Introduction to Elliptic Curves and Modular Forms

Abstract

Library of Congress Cataloging-in-Publieation Data Koblitz, Neal. Introduetion to elliptie eurves and modular forms I Neal Koblitz. -2nd ed. p. em. -(Graduate texts in mathematies; 97

A strong coupling test of S-duality

“…f M may still fail to be a modular form in the usual sense if it has poles at some cusps, which may occur -as we discussed in connection with (2.71)-if bundles of negative instanton number contribute due to a failure of the vanishing theorem. 23 See [47], p. 231, solutions to exercises 12 and 14, for this and subsequent assertions.…”

“…For a relatively elementary introduction to such series see [47], section IV.2, especially p. 194; see also [55].…”

Geometry of 2D topological field theories

“…In the terminology of the theory of automorphic functionsf is an automorphic form of the modular group of the weight 2 k . [Also some assumptions about behaviour off in the orbifold points are needed in the de nition of an automorphic form; we refer the reader to a textbook in automorphic functions (e.g., [80] The FS-connection was rediscovered in the theory of automorphic forms by Rankin [119] (see also [80, page 123] In Appendix J I will explain the relation of this example to geometry of complex crystallographic group.…”

Higher-Level Appell Functions, Modular Transformations, and Characters

“…Here, [30] for the details (we note that c and d are coprime because of the determinant condition in SL(2, Z), and we assume 0 ±1 = 1). As a simple corollary of the Lemma, we have the formula…”