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Quaternionic Kolyvagin systems and Iwasawa theory for Hida families

Francesco Zerman

Abstract

We build a modified universal Kolyvagin system for the Galois representation attached to a Hida family of modular forms, starting from the big Heegner point Euler system of Longo--Vigni built in towers of Shimura curves. We generalize the work of Büyükboduk to a quaternionic setting, relaxing the classical \emph{Heegner hypothesis} on the tame conductor of the family. As a byproduct of this construction, we give a proof of one divisibility of the anticyclotomic Iwasawa main conjecture for Hida families.

Quaternionic Kolyvagin systems and Iwasawa theory for Hida families

Abstract

We build a modified universal Kolyvagin system for the Galois representation attached to a Hida family of modular forms, starting from the big Heegner point Euler system of Longo--Vigni built in towers of Shimura curves. We generalize the work of Büyükboduk to a quaternionic setting, relaxing the classical \emph{Heegner hypothesis} on the tame conductor of the family. As a byproduct of this construction, we give a proof of one divisibility of the anticyclotomic Iwasawa main conjecture for Hida families.

Paper Structure

This paper contains 29 sections, 35 theorems, 112 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments
  3. Big Galois representations and big Heegner classes
  4. Setting
  5. Hida families
  6. Arithmetic primes
  7. Big Galois representations
  8. Big Heegner classes
  9. Anticyclotomic twists
  10. Quotients, Galois groups and primes
  11. Galois invariants over abelian extensions of $K$
  12. Shapiro's lemma
  13. Selmer structures and Kolyvagin systems
  14. Local conditions and Selmer structures
  15. Greenberg conditions
  16. ...and 14 more sections

Key Result

Theorem A

Under Assumptions ass:residual-representation-irreducible, ass:field-K, ass:tamagawa-factors-at-N, ass:H.stz and ass:big-image, there is an infinite set of primes $\mathcal{P}'$, a set of automorphisms $\{\chi_{n,\ell}\}$ of $\boldsymbol{\mathrm{T}}^{\mathrm{ac}}$ and a universal Kolyvagin system $\ where $U_p\in\mathcal{R}^\times$ is the $p$-th Hecke operator.

Theorems & Definitions (106)

  • Theorem A: Theorem \ref{['thm:main-thm']}
  • Theorem B: Theorem \ref{['thm:iwasawa-main-conjecture']}
  • Remark 2.3
  • Definition 2.4
  • Theorem 2.5: Hida
  • proof
  • Definition 2.6
  • Remark 2.7
  • Lemma 2.8
  • proof
  • ...and 96 more