Trihan, Fabien; Vauclair, David

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

Doc. Math. 24, 473-522 (2019)
DOI: 10.25537/dm.2019v24.473-522

Summary

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 \textit{J.-Y. Etesse} and \textit{B. Le Stum} [Math. Ann. 296, No. 3, 557--576 (1993; Zbl 0789.14015)].

Mathematics Subject Classification

11S40, 11R23, 11R34, 11R42, 11R58, 11G05, 11G10

Keywords/Phrases

abelian variety, Iwasawa theory, \(p\)-adic cohomology, syntomic, non-commutative

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Affiliation

Trihan, Fabien
Department of Information and Communication Sciences, Sophia University, Chiyoda-ku, Tokyo, 102-0081, Japan
Vauclair, David
Laboratoire de Mathématiques Nicolas Oresme, Université de Caen Normandie, BP 5186, 14032 Caen Cedex, France

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