A geometric view on Iwasawa theory

TL;DR

This paper explores the geometry of the $p$-adic eigencurve at a special point associated with a weight 1 cuspform with complex multiplication, revealing new insights into Iwasawa and Hida theories.

Contribution

It introduces new methods to determine Fourier coefficients of non-classical $p$-adic modular forms and computes cross-ratios of $p$-ordinary filtrations in Hida families.

Findings

Fourier coefficients expressed via $p$-adic logarithms of algebraic numbers.

Computed cross-ratios of $p$-ordinary filtrations in Hida families.

Extended understanding of the geometry of the $p$-adic eigencurve at CM points.

Abstract

This article extends our study of the geometry of the $p$-adic eigencurve at a point defined by a weight $1$ cuspform $f$ irregular at $p$ and having complex multiplication, and the implications in Iwasawa and in Hida theories. The novel results include the determination of the Fourier coefficients of certain non-classical $p$-adic modular forms belonging to the generalized eigenspace of $f$, in terms of $p$-adic logarithms of algebraic numbers. We also compute the "mysterious" cross-ratios of the $p$-ordinary filtrations of the Hida families containing $f$.