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The Equivariant Tamagawa Number Conjectures for modular motives with coefficients in Hecke algebra

Olivier Fouquet

TL;DR

The paper establishes an equivariant Tamagawa number conjecture framework for modular motives with coefficients in Hecke algebras and proves that, under broad ramification hypotheses, the conjectures interpolate coherently in $p$-adic families. It combines zeta-morphism formalism, Euler systems, and Taylor–Wiles–Kisin patching to extend Iwasawa main conjectures from single forms to universal deformation settings, ensuring compatibility at motivic points. The results imply that the equivariant Tamagawa framework governs special values across congruent modular forms and yields universal Iwasawa theory statements with concrete consequences for classical and non-classical points. Numerical examples at $p=3$ and $p=5$ illustrate how congruences predict the behavior of $H^{2}_{\text{ét}}$ and Tamagawa factors, offering evidence for the interaction between ETNC and BSD-type conjectures in families.

Abstract

Modular motives have coefficients in Hecke algebras. According to the equivariant philosophy, special values of $L$-functions of eigencuspforms should therefore exhibit equivariant properties with respect to various Hecke actions. This manuscript shows that this is indeed the case at least under broad conditions on ramification and deduce from them new properties of the Iwasawa Main Conjecture for modular forms. This manuscript is dedicated to the memory of Joël Bellaïche.

The Equivariant Tamagawa Number Conjectures for modular motives with coefficients in Hecke algebra

TL;DR

The paper establishes an equivariant Tamagawa number conjecture framework for modular motives with coefficients in Hecke algebras and proves that, under broad ramification hypotheses, the conjectures interpolate coherently in -adic families. It combines zeta-morphism formalism, Euler systems, and Taylor–Wiles–Kisin patching to extend Iwasawa main conjectures from single forms to universal deformation settings, ensuring compatibility at motivic points. The results imply that the equivariant Tamagawa framework governs special values across congruent modular forms and yields universal Iwasawa theory statements with concrete consequences for classical and non-classical points. Numerical examples at and illustrate how congruences predict the behavior of and Tamagawa factors, offering evidence for the interaction between ETNC and BSD-type conjectures in families.

Abstract

Modular motives have coefficients in Hecke algebras. According to the equivariant philosophy, special values of -functions of eigencuspforms should therefore exhibit equivariant properties with respect to various Hecke actions. This manuscript shows that this is indeed the case at least under broad conditions on ramification and deduce from them new properties of the Iwasawa Main Conjecture for modular forms. This manuscript is dedicated to the memory of Joël Bellaïche.
Paper Structure (23 sections, 17 theorems, 131 equations)

This paper contains 23 sections, 17 theorems, 131 equations.

Table of Contents

  1. Introduction
  2. Equivariant conjectures for modular motives
  3. Statement of results
  4. Outline of the proof
  5. Notations
  6. The ETNC for modular motives
  7. Modular motives
  8. The Tamagawa Number Conjectures
  9. The motivic fundamental line
  10. Zeta morphisms
  11. The classical Iwasawa Main Conjecture
  12. The universal Iwasawa Main Conjecture
  13. Deformations
  14. The Iwasawa Main Conjecture
  15. Compatibility of the ETNC at Iwasawa-suitable points
  16. ...and 8 more sections

Key Result

Theorem 1.6

Assume $R_{\Sigma}(\bar{\rho})$ to be minimal. Then the following assertions are equivalent.

Theorems & Definitions (43)

  • Conjecture 1.1
  • Conjecture 1.2
  • Conjecture 1.3
  • Conjecture 1.4
  • Theorem 1.6
  • Theorem 1.7
  • Corollary 1.8
  • Definition 2.1
  • Definition 2.2
  • Definition 2.3
  • ...and 33 more