Theta correspondence and the Borisov-Gunnells relations

TL;DR

The paper builds a geometric theta lift $\mathcal{E}: H_1(Y_1(N);\mathbb{Z}) \to M_2(\Gamma_1(N))$ via a differential form $\mathcal{E}(z,\tau)$ on $Y_1(N)$ whose Fourier coefficients are Poincaré duals of modular symbols. It shows that modular caps map to weight-2 Eisenstein series while unimodular symbols map to products of weight-1 Eisenstein, yielding the Borisov-Gunnells spanning set and Li's diagonal Hilbert-Eisenstein results; the image contains $\mathcal{H}^{(2)}$ plus $S^{new}_{2,\mathrm{rk}=0}(\Gamma_1(N))$, and for prime $N$ these pieces coincide. A key outcome is a concrete linear relation among Eisenstein series $G^{(1)}$ and $G^{(2)}$ of weight 1 and 2, namely $G_a^{(1)}G_b^{(1)}+G_b^{(1)}G_c^{(1)}+G_c^{(1)}G_a^{(1)}=G_a^{(2)}+G_b^{(2)}+G_c^{(2)}$ when $a+b+c\equiv0\pmod N$, with generalizations to higher terms via polygonal decompositions. The framework naturally extends to a higher rank theta correspondence from $(n-1)$-st homology of $\mathrm{SL}_n(\mathbb{R})/\mathrm{SO}(n)$ to weight $n$ modular forms, linking geometric cycles to Eisenstein theory and $L$-value phenomena.

Abstract

We consider a geometric theta correspondence from the first homology of a modular curve, to modular forms of weight $2$. Using Stevens' description of the homology, we find that this map sends modular symbols to product of weight one Eisenstein series, modular caps to weight $2$ Eisenstein series, and hyperbolic cycles to diagonal restrictions of Hilbert-Eisenstein series. We use it to revisit work of Borisov and Gunnells, and explain its connection to a theorem of Li. In particular, we give a geometric proof of certain relations between Eisenstein series.

Figures (5)

• Figure 1: We represent $\mathbb{H}$ in the Poincaré disc model. The polygon $\pazocal{P}$ is closed in the Borel-Serre compactification. The circles around the cusps are the horocycles at infinity containing the modular caps. The theta lift sends the modular symbols on each side to a product of two weight $1$ Eisenstein series and each modular cap to a weight $2$ Eisenstein series. The two segments between the interior and the cusp cancel out.
• Figure 2: The hyperbolic triangle closes in the Borel–Serre compactification. Its vertices $r_1, r_2, r_3$ are cusps connected by unimodular sides and completed by modular caps. Each side is mapped to a product of two weight $1$ Eisenstein series, while each modular cap is mapped to a weight $2$ Eisenstein series.
• Figure 3: We visualize the hyperbolic $2$-space in the disk model. The cycle $\pazocal{Z}_\gamma$ is moved to the boundary component at the cusp $r$. The two sides are $\Gamma$-translates and cancel out.
• Figure 4: The polygon $\pazocal{P}$.
• Figure 5: A hyperbolic triangle with unimodular sides and closed by modular caps.

Theorems & Definitions (52)

• Theorem 1.1
• Corollary 1.1.1
• Theorem 1.2
• Theorem 1.3
• Remark 1.1
• Theorem 1.4
• Remark 1.2
• Remark 1.3
• Theorem 1.5
• Remark 1.4