The -parity conjecture for abelian varieties over function fields of characteristic

Abstract

Let A/K be an abelian variety over a function field of characteristic p > 0 and let be a prime number ( = p allowed). We prove the following: the parity of the corank r of the -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 -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.

Positivity of Hodge bundles of abelian varieties over some function fields

“…With a little extra work (cf. [TY14, Theorem 5.5]), the above results imply that for all projective and smooth surfaces over finite fields is equivalent to for all abelian varieties over global function fields.…”

Noncommutative Main Conjectures of Geometric Iwasawa Theory

“…Completing the work of Ochiai and Trihan [OT09], a noncommutative main conjecture for abelian varieties over F in the case ℓ = p is formulated in [TV11] and there is progress towards a proof, at least under certain hypotheses. The conjectured interpolation property is only for the leading terms of the L-functions of Artin twists of the abelian variety in s = 1.…”