On the Non Commutative Iwasawa Main Conjecture for Abelian Varieties over Function Fields

Abstract

We establish the Iwasawa main conjecture for semistable 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 usual $\mu =0$ hypothesis, we give a proof which mainly relies on the interpretation of the Selmer complex in terms of $p$-adic cohomology [F. Trihan, D. Vauclair, A comparison theorem for semi abelian schemes over a smooth curve, preprint arXiv:1505.02942, 2015] together with the trace formulas of J.-Y. Etesse and B. Le Stum [Math. Ann. 296, No. 3, 557–576 (1993; Zbl 0789.14015)].