The Iwasawa Main Conjectures for ${\rm GL}_2$ and derivatives of $p$-adic $L$-functions

TL;DR

This paper proves a three-variable Iwasawa main conjecture for p-ordinary modular forms, advancing understanding of p-adic L-functions and their derivatives, with implications for nonvanishing conjectures in number theory.

Contribution

It establishes the Iwasawa main conjecture in a new setting, complementing prior work and enabling applications to key nonvanishing conjectures for p-adic L-functions.

Findings

Proved the three-variable Iwasawa main conjecture under mild hypotheses.

Applied results to Greenberg's nonvanishing conjecture for derivatives of p-adic L-functions.

Contributed to Howard's horizontal nonvanishing conjecture.

Abstract

We prove under mild hypotheses the three-variable Iwasawa main conjecture for $p$-ordinary modular forms in the indefinite setting. Our result is in a setting complementary to that in the work of Skinner-Urban, and it has applications to Greenberg's nonvanishing conjecture for the first derivatives at the center of $p$-adic $L$-functions of cusp forms in Hida families with root number $-1$ and to Howard's horizontal nonvanishing conjecture.