The Iwasawa Main Conjecture for universal families of modular motives

TL;DR

This paper proves the cyclotomic Iwasawa Main Conjecture for modular motives and extends it to p-adic families, using novel techniques involving Eisenstein series, universal deformation spaces, and p-adic L-functions.

Contribution

It establishes the Iwasawa Main Conjecture for modular motives with arbitrary reduction at p and generalizes it to p-adic families, introducing new methods in the process.

Findings

Proved the cyclotomic Iwasawa Main Conjecture for modular motives.

Extended the conjecture to p-adic families of modular forms.

Developed new techniques involving Eisenstein series and universal deformation spaces.

Abstract

Let $p$ be an odd prime. We prove the cyclotomic Iwasawa Main Conjecture of K.Kato for the motive attached to an eigencuspform $f\in S_{k}(\Gamma_{0}(N))$ with arbitrary reduction type at $p$ under mild assumptions on the residual Galois representation $\bar{\rho}_{f}$. Under the same hypotheses, we also prove the generalized Iwasawa Main Conjecture for $p$-adic families of modular forms. The Iwasawa Main Conjecture for $f$ is deduced by a limit argument involving fundamental lines from a universal Iwasawa Main Conjecture over the universal deformation space of $\bar{\rho}_{f}$, which itself follows from the cyclotomic Iwasawa Main Conjecture for crystalline eigencuspforms and hence from results on the Iwasawa-Greenberg Main Conjecture for Rankin-Selberg products. The main novel ingredients in our proof are as follows: a new way to study the arithmetic of the Fourier-Jacobi coefficients of Eisenstein series for the group $\operatorname{U}(3,1)$, an explicit description of the exponential map in a well-chosen family with prescribed ramification to obtain integral comparisons of various $p$-adic $L$-functions and Selmer modules, Iwasawa theory for the universal zeta elements constructed by K.Nakamura, descent techniques for fundamental lines over the universal regular ring underlying the universal deformation space and ramification properties of the latter over the former.