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2015 | OriginalPaper | Chapter

Introduction to Modular Forms

Author: Simon C. F. Rose

Published in: Calabi-Yau Varieties: Arithmetic, Geometry and Physics

Publisher: Springer New York

Abstract

We introduce the notion of modular forms, focusing primarily on the group \(PSL_{2}\mathbb{Z}\). We further introduce quasi-modular forms, as well as discuss their relation to physics and their applications in a variety of enumerative problems. These notes are based on a lecture given at the Field’s Institute during the thematic program on Calabi-Yau Varieties: Arithmetic, Geometry, and Physics.

previous chapter Enumerative Aspects of the Gross-Siebert Program

next chapter Lectures on BCOV Holomorphic Anomaly Equations

Note that in this double sum we should exclude the (m, n) = (0, 0) term.

This is why we choose this particular normalization.

This actually requires an extension of the notion of modularity to (a) deal with characters of the group Γ and (b) to deal with forms of half-integer wieght. However, for the case at hand (r = 8), no such generalization is needed.

Note that we are using the non-holomorphic extension of \(E_{2}(\tau )\) so that this is well-defined on \(\mathcal{M}_{\mathbb{C}}^{E}\).

There is of course the factor of q ^1∕2 in the first term which does break modularity. However, we can easily include this into the definition of the function, and end up with a modular form as we desire.

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Title: Introduction to Modular Forms
Author: Simon C. F. Rose
Publisher: Springer New York
Book: Calabi-Yau Varieties: Arithmetic, Geometry and Physics
Print ISBN: 978-1-4939-2829-3

Electronic ISBN: 978-1-4939-2830-9

Copyright Year: 2015
DOI: https://doi.org/10.1007/978-1-4939-2830-9_12

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