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

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.

33 pages

References in corpus (1)

• On the non-triviality of the $p$-adic Abel-Jacobi image of generalised Heegner cycles modulo $p$, II: Shimura curves