Iwasawa Main Conjecture for ordinary semistable elliptic curves over global function fields

TL;DR

This paper proves the Iwasawa Main Conjecture for ordinary semistable elliptic curves over certain global function fields, using a new $ ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright}$-formula and establishing the non-vacuous nature of a key $ ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright}$-hypothesis for a large class of curves.

Contribution

The paper introduces a $ ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright}$-formula comparing Selmer modules and $p$-adic $L$-functions, and proves the conjecture under a $ ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright}$-hypothesis that is shown to be generally valid.

Findings

The $ ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} ext{ extquoteright} extquoteright}$-hypothesis holds on a Zariski open dense locus for $p>3$.

The conjecture is proved under the specified conditions.

The approach extends Iwasawa theory to a new class of function fields.

Abstract

Let $A$ be an ordinary elliptic curve over a global function field $K$ of characteristic $p$, assumed semistable at every place, and let $L/K$ be a $\mathbb{Z}_p^d$-extension ramified only at finitely many places where $A$ has ordinary reduction. Building on the framework of [Tan26] (arXiv:2603.10576), we prove the Iwasawa Main Conjecture for $A$ over $L$, subject to a technical $\mu$-invariant hypothesis that is already detected after specialization to the unramified $\mathbb{Z}_p$-extension. The principal new input is a `$\chi$-formula' that compares appropriate $\chi$-isotypic characteristic ideals of Selmer modules with the corresponding specializations of the $p$-adic $L$-function. Finally, to show that our $\mu$-hypothesis is non-vacuous, we prove, for $p>3$, that the hypothesis holds on a Zariski open dense locus in the moduli of semistable elliptic curves.