Twisted periods of modular forms
Tianyu Ni, Hui Xue
TL;DR
The paper introduces twisted periods $r_{t,\chi}$ for cusp forms in $S_k$ and proves linear independence results for families with the same twist or the same index as the weight grows, by linking periods to cusp forms through Trace and Rankin–Selberg methods. Central to the approach is the study of traces of products and Rankin–Cohen brackets of Eisenstein series at level $D$, yielding explicit Fourier expansions and asymptotics for kernel forms $\mathcal F_{K,\ell,\chi}$ and $\mathcal G_{K,\ell,\chi}$, which in turn produce non-singular coefficient matrices that certify independence. The method yields two applications: identities for convolution sums of twisted divisor functions expressed via generalized Bernoulli numbers, and a Maeda-conjecture–driven non-vanishing result for twisted central $L$-values $L(f,\chi,K/2)$. The authors also propose conjectures on spanning $S_K^*$ by twisted periods, with supporting numerical evidence and potential extensions to multiple twists.
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 $χ$ mod $D$, we introduce twisted periods $r_{t,χ}$ 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.