$r$-tuple error functions and indefinite theta series of higher-depth

Theta functions for definite signature lattices constitute a rich source of modular forms. A natural question is then their generalization to indefinite signature lattices. One way to ensure a convergent theta series while keeping the holomorphicity property of definite signature theta series is to restrict the sum over lattice points to a proper subset. Although such series do not generally have the modular properties that a definite signature theta function has, as shown by Zwegers for signature $(1, n-1)$ lattices, they can be completed to a function that has these modular properties by compromising on the holomorphicity property in a certain way. This construction has recently been generalized to signature $(2, n-2)$ lattices by Alexandrov, Banerjee, Manschot, and Pioline. A crucial ingredient in this work is the notion of double error functions which naturally lends itself to generalizations. In this work we study the properties of such error functions which we will call $r$-tuple error functions. We then construct an indefinite theta series for signature $(r, n-r)$ lattices and show they can be completed to modular forms by using these $r$-tuple error functions.

Keywords

generalized error functions, indefinite theta series, modular forms, mock modular forms

2010 Mathematics Subject Classification

Primary 11F27. Secondary 11F50.

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The author acknowledges and thanks the support of NSF grant 1520748.

Any opinions, findings, and conclusions or recommendations expressed in this material are those of the author and do not necessarily reflect the views of the National Science Foundation.

Received 15 November 2016

Accepted 12 April 2018

Published 25 September 2018