Abstract
The notion of a modular form is introduced. The dimension of the space of modular forms is determined and Fourier-expansions of Eisenstein series are determined leading to the first number theoretical applications. The Fourier-coefficients of a modular form are fed into a Dirichlet series thus forming the associated L-function, which is shown to extend to an entire function. Hecke operators acting on modular forms are introduced and it is shown that L-functions of Hecke eigenforms admit Euler products. Finally, non-holomorphic Eisenstein series and Maaß-wave forms, the real-analytic counterparts of modular forms, and there L-functions, are introduced.
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Deitmar, A. (2013). Modular Forms for SL2(ℤ). In: Automorphic Forms. Universitext. Springer, London. https://doi.org/10.1007/978-1-4471-4435-9_2
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DOI: https://doi.org/10.1007/978-1-4471-4435-9_2
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