The p-adic Shintani modular symbol and evil Eisenstein series

TL;DR

This paper constructs a p-adic Shintani modular symbol using Solomon’s cocycle to compute the p-adic L-function of evil twin Eisenstein series, providing new proofs of recent results in the field.

Contribution

It introduces a novel construction of a p-adic Shintani modular symbol that specializes to overconvergent symbols with the same p-adic L-function as evil Eisenstein series.

Findings

Computed the p-adic L-function of evil twin Eisenstein series.

Connected Solomon's Shintani cocycle to overconvergent modular symbols.

Provided a new proof of recent results by Bellaiche and Dasgupta.

Abstract

We compute the p-adic L-function (in the sense of Pollack-Stevens and Bellaiche) of the "evil twin" Eisenstein series. We use Solomon's Shintani cocycle to construct a p-adic Shintani modular symbol, and show how to specialize this modular symbol to overconvergent modular symbols with the same p-adic L-function as evil/critical-slope Eisenstein series. This gives a new proof of recent results of Bellaiche and Dasgupta.