On the Selmer groups of abelian varieties over function fields of characteristic p>0

In this paper, we study a (p-adic) geometric analogue for abelian varieties over a function field of characteristic p of the cyclotomic Iwasawa theory and the non-commutative Iwasawa theory for abelian varieties over a number field initiated by Mazur and Coates respectively. We will prove some analogue of the principal results obtained in the case over a number field and we study new phenomena which did not happen in the case of number field case. We propose also a conjecture which might be considered as a counterpart of the principal conjecture in the case over a number field. \par This is a preprint which is distributed since 2005 which is still in the process of submision. Following a recent modification of some technical mistakes in the previous version of the paper as well as an amelioration of the presentation of the paper, we decide wider distribution via the archive.Comment: 21 page

The Iwasawa main conjecture for semistable abelian varieties over function fields

We prove the Iwasawa main conjecture over the arithmetic $\mathbb{Z}_p$-extension for semistable abelian varieties over function fields of characteristic $p>0$.Comment: arXiv admin note: substantial text overlap with arXiv:1205.594

On the Iwasawa Main conjecture of abelian varieties over function fields

We study a geometric analogue of the Iwasawa Main Conjecture for abelian varieties in the two following cases: constant ordinary abelian varieties over $Z_p^d$-extensions of function fields ($d\geq 1$) ramified at a finite set of places, and semistable abelian varieties over the arithmetic $Z_p$-extension of a function field. One of the tools we use in our proof is a pseudo-isomorphism relating the duals of the Selmer groups of $A$ and its dual abelian variety $A^t$. This holds as well over number fields and is a consequence of a quite general algebraic functional equation.Comment: 80 pages; many relevant changes all over the paper from v1. Among the most significant ones: new introduction; proof of the functional equation for Gamma systems in more cases and some applications to CM abelian varietie

Pontryagin duality for Iwasawa modules and abelian varieties

We prove a functional equation for two projective systems of finite abelian $p$-groups, \{\fa_n\} and \{\fb_n\}, endowed with an action of \ZZ_p^d such that \fa_n can be identified with the Pontryagin dual of \fb_n for all $n$. Let $K$ be a global field. Let $L$ be a \ZZ_p^d-extension of $K$ ($d\geq 1$), unramified outside a finite set of places. Let $A$ be an abelian variety over $K$. We prove an algebraic functional equation for the Pontryagin dual of the Selmer group of $A$.Comment: 30 pages. arXiv admin note: substantial text overlap with arXiv:1205.594

Parity conjectures for elliptic curves over global fields of positive characteristic

We prove the $p$-parity conjecture for elliptic curves over global fields of characteristic $p > 3$. We also present partial results on the $\ell$-parity conjecture for primes $\ell \neq p$.Comment: to be published in Compositio Mathematica. This version differs slightly from the one to be publishe

The μ\mu-invariant change for abelian varieties over finite pp-extensions of global fields

We extend the work of Lai, Longhi, Suzuki, the first two authors and study the change of $\mu$-invariants, with respect to a finite Galois p-extension $K'/K$, of an ordinary abelian variety $A$ over a $\mathbb{Z}_p^d$-extension of global fields $L/K$ (whose characteristic is not necessarily positive) that ramifies at a finite number of places at which $A$ has ordinary reductions. We obtain a lower bound for the $\mu$-invariant of $A$ along $LK'/K'$ and deduce that the $\mu$-invariant of an abelian variety over a global field can be chosen as big as needed. Finally, in the case of elliptic curve over a global function field that has semi-stable reduction everywhere we are able to improve the lower bound in terms of invariants that arise from the supersingular places of $A$ and certain places that split completely over $L/K$.Comment: 31 page