On the non commutative Iwasawa main conjecture for abelian varieties over function fields

Abstract

We establish the Iwasawa main conjecture for semi-stable abelian varieties over a function field of characteristic $p$ under certain restrictive assumptions. Namely we consider $p$-torsion free $p$-adic Lie extensions of the base field which contain the constant ${\mathbb{Z}}_p$-extension and are everywhere unramified. Under the classical $\mu =0$ hypothesis we give a proof which mainly relies on the interpretation of the Selmer complex in terms of $p$-adic cohomology [TV] together with the trace formulas of [EL1].

Revised version, 44 pages. One of the two possible constructions of the normic section functor has disappeared