Abstract
In this chapter, we explain the general theory of modular forms. In §4.1, we discuss the full modular group SL 2(ℤ) and modular forms with respect to SL 2(ℤ), as an introduction to the succeeding sections. We define and study congruence modular groups in §4.2. In §4.3, we explain the relation between modular forms and Dirichlet series obtained by Hecke and Weil. As an application of §4.3, we prove the transformation equation of η(z) in §4.4. We explain Hecke’s theory of Hecke operators in §4.5 and define primitive forms in §4.6. In §4.7 and §4.8, we construct Eisenstein series and some cusp forms from Dirichlet series of number fields. In §4.9, we explain theta functions which are also useful for constructing modular forms.
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© 1989 Springer-Verlag Berlin Heidelberg
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Miyake, T. (1989). Modular Groups and Modular Forms. In: Modular Forms. Springer Monographs in Mathematics. Springer, Berlin, Heidelberg. https://doi.org/10.1007/3-540-29593-3_4
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DOI: https://doi.org/10.1007/3-540-29593-3_4
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-662-22188-4
Online ISBN: 978-3-540-29593-8
eBook Packages: Springer Book Archive