Twisted periods of modular forms

TL;DR

This paper introduces twisted periods for cusp forms, establishes their linear independence under certain conditions, and applies these results to evaluate divisor sums and relate Maeda's conjecture to twisted L-value non-vanishing.

Contribution

It develops a new framework for twisted periods of modular forms, proving their linear independence and connecting these properties to divisor sums and L-value non-vanishing.

Findings

Proved linear independence of twisted periods with same twist and different indices for large weight.

Established linear independence of twisted periods with same index and different twists for large weight.

Connected Maeda's conjecture to non-vanishing of twisted central L-values.

Abstract

Let $S_k$ denote the space of cusp forms of weight $k$ and level one. For $0\leq t\leq k-2$ and primitive Dirichlet character $\chi$ mod $D$, we introduce twisted periods $r_{t,\chi}$ on $S_k$. We show that for a fixed natural number $n$, if $k$ is sufficiently large relative to $n$ and $D$, then any $n$ periods with the same twist but different indices are linearly independent. We also prove that if $k$ is sufficiently large relative to $D$ then any $n$ periods with the same index but different twists mod $D$ are linearly independent. These results are achieved by studying the trace of the products and Rankin-Cohen brackets of Eisenstein series of level $D$ with nebentypus. Moreover, we give two applications of our method. First, we prove certain identities that evaluate convolution sums of twisted divisor functions. Second, we show that Maeda's conjecture implies a non-vanishing result on twisted central $L$-values of normalized Hecke eigenforms.