. / Pries, Rachel; Ulmer, Douglas.

In: Transactions of the American Mathematical Society, Vol. 368, No. 12, 01.01.2016, p. 8553-8595.

Research output: Contribution to journalArticle

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