Twisted periods of modular forms

Technical Summary: Twisted Periods of Modular Forms

Problem Statement
The paper investigates the linear independence of "twisted periods" associated with cusp forms of weight $k$ and level one, denoted by the space $S_k$. While the classical Eichler-Shimura theory establishes that periods span the dual space $S_k^*$, these periods satisfy numerous linear dependence relations (Eichler-Shimura relations), meaning they are not linearly independent. Previous works by Fukuhara, Lei et al., and others have addressed the linear independence of specific subsets of odd or even periods. This paper extends these inquiries to twisted periods, defined for a primitive Dirichlet character $\chi$ modulo $D$ and an integer $0 \le n \le k-2$ as:
$$r_{n,\chi}(f) := \int_0^{i\infty} f_\chi(z) z^n dz = \frac{n!}{(-2\pi i)^{n+1}} L(f, \chi, n+1),$$
where $f_\chi$ is the twist of the cusp form $f$. The central problem is to determine conditions under which sets of twisted periods are linearly independent. Specifically, the authors address two scenarios:

1. Fixed Character, Different Indices: Are periods $\{r_{\ell_i-1, \chi}\}$ (or $\{r_{\ell_i, \chi}\}$) linearly independent for a fixed $\chi$ and distinct indices $\ell_i$?
2. Fixed Index, Different Characters: Are periods $\{r_{\ell-1, \chi_i}\}$ (or $\{r_{\ell, \chi_i}\}$) linearly independent for a fixed index $\ell$ and distinct primitive characters $\chi_i$ modulo $D$?

Methodology
The authors employ a strategy analogous to recent works on untwisted periods, relying on the Rankin-Selberg method and the study of kernel functions. The core logical flow is as follows:

1. Kernel Form Construction: The twisted period $r_{n,\chi}(f)$ is realized as a Petersson inner product $\langle f, K_{n,\chi} \rangle$. Instead of analyzing the kernel $K_{n,\chi}$ directly (which has transcendental Fourier coefficients), the authors construct explicit cusp forms $F_{K,\ell,\chi}$ and $G_{K,\ell,\chi}$ in $S_K$ that are related to these periods.

   • For odd periods (indices $\ell-1$), $F_{K,\ell,\chi}$ is derived from the cuspidal projection of the trace of the product of Eisenstein series: $\text{Tr}_1^D(G_{\ell,\chi} G_{K-\ell,\chi})$.
   • For even periods (indices $\ell$), $G_{K,\ell,\chi}$ is derived from the Rankin-Cohen bracket of Eisenstein series: $\text{Tr}_1^D([G_{\ell,\chi}, G_{K-2-\ell,\chi}]_1)$.

2. Equivalence of Independence: Using Rankin-Selberg convolution formulas, the authors prove that the linear independence of the set of twisted periods is equivalent to the linear independence of the corresponding set of cusp forms $\{F_{K,\ell_i,\chi}\}$ or $\{G_{K,\ell,\chi_i}\}$.

3. Fourier Coefficient Analysis: To prove the linear independence of the cusp forms, the authors analyze their Fourier coefficients. They derive explicit formulas for the coefficients of the Eisenstein series products and Rankin-Cohen brackets under the trace map.

   • They establish asymptotic behaviors for these coefficients as the weight $K$ becomes sufficiently large relative to the indices and the level $D$.
   • Specifically, they show that the normalized $q^{2j}$-Fourier coefficients of these cusp forms behave asymptotically like $\sigma_{\ell-1,\chi}(2^j)$ (for $F$) and $2^j \sigma_{\ell-1,\chi}(2^j)$ (for $G$), with error terms vanishing as $K \to \infty$.

4. Matrix Non-Singularity: The linear independence of the cusp forms is reduced to the non-singularity of matrices formed by these Fourier coefficients.

   • For fixed characters and varying indices, the relevant matrices are shown to be perturbations of Vandermonde-like matrices involving powers of $\chi(2)$.
   • For fixed indices and varying characters, the matrices involve powers of $\chi_i(2)$.
   • By bounding the error terms and comparing the determinant of the perturbed matrix to the main term (using properties of the Vandermonde determinant), the authors demonstrate that these matrices are non-singular when $K$ is sufficiently large.

Key Contributions and Results
The paper establishes four main theorems regarding the linear independence of twisted periods, assuming $D$ is an odd square-free integer and $K$ is sufficiently large relative to $n$ and $D$:

• Theorem 1.1 & 1.2: For a fixed primitive character $\chi$, any $n$ twisted periods with the same twist but different indices (either odd or even types) are linearly independent.
• Theorem 1.3 & 1.4: For a fixed index, any $n$ twisted periods with different primitive characters $\chi_i$ (satisfying $\chi_i(-1) = (-1)^\ell$ and distinct values at a fixed prime, e.g., $\chi_i(2)$) are linearly independent.

Applications
The authors demonstrate two applications of their method:

1. Convolution Sums: They derive explicit identities evaluating convolution sums of twisted divisor functions in terms of generalized Bernoulli numbers. These identities arise from the fact that for certain weights where $\dim S_K = 0$, the constructed cusp forms must vanish, forcing relations among their Fourier coefficients.
2. Non-vanishing of L-values: Assuming Maeda's conjecture (specifically that the Galois conjugates of a Hecke eigenform span $S_K$), the authors show that the twisted central L-values $L(f, \chi, K/2)$ are non-vanishing for normalized Hecke eigenforms $f$ when $K \ge 10D + 2$. This relies on proving that the associated kernel form $F_{K, K/2, \chi}$ is not identically zero.

Significance and Claims
The paper claims to provide the first systematic evidence for the linear independence of twisted periods, extending previous results on untwisted periods to the twisted setting. The authors explicitly state that the question of whether twisted periods span the entire space $S_k^*$ and the existence of Eichler-Shimura type relations for them are beyond the scope of this work, though they propose conjectures (Conjectures 6.1–6.4) suggesting that twisted periods do indeed span $S_k^*$ under certain conditions. The results are conditional on the weight $K$ being "sufficiently large," and the authors provide numerical verification for small weights ($K, D \le 40$) supporting their conjectures. The work connects the arithmetic of L-values, the structure of modular forms, and the properties of Dirichlet characters through the lens of linear algebra on Fourier coefficients.