Modular forms from Noether–Lefschetz theory

Greer, François

doi:10.2140/ant.2020.14.2335

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

Modular forms from Noether–Lefschetz theory

François Greer

Vol. 14 (2020), No. 9, 2335–2368

DOI: 10.2140/ant.2020.14.2335

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.

Keywords

modular forms, elliptic fibrations, Calabi–Yau threefolds, rational curves

Mathematical Subject Classification 2010

Primary: 14NXX

Milestones

Received: 31 January 2019

Revised: 27 March 2020

Accepted: 29 April 2020

Published: 13 October 2020

Authors

François Greer
Simons Center for Geometry and Physics Stony Brook University Stony Brook, NY United States

Vol. 14, No. 9, 2020