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

Ki-Seng Tan, Fabien Trihan, Kwok-Wing Tsoi

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 $μ$-invariant hypothesis that is already detected after specialization to the unramified $\mathbb{Z}_p$-extension. The principal new input is a `$χ$-formula' that compares appropriate $χ$-isotypic characteristic ideals of Selmer modules with the corresponding specializations of the $p$-adic $L$-function. Finally, to show that our $μ$-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.

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

Abstract

Let be an ordinary elliptic curve over a global function field of characteristic , assumed semistable at every place, and let be a -extension ramified only at finitely many places where has ordinary reduction. Building on the framework of [Tan26] (arXiv:2603.10576), we prove the Iwasawa Main Conjecture for over , subject to a technical -invariant hypothesis that is already detected after specialization to the unramified -extension. The principal new input is a `-formula' that compares appropriate -isotypic characteristic ideals of Selmer modules with the corresponding specializations of the -adic -function. Finally, to show that our -hypothesis is non-vacuous, we prove, for , that the hypothesis holds on a Zariski open dense locus in the moduli of semistable elliptic curves.
Paper Structure (33 sections, 30 theorems, 191 equations, 1 figure)

This paper contains 33 sections, 30 theorems, 191 equations, 1 figure.

Table of Contents

  1. Introduction
  2. The main conjecture
  3. Properties (I): Functional equations
  4. Properties (II): Specialization formulae
  5. Properties (III): Restriction formulae
  6. The $\chi$-formula
  7. The technical hypothesis on $\mu$-invariants
  8. Notations
  9. The global factor $\varrho_{L/L'}$
  10. Local factors $\vartheta_{L/L',v}$
  11. The element $\dag_{A/L}$
  12. The fudge factor $\star_\omega$
  13. The $\chi$-formula
  14. The proof of Theorem \ref{['t:chi']}
  15. The characteristic ideal of ${\mathbb Q}_\chi X_L/(\psi_0-\chi(\psi_0))$
  16. ...and 18 more sections

Key Result

Theorem 1

Let the notations be as above. In ${\mathbb Q}_\chi\Lambda_{\Gamma_0}$, the ideal such that for all $\omega\in\hat{\Gamma}_0$,

Figures (1)

  • Figure 1: Field extension diagram

Theorems & Definitions (54)

  • Theorem 1
  • Remark 1.5.1
  • Theorem 2
  • Theorem 3
  • Remark 1.7.1
  • Lemma 2.1.1
  • proof
  • Lemma 2.1.2
  • proof
  • Lemma 2.1.3
  • ...and 44 more