Aspects of Iwasawa theory over function fields

TL;DR

This paper explores Iwasawa theory over function fields, formulating main conjectures for Selmer groups and class groups, and relating cyclotomic units to Bernoulli-Carlitz numbers in the context of $Z_p^{N}$-extensions.

Contribution

It develops Iwasawa main conjectures for function fields, connecting Selmer groups, class groups, and $p$-adic $L$-functions, extending number field results to the function field setting.

Findings

Formulation of Iwasawa main conjecture for Selmer groups of abelian varieties.

Relation between cyclotomic units and Bernoulli-Carlitz numbers.

Results on characteristic ideals and Stickelberger elements for class groups.

Abstract

We consider $\mathbb{Z}_p^{\mathbb{N}}$-extensions $\mathcal{F}$ of a global function field $F$ and study various aspects of Iwasawa theory with emphasis on the two main themes already (and still) developed in the number fields case as well. When dealing with the Selmer group of an abelian variety $A$ defined over $F$, we provide all the ingredients to formulate an Iwasawa Main Conjecture relating the Fitting ideal and the $p$-adic $L$-function associated to $A$ and $\mathcal{F}$. We do the same, with characteristic ideals and $p$-adic $L$-functions, in the case of class groups (using known results on characteristic ideals and Stickelberger elements for $\mathbb{Z}_p^d$-extensions). The final section provides more details for the cyclotomic $\mathbb{Z}_p^{\mathbb{N}}$-extension arising from the torsion of the Carlitz module: in particular, we relate cyclotomic units with Bernoulli-Carlitz numbers by a Coates-Wiles homomorphism.