# Geometry of the eigencurve at CM points and trivial zeros of Katz $p$-adic $L$-functions

## TL;DR

This paper explores the local geometry of the $p$-adic eigencurve at CM points, linking intersection multiplicities to $ ext{L}$-invariants and trivial zeros of Katz $p$-adic $L$-functions, extending conjectures of Gross.

## Contribution

It provides a detailed analysis of the eigencurve's structure at CM points and establishes new connections between $ ext{L}$-invariants, trivial zeros, and the geometry of $p$-adic $L$-functions.

## Key findings

- The eigencurve at CM points has 3 or 4 irreducible components.
- A simple zero of the congruence ideal corresponds to a non-vanishing $ ext{L}$-invariant.
- At least one of the $ ext{L}$-invariants $ ext{L}_-( ext{varphi})$ or $ ext{L}_-( ext{varphi}^{-1})$ is non-zero.

## Abstract

The primary goal of this paper is to investigate the geometry of the $p$-adic eigencurve at a point $f$ corresponding to a weight one cuspidal theta series irregular at the prime number $p$. We show that $f$ belongs to exactly three or four irreducible components and study their intersection multiplicities. In particular, we show that the congruence ideal of a CM component has a simple zero at $f$ if and only if a certain anti-cyclotomic $\mathscr{L}$-invariant $\mathscr{L}_-(\varphi)$ does not vanish. Further, using Roy's Strong Six Exponential Theorem we show that at least one amongst $\mathscr{L}_-(\varphi)$ and $\mathscr{L}_-(\varphi^{-1})$ is non-zero. Combined with a divisibility proved by Hida and Tilouine, we deduce that the anti-cyclotomic Katz $p$-adic $L$-function of $\varphi$ has a simple (trivial) zero at $s=0$ if $\mathscr{L}_-(\varphi)$ is non-zero, which can be seen as an anti-cyclotomic analogue of a result of Ferrero and Greenberg. Finally, we propose a formula for the linear term of the two-variable Katz $p$-adic $L$-function of $\varphi$ at $s=0$ extending a conjecture of Gross.