Communications in Number Theory and Physics

Volume 11 (2017)

Number 4

Calabi-Yau modular forms in limit: Elliptic fibrations

Pages: 879 – 912



Babak Haghighat (Yau Mathematical Sciences Center, Tsinghua University, Beijing, China)

Hossein Movasati (Instituto de Matemática Pura e Applicado (IMPA), Rio de Janeiro, Brazil)

Shing-Tung Yau (Department of Mathematics, Harvard University, Cambridge, Massachusetts, U.S.A.)


We study the limit of Calabi–Yau modular forms, and in particular, those resulting in classical modular forms. We then study two parameter families of elliptically fibred Calabi–Yau fourfolds and describe the modular forms arising from the degeneracy loci. In the case of elliptically fibred Calabi–Yau threefolds our approach gives a mathematical proof of many observations about modularity properties of topological string amplitudes starting with the work of Candelas, Font, Katz and Morrison. In the case of Calabi–Yau fourfolds we derive new identities not computed before.


modular forms, Hodge filtration, Picard–Fuchs system

2010 Mathematics Subject Classification

14J15, 14N35, 32G20

Received 12 February 2016

Accepted 13 July 2017

Published 29 November 2017