Theta correspondence and the Borisov-Gunnells relations

Technical Summary: Theta Correspondence and the Borisov-Gunnells Relations

Problem Statement
This paper investigates the geometric theta correspondence from the first homology of a modular curve $Y_1(N)$ to the space of holomorphic modular forms of weight 2, $M_2(\Gamma_1(N))$. The primary objective is to explicitly evaluate this correspondence on specific generators of the homology group—namely modular symbols, modular caps, and hyperbolic cycles—to recover and explain known results regarding the spanning sets of modular forms. Specifically, the author seeks to clarify the relationship between the work of Borisov and Gunnells, who described the space of newforms of rank 0 and Eisenstein series using products of weight 1 Eisenstein series, and the work of Li, who described similar spaces using diagonal restrictions of Hilbert-Eisenstein series. A secondary goal is to utilize this geometric framework to derive linear relations among Eisenstein series.

Methodology
The methodology relies on the construction of a closed differential form $E(z, \tau)$ on the modular curve, valued in modular forms of weight 2. This construction combines ideas from Bergeron-Charollois-Garcia and Kudla-Millson, utilizing the Mathai-Quillen formalism and the Weil representation.

1. Geometric Setup: The author considers the Borel-Serre compactification $Y_1(N)^{BS}$ of the modular curve. The homology $H_1(Y_1(N); \mathbb{C})$ is decomposed, following Stevens, into a direct sum of the space of closed modular caps $C(C)$ and the space of degree 0 modular symbols $MS_0(\mathbb{C})$.
2. Theta Lift: The map $E: H_1(Y_1(N); \mathbb{C}) \to M_2(\Gamma_1(N))$ is defined by integrating the differential form $E(z, \tau)$ over 1-cycles. The form is constructed via the pushforward of a theta series associated with a lattice coset $L_{1,0}$ in the space of $2 \times 2$ matrices.
3. Explicit Evaluation:
   • Modular Caps: The integral over a closed modular cap $C_r$ at a cusp $r$ is computed by analyzing the constant term of the form at the boundary.
   • Modular Symbols: The integral over unimodular symbols is computed using Poisson summation and the restriction of the theta series to geodesics.
   • Hyperbolic Cycles: Using Stevens' continued fraction algorithm, hyperbolic cycles $Z_\gamma$ are decomposed into linear combinations of modular caps and unimodular symbols. The integral is then evaluated term-by-term.
4. Spectral Analysis: The image of the theta lift is analyzed using the spectral decomposition of the cohomology, relating the lift to newforms with non-vanishing $L$-values at $s=1$ and Eisenstein series.

Key Contributions and Results

• Evaluation on Homology Generators (Theorem 1.4):

  • The integral of the theta lift over a closed modular cap $C_r$ (where $r = \gamma \infty$) yields a weight 2 Eisenstein series $H^{(2)}_{d, -c}(\tau)$.
  • The integral over a unimodular symbol $M = \gamma\{0, \infty\}$ yields the product of two weight 1 Eisenstein series, specifically $-G^{(1)}_d(\tau)G^{(1)}_c(\tau)$.

• Spanning Sets and Li's Result (Theorems 1.2, 1.3, and 3.15):

  • The image of the theta lift, $\text{Im}(E)$, contains the subspace spanned by weight 2 Eisenstein series $H^{(2)}$ and the subspace of newforms of rank 0 ($S^{new}_{2, rk=0}$). When $N$ is prime, this inclusion is an equality.
  • By evaluating the lift on hyperbolic cycles $Z_\gamma$, the author recovers Li's result: the space $S^{new}_{2, rk=0}$ is spanned by the diagonal restrictions of Hilbert-Eisenstein series $E^{(1)}_{1, f_\gamma}(\tau, \tau)$.
  • The paper establishes that $\text{Im}(E)$ is contained in the span of weight 2 Eisenstein series and products of weight 1 Eisenstein series ($H^{(1,1)}$), thereby providing a geometric explanation for the spanning set described by Borisov and Gunnells.

• Relations Among Eisenstein Series (Theorem 1.6 and 4.10):

  • By considering a hyperbolic triangle $T$ in the Borel-Serre compactification whose boundary consists of three unimodular symbols and three modular caps, the author exploits the fact that the boundary is homologous to zero ($\partial T = 0$).
  • This yields the relation:
    $$G^{(1)}_a G^{(1)}_b + G^{(1)}_b G^{(1)}_c + G^{(1)}_c G^{(1)}_a = G^{(2)}_a + G^{(2)}_b + G^{(2)}_c$$
    for integers $a, b, c$ satisfying $a+b+c \equiv 0 \pmod N$ and non-zero modulo $N$. This provides a geometric proof of relations previously established by Borisov and Gunnells.

Significance
The paper claims significance in providing a unified geometric framework that connects two distinct descriptions of modular form spaces:

1. It explains the Borisov-Gunnells relations (products of weight 1 Eisenstein series) and Li's results (diagonal restrictions of Hilbert-Eisenstein series) as natural consequences of evaluating a single geometric theta lift on different homology generators.
2. It offers a geometric proof of specific linear relations between Eisenstein series of weight 1 and 2, derived directly from the topology of the modular curve and the properties of the theta lift, rather than purely analytic manipulations.
3. It clarifies the role of the Borel-Serre compactification and modular caps in the theory of modular symbols, demonstrating how the boundary terms (caps) correspond to Eisenstein series while the interior terms (symbols) correspond to products of lower-weight series.

The author notes that while the results regarding spanning sets are similar to existing literature (specifically [BG01] and [Li17]), the contribution lies in the derivation via the theta correspondence and the explicit geometric interpretation of the Fourier coefficients and integrals.