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On the Iwasawa invariants of Kato's zeta elements for modular forms

Chan-Ho Kim, Jaehoon Lee, Gautier Ponsinet

Abstract

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without $p$-adic $L$-functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim, Greenberg-Iovita-Pollack, and one of us simultaneously. As a consequence, we establish the propagation of Kato's main conjecture for modular forms of higher weight at arbitrary good prime under the assumption on the mod $p$ non-vanishing of Kato's zeta elements. The application to the $\pm$ and $\sharp/\flat$-Iwasawa theory for modular forms is also discussed.

On the Iwasawa invariants of Kato's zeta elements for modular forms

Abstract

We study the behavior of the Iwasawa invariants of the Iwasawa modules which appear in Kato's main conjecture without -adic -functions under congruences. It generalizes the work of Greenberg-Vatsal, Emerton-Pollack-Weston, B.D. Kim, Greenberg-Iovita-Pollack, and one of us simultaneously. As a consequence, we establish the propagation of Kato's main conjecture for modular forms of higher weight at arbitrary good prime under the assumption on the mod non-vanishing of Kato's zeta elements. The application to the and -Iwasawa theory for modular forms is also discussed.

Paper Structure

This paper contains 21 sections, 24 theorems, 65 equations.

Table of Contents

  1. Introduction
  2. Overview
  3. Galois representations
  4. Construction
  5. Properties
  6. Iwasawa cohomologies
  7. Kato's zeta elements and the main conjecture
  8. Main results and applications
  9. The statement of main theorem
  10. Applications to main conjectures with $L$-functions
  11. "Prime-to-$p$ local" Iwasawa theory
  12. Euler factors
  13. Local cohomologies
  14. The zeta element side
  15. $\Sigma_0$-imprimitive zeta elements
  16. ...and 6 more sections

Key Result

Theorem 1.5

For all $i = 0, \cdots , p-2$,

Theorems & Definitions (61)

  • Definition 1.3
  • Remark 1.4
  • Theorem 1.5: Kato
  • proof
  • Proposition 1.6
  • proof
  • Definition 1.7
  • Remark 1.8
  • Conjecture 1.9: Kato's main conjecture
  • Remark 1.10
  • ...and 51 more