$p$-adic moments of $L$-functions

TL;DR

This paper derives a formula for the p-adic valuation of weighted moments of central L-values of cusp forms twisted by Dirichlet characters of order p, providing arithmetic interpretations in some cases.

Contribution

It introduces a new formula for p-adic valuations of L-value moments and interprets constants arithmetically, advancing understanding of p-adic L-functions.

Findings

Derived a formula for p-adic valuation of L-value moments

Provided arithmetic interpretation of constants in the formula

Applied digit map to relate horizontal and vertical p-adic measures

Abstract

We obtain a formula for the $p$-adic valuation of weighted moments of central $L$-values of holomorphic cusp forms twisted by Dirichlet characters of order $p$. In some cases we give an arithmetic interpretation of the constants in the formula. The result is obtained via the study of the digit map, turning a horizontal $p$-adic measure into a vertical one, applied to the horizontal $p$-adic $L$-functions as defined by the authors in previous work.