Anticyclotomic Iwasawa main conjectures for modular forms

TL;DR

This paper proves Iwasawa main conjectures for modular forms over anticyclotomic extensions of imaginary quadratic fields, extending previous results to both definite and indefinite cases, and includes new results when $p$ splits in $K$.

Contribution

It establishes the Iwasawa main conjectures for modular forms in the anticyclotomic setting, including the indefinite case and cases where $p$ splits in $K$, using congruence methods and Heegner points.

Findings

Proved Iwasawa main conjectures in the definite setting.

Established main conjectures in the indefinite setting, including a Perrin-Riou type conjecture.

Proved an Iwasawa-Greenberg main conjecture when $p$ splits in $K$.

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 \`a 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.