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The $l$-parity conjecture for abelian varieties over function fields of characteristic $p>0$

Part of: Arithmetic algebraic geometry

Published online by Cambridge University Press:  10 March 2014

Fabien Trihan  and
Seidai Yasuda
Fabien Trihan
Affiliation:
School of Engineering, Computing and Mathematics, University of Exeter, EX4 4QF, UK email f.trihan@exeter.ac.uk
Seidai Yasuda
Affiliation:
Department of Mathematics, Graduate School of Science, Osaka University, Toyonaka 560-0043, Japan email s-yasuda@math.sci.osaka-u.ac.jp
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Abstract

Let $A/K$ be an abelian variety over a function field of characteristic $p>0$ and let $\ell $ be a prime number ($\ell =p$ allowed). We prove the following: the parity of the corank $r_\ell $ of the $\ell $-discrete Selmer group of $A/K$ coincides with the parity of the order at $s=1$ of the Hasse–Weil $L$-function of $A/K$. We also prove the analogous parity result for pure $\ell $-adic sheaves endowed with a nice pairing and in particular for the congruence Zeta function of a projective smooth variety over a finite field. Finally, we prove that the full Birch and Swinnerton-Dyer conjecture is equivalent to the Artin–Tate conjecture.

Keywords

elliptic curvesSelmer groupsparity conjecture

MSC classification

Secondary: 11G20: Curves over finite and local fields 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Compositio Mathematica , Volume 150 , Issue 4 , April 2014 , pp. 507 - 522
Copyright
© The Author(s) 2014 

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