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Anticyclotomic Iwasawa main conjectures for modular forms

Matteo Longo, Maria Rosaria Pati, Stefano Vigni

Abstract

Let $f$ be a newform of even weight at least $4$, level $N$ and trivial character. Let $p\nmid N$ be an odd prime number that is ordinary for $f$ and let $K$ be an imaginary quadratic field satisfying a generalized Heegner hypothesis relative to $N$. In this paper, we prove (under mild arithmetic assumptions) Iwasawa main conjectures for $f$ over the anticyclotomic $\mathbb Z_p$-extension of $K$ both in the definite setting and in the indefinite setting (in the second case, we prove a main conjecture à la Perrin-Riou for modular forms). Our strategy of proof follows the approach of Bertolini-Darmon via congruences combined with our previous results on an analogue for $f$ of Kolyvagin's conjecture on the non-triviality of his $p$-adic system of derived Heegner points on elliptic curves. As a second contribution, when $p$ splits in $K$ we prove an Iwasawa-Greenberg main conjecture for the $p$-adic $L$-functions of Bertolini-Darmon-Prasanna and Brooks.

Anticyclotomic Iwasawa main conjectures for modular forms

Abstract

Let be a newform of even weight at least , level and trivial character. Let be an odd prime number that is ordinary for and let be an imaginary quadratic field satisfying a generalized Heegner hypothesis relative to . In this paper, we prove (under mild arithmetic assumptions) Iwasawa main conjectures for over the anticyclotomic -extension of both in the definite setting and in the indefinite setting (in the second case, we prove a main conjecture à la Perrin-Riou for modular forms). Our strategy of proof follows the approach of Bertolini-Darmon via congruences combined with our previous results on an analogue for of Kolyvagin's conjecture on the non-triviality of his -adic system of derived Heegner points on elliptic curves. As a second contribution, when splits in we prove an Iwasawa-Greenberg main conjecture for the -adic -functions of Bertolini-Darmon-Prasanna and Brooks.
Paper Structure (39 sections, 30 theorems, 217 equations)

This paper contains 39 sections, 30 theorems, 217 equations.

Table of Contents

  1. Introduction
  2. Preliminaries on quaternion algebras
  3. Quaternion algebras
  4. Special elements
  5. Polynomial representations
  6. Anticyclotomic $p$-adic $L$-functions
  7. Shimura curves
  8. Modular forms
  9. Theta elements
  10. $p$-adic $L$-functions
  11. Heegner cycles in Iwasawa theory
  12. Shimura curves
  13. Bad reduction of Shimura curves
  14. Kuga--Sato varieties
  15. CM points on Shimura curves
  16. ...and 24 more sections

Key Result

Theorem 1

Suppose we are in the definite case and that Assumption introass holds. Then $\mathop{\mathrm{Sel}}\nolimits(K_\infty,A)^\vee$ is a torsion $\Lambda$-module whose characteristic ideal coincides with the ideal generated by $\mathcal{L}_\mathfrak{p}(f)$.

Theorems & Definitions (74)

  • Theorem 1: Definite IMC
  • Theorem 2: Indefinite IMC
  • Theorem 3: Indefinite IMC
  • Remark 3.1
  • Remark 4.1
  • Remark 4.2
  • Definition 4.3
  • Lemma 4.4
  • proof
  • Definition 4.5
  • ...and 64 more