A note on critical $p$-adic $L$-functions

TL;DR

This paper investigates the properties of $p$-adic $L$-functions near critical points on the eigencurve, using Emerton's representation theory to construct two-variable $p$-adic $L$-functions.

Contribution

It introduces a new approach to constructing two-variable $p$-adic $L$-functions around critical points via the Jacquet-Emerton functor.

Findings

Analysis of the adjunction property of the Jacquet-Emerton functor near critical points

Construction of two-variable $p$-adic $L$-functions using Emerton's representation theoretic approach

Insights into the behavior of $p$-adic $L$-functions at critical points

Abstract

We study the adjunction property of the Jacquet-Emerton functor in certain neighborhoods of critical points in the eigencurve. As an application, we construct two-variable $p$-adic $L$-functions around critical points via Emerton's representation theoretic approach.