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On some incomplete theta integrals.

Funke, Jens and Kudla, Stephen (2019) 'On some incomplete theta integrals.', Compositio Mathematica., 155 (9). pp. 1711-1746.

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.

Item Type:Article
Full text:(AM) Accepted Manuscript
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Status:Peer-reviewed
Publisher Web site:https://doi.org/10.1112/S0010437X19007504
Publisher 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.
Date accepted:02 April 2019
Date deposited:02 May 2019
Date of first online publication:02 August 2019
Date first made open access:02 May 2019

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