Critical slope p-adic L-functions of CM modular forms

TL;DR

This paper proves the conjecture that two constructions of critical-slope p-adic L-functions coincide for CM modular forms by comparing Kato's Euler system approach with Bellaiche's overconvergent modular symbols method.

Contribution

It establishes the equality of two different critical-slope p-adic L-functions for CM modular forms, confirming a key conjecture in the field.

Findings

Proved the conjecture for CM modular forms.

Calculated the critical-slope L-function from Kato's Euler system.

Compared and matched it with Bellaiche's overconvergent modular symbols approach.

Abstract

For ordinary modular forms, there are two constructions of a p-adic L-function attached to the non-unit root of the Hecke polynomial, which are conjectured but not known to coincide. We prove this conjecture for modular forms of CM type, by calculating the the critical-slope L-function arising from Kato's Euler system and comparing this with results of Bellaiche on the critical-slope L-function defined using overconvergent modular symbols.