On some incomplete theta integrals

Funke, Jens; Kudla, Stephen

doi:10.1112/s0010437x19007504

On some incomplete theta integrals

Funke, Jens; Kudla, Stephen

Authors

Professor Jens Funke jens.funke@durham.ac.uk
Professor

Stephen Kudla

Abstract

In this paper we construct indefinite theta series for lattices of arbitrary signature as ‘incomplete’ theta integrals, that is, by integrating the theta forms constructed by the second author with J. Millson over certain singular -chains in the associated symmetric space . These chains typically do not descend to homology classes in arithmetic quotients of , and consequently the theta integrals do not give rise to holomorphic modular forms, but rather to the non-holomorphic completions of certain mock modular forms. In this way we provide a general geometric framework for the indefinite theta series constructed by Zwegers and more recently by Alexandrov, Banerjee, Manschot, and Pioline, Nazaroglu, and Raum. In particular, the coefficients of the mock modular forms are identified with intersection numbers.

Citation

Funke, J., & Kudla, S. (2019). On some incomplete theta integrals. Compositio Mathematica, 155(9), 1711-1746. https://doi.org/10.1112/s0010437x19007504

Journal Article Type	Article
Acceptance Date	Apr 2, 2019
Online Publication Date	Aug 2, 2019
Publication Date	Sep 30, 2019
Deposit Date	May 1, 2019
Publicly Available Date	Feb 2, 2020
Journal	Compositio Mathematica
Print ISSN	0010-437X
Electronic ISSN	1570-5846
Publisher	Cambridge University Press
Peer Reviewed	Peer Reviewed
Volume	155
Issue	9
Pages	1711-1746
DOI	https://doi.org/10.1112/s0010437x19007504

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Copyright Statement
This article has been published in a revised form in Compositio Mathematica https://doi.org/10.1112/S0010437X19007504. This version is free to view and download for private research and study only. Not for re-distribution, re-sale or use in derivative works. © The Authors 2019.