Rational spanning sets of level-one cusp forms from Eisenstein series at prime levels

Technical Summary: Rational Spanning Sets of Level-One Cusp Forms from Eisenstein Series at Prime Levels

Problem Statement
The paper addresses the construction of explicit rational spanning sets for the space of cusp forms of weight $k$ and level one, denoted $S_k$. While it is well-established that $S_k$ can be spanned by products of level-one Eisenstein series (e.g., $E_l \cdot E_{k-l} - E_k$), finding an explicit basis from these forms is difficult. Recent works have provided spanning sets using Rankin-Cohen brackets of level-one Eisenstein series. The central question motivating this work is whether one can construct an explicit rational spanning subset of $S_k$ using only Eisenstein series of higher levels (specifically prime levels), rather than relying on lower-weight level-one forms.

Methodology
The author employs a strategy involving the trace map and Rankin-Cohen brackets to lift forms from higher levels down to level one. The core methodology proceeds as follows:

1. Construction of Forms: The paper defines a family of modular forms $F_{p, k_1, k_2, c}(z)$ by taking the $c$-th Rankin-Cohen bracket of two Eisenstein series, $G_{k_1, \chi_p}$ and $G_{k_2, \chi_p}$, of level $p$ (where $p$ is an odd prime and $\chi_p$ is the Legendre symbol). These forms are then traced down to the level-one space using the trace map $\text{Tr}_1^p$.
2. Rankin-Selberg Convolution: The relationship between these constructed forms and cusp forms is established via the Rankin-Selberg convolution. Specifically, the Petersson inner product of a normalized Hecke eigenform $f \in S_k$ with $F_{p, k_1, k_2, c}$ is shown to be proportional to the product of a central $L$-value and a twisted $L$-value: $L(f, k_1+k_2+c-1)L(f, \chi_p, k_2+c)$.
3. Determination by Noncentral $L$-Values: The proof relies on a novel result (Theorem 3.2) demonstrating that a cusp form is uniquely determined (and must be zero if all relevant coefficients vanish) by its noncentral quadratic-twist $L$-values $L(f, \chi_p, s)$ for primes $p$ in a specific arithmetic progression. This result utilizes:
   • The orthogonality of Dirichlet characters and the Siegel-Walfisz theorem to average over primes.
   • Properties of symmetric square $L$-functions to relate twisted $L$-values to Fourier coefficients of square-free indices.
   • A lemma by Anamby and Das stating that a cusp form vanishing on all square-free Fourier coefficients is identically zero.

Key Contributions and Results

• Theorem 1.1 (Main Result): The paper proves that the set of forms $\{F_{p, k_1, k_2, c}(z)\}_{p \in P_{4, (-1)^{k_1}}}$ spans the entire space $S_{k_1+k_2+2c}$. Here, $P_{4, (-1)^{k_1}}$ denotes the set of primes in the residue class $(-1)^{k_1} \pmod 4$. This provides an affirmative answer to the motivating question, constructing a rational spanning set entirely from Eisenstein series at varying prime levels.
• Theorem 3.2 (Determination Theorem): A significant intermediate result establishes that if a cusp form $f \in S_k$ satisfies $L(f, \chi_p, s) = 0$ for all primes $p$ in a fixed residue class modulo 4 (with $\text{Re}(s) > k + 1/2$), then $f$ must be the zero form. This generalizes previous work on central $L$-values to noncentral values and general cusp forms.
• Numerical Evidence and Conjecture: The paper provides numerical examples (e.g., for $S_{24}$) showing that for distinct primes $p_1, p_2$, the forms $F_{p_1}$ and $F_{p_2}$ are linearly independent. Based on this, the author proposes Conjecture 1.4, suggesting that any subset of $d = \dim S_k$ such forms (where the primes are distinct and in the correct residue class) forms a basis for $S_k$.

Significance and Claims
The paper claims to provide the first explicit construction of a rational spanning set for level-one cusp forms derived exclusively from Eisenstein series of higher levels. Unlike previous constructions that rely on level-one Eisenstein series of lower weights, this method fixes the indices of the Eisenstein series while varying the level.

The significance of the work lies in:

1. Explicit Construction: It offers a concrete, computable set of rational forms spanning $S_k$, derived from the trace of Rankin-Cohen brackets of Eisenstein series.
2. Theoretical Advancement: The proof introduces a method for determining cusp forms via noncentral quadratic-twist $L$-values, which is technically simpler than methods relying on central values.
3. Connection to Twisted Periods: In the discussion (Section 5), the author connects these results to the theory of twisted periods of modular forms, answering a specific question regarding whether twisted periods with fixed indices but varying quadratic characters span the dual space $S_k^*$.

The author remains modest regarding the finiteness of the spanning set; while the theorem guarantees that the infinite set of primes spans the space, the paper does not prove that a finite subset of these primes is sufficient to form a basis, though numerical data suggests this is likely true.