Short Second Moment Bound for GL(2) $L$-functions in $q$-Aspect

TL;DR

This paper establishes a new upper bound for the second moment of certain $L$-functions in the $q$-aspect, extending previous results to higher prime powers and providing insights into the behavior of these functions in short intervals.

Contribution

It introduces a Lindel"of-on-average bound for the second moment of $L$-functions associated with level 1 cusp forms twisted along specific character cosets, generalizing to higher prime powers.

Findings

Proves a $q$-aspect upper bound for the second moment of $L$-functions.

Extends Good's 1982 results to higher prime power moduli.

Applicable to twisted $L$-functions in short intervals.

Abstract

We prove a Lindel\"{o}f-on-average upper bound for the second moment of the $L$-functions associated to a level 1 holomorphic cusp form, twisted along a coset of subgroup of the characters modulo $q^{2/3}$ (where $q = p^3$ for some odd prime $p$). This result should be seen as a $q$-aspect analogue of Anton Good's (1982) result on upper bounds of the second moment of cusp forms in short intervals. The results generalize easily to higher prime powers as well.