Theta correspondence and the Borisov-Gunnells relations

TL;DR

This paper explores a geometric theta correspondence linking homology of modular curves to modular forms, revealing new relations among Eisenstein series and revisiting Borisov-Gunnells work with a geometric proof.

Contribution

It introduces a geometric theta correspondence that maps homology classes to Eisenstein series, providing new insights and proofs of relations among these series.

Findings

Mapped modular symbols to weight one Eisenstein series

Connected hyperbolic cycles to Hilbert-Eisenstein series

Provided a geometric proof of Eisenstein series relations

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.