Modular forms from Noether-Lefschetz theory

TL;DR

This paper connects the enumeration of rational curves on certain algebraic varieties with classical modular forms, using advanced intersection theory and theta correspondence techniques.

Contribution

It introduces a novel approach to counting rational curves via topological and cohomological methods, linking these counts to modular forms.

Findings

Generating functions are modular forms.

Results are currently limited to base degree 1.

Heuristics suggest extensions to higher degrees.

Abstract

We enumerate smooth rational curves on very general Weierstrass fibrations over hypersurfaces in projective space. The generating functions for these numbers lie in the ring of classical modular forms. The method of proof uses topological intersection products on a period stack and the cohomological theta correspondence of Kudla and Millson for special cycles on a locally symmetric space of orthogonal type. The results here apply only in base degree 1, but heuristics for higher base degree match predictions from the topological string partition function.