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

Abstract

Abstract. 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 noncommutative 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 (Conjecture 1.6) which might be considered as a counterpart of the principal conjecture in the case over a number field. 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.

Characteristic ideals and Selmer groups

“…Note that Assumption 2.2.2 is verified in this case with e = 1, thanks to [12,Theorem 1.7], hence our next results hold for all these extensions as well.…”

“…Let F n denote the layers of the Z d p -extensions F d : note that Gal(F n /F ) (Z/p n ) d and F and F (p) are disjoint. By [12,Theorem 1.7], Sel(F n F (p) ) is a finitely generated torsion Λ(F (p) )-module and this implies that the Z p -coranks of Sel(F n F (p) t ) are bounded (see, e.g., the proof of [3, Corollary 4.14]). Moreover for any s t, the restriction maps…”

On a noncommutative Iwasawa main conjecture for varieties over finite fields

“…The results of this article together with the descent formalism developed by Venjakob and Burns in [BV11] can also be used to generalise the proof of an analogue of the equivariant Tamagawa number conjecture given in [Bur04]. An analogue of the noncommutative Iwasawa main conjecture for elliptic curves over function fields in the case that ℓ is equal to the characteristic p of the field in question has been considered in [OT09]. F. Trihan and D. Vauclair have announced proofs of more general main conjectures in this case.…”

A generalized Mazur’s theorem and its applications

“…However, lacking our local control theorem, to make sure the above-mentioned assertion holds (over function fields) they need to depend on additional assumptions. For example, in Ochiai and Trihan ( [OTr06,OTr08]), they assume that L/K is the constant Z p -extension unramified at every place of K, while Bandini and Longhi ( [BL06]) treat the case of an elliptic curve with split multiplicative reduction at every place of S.…”