$p$-adic $L$-functions on Hida Families

TL;DR

This paper investigates the relationship between $p$-adic $L$-functions and the geometry of Hida families, revealing how their Taylor expansions can detect geometric phenomena like ramification and component intersections.

Contribution

It establishes a connection between $p$-adic $L$-functions' Taylor expansions and geometric features of Hida families, including a converse to Vatsal's congruence results.

Findings

Taylor expansions detect ramification over weight space

Crossing components are characterized by intersection multiplicity

A new interpolation of congruences relates to algebraic $L$-values

Abstract

A major theme in the theory of $p$-adic deformations of automorphic forms is how $p$-adic $L$-functions over eigenvarieties relate to the geometry of these eigenvarieties. In this article we prove results in this vein for the ordinary part of the eigencurve (i.e. Hida families). We address how Taylor expansions of one variable $p$-adic $L$-functions varying over families can detect "bad" geometric phenomena: crossing components of a certain intersection multiplicity and ramification over the weight space. Our methods involve proving a converse to a result of Vatsal relating congruences between eigenforms to their algebraic special $L$-values and then $p$-adically interpolating congruences using formal models.