Explicit generators of the space of modular forms

Technical Summary: Explicit Generators of the Space of Modular Forms

Problem Statement
The paper addresses two interconnected problems regarding the space of cusp forms $S_\kappa$ of even weight $\kappa \ge 12$ and level one, and its dual space $S^*_\kappa$.

1. Generators for $S_\kappa$: While it is known that spaces of modular forms are spanned by forms with rational Fourier coefficients, the specific question remains whether the subspace of cusp forms $S_\kappa$ can be generated by Rankin-Cohen brackets of Eisenstein series. Previous work by Xue [17] established this for specific cases of the bracket order $n$ (e.g., $n=0, 1, 2, 3$) and conjectured that the set of brackets $\{[E_k, E_\ell]_n\}$ spans $S_\kappa$ for any $n$, provided $\kappa$ is sufficiently large relative to $n$.
2. Generators for $S^*_\kappa$: The dual space $S^*_\kappa$ is spanned by periods of modular forms. While the Eichler-Shimura relations imply that the even periods $\{r_0, r_2, \dots, r_{\kappa-2}\}$ and the odd periods $\{r_1, r_3, \dots, r_{\kappa-3}\}$ both span $S^*_\kappa$, these sets are not linearly independent (their size is approximately $6 \dim S_\kappa$). The problem is to identify explicit, smaller subsets of periods that form a spanning set (or basis) for $S^*_\kappa$.

Methodology
The authors employ a combination of analytic number theory and linear algebra:

• Rankin-Selberg Method: To prove the spanning property for $S_\kappa$, the authors utilize the Rankin-Selberg method to relate the inner product of a cusp form with a Rankin-Cohen bracket to the periods of the cusp form. Specifically, they show that if a cusp form $g$ is orthogonal to the span of the brackets, its associated linear combination of periods must vanish.
• Eichler-Shimura Relations: The core of the proof relies on the linear dependence relations among periods (equations (1t), (2t), and (3t) in the paper). The authors construct a matrix $A$ representing a subsystem of these relations.
• Matrix Analysis: To prove that a specific subset of periods spans the dual space, the authors must show that the complementary subset of periods can be expressed as linear combinations of the chosen subset. This requires proving that the matrix $A$ (formed by the coefficients of the "unselected" periods in the relations) is non-singular.
• Combinatorial Techniques: The proof of non-singularity involves row-reducing matrix $A$ using a specific elimination matrix $P$. Unlike standard Gaussian elimination targeting upper triangular matrices, the authors reduce $A$ to a lower anti-triangular matrix. They utilize properties of Stirling numbers of the first and second kinds (Lemma 3.1) and combinatorial identities to demonstrate that the entries strictly above the anti-diagonal of $PA$ vanish and that the anti-diagonal entries are non-zero.

Key Contributions and Results

1. Resolution of the Spanning Conjecture for $S_\kappa$ (Theorem 1.1):
   The authors provide an affirmative answer to the conjecture in [17]. They prove that for any integer $n \ge 1$, if $\kappa \ge 4n + 18$, the set of Rankin-Cohen brackets $\{[E_k, E_\ell]_n\}$ with even $k, \ell \ge 4$ and $k + \ell + 2n = \kappa$ spans the vector space $S_\kappa$.

   • Mechanism: They define a function $G_g$ based on the periods of a hypothetical orthogonal cusp form $g$. By analyzing the range of indices for the periods involved, they show that $G_g$ satisfies the vanishing criteria of Proposition 2.2, implying $G_g = 0$ and consequently $g=0$.

2. Explicit Spanning Subsets for $S^*_\kappa$ (Theorem 1.2):
   The paper identifies explicit subsets of periods that span $S^*_\kappa$. Let $\kappa = 12m + 2a$ with $0 \le a \le 5$. The authors define an index set $I = \{i \in \mathbb{Z} : \lceil \frac{3m}{2} \rceil + 1 \le i \le 3m + \lfloor \frac{a}{2} \rfloor - 1\}$.

   • They prove that the even periods $\{r_{2i}\}_{i \in I}$ span $S^*_\kappa$.
   • Similarly, the odd periods $\{r_{2i-1}\}_{i \in I}$ span $S^*_\kappa$.
   • The size of these spanning sets is approximately $\lceil \frac{3}{2} \dim S_\kappa \rceil$, which is distinct from the basis found by Fukuhara [4].

3. Non-Singularity of the Relation Matrix:
   A significant technical contribution is the proof that the matrix $A$ derived from the Eichler-Shimura relations is non-singular. This is achieved by constructing an elimination matrix $P$ that transforms $A$ into a lower anti-triangular matrix with non-zero anti-diagonal entries. This result validates the linear independence of the specific periods excluded from the spanning set, thereby confirming the spanning property of the selected subset.

Significance and Claims
The paper claims to resolve the conjecture regarding the generation of cusp forms by Rankin-Cohen brackets of Eisenstein series for sufficiently large weights. Furthermore, it provides new, explicit spanning sets for the dual space of cusp forms, offering a more refined understanding of the linear dependencies among periods than previously available.

The authors note that their results for level one are somewhat mysterious compared to higher levels (e.g., $\Gamma_0(N)$), where periods generally do not form a basis due to dimension constraints. The paper concludes by linking their findings to broader conjectures (Conjectures 4.2 and 4.3) regarding the linear independence of arbitrary sets of periods, suggesting that Theorem 1.2 is a corollary of these conjectures. The work relies on the assumption that $\kappa$ is large enough ($\kappa \ge 4n + 18$) to ensure the necessary inequalities for the period indices hold.