Titlebook: Siegel Modular Forms; A Classical and Repr Ameya Pitale Book 2019 Springer Nature Switzerland AG 2019 Siegel modular forms.Symplectic group

BRIDE

0075-8434 basic concepts can be applied to solve contemporary problemsThis monograph introduces two approaches to studying Siegel modular forms: the classical approach as holomorphic functions on the Siegel upper half space, and the approach via representation theory on the symplectic group. By illustrating t

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Integral Representation of the Standard ,-Function,ence subgroup. Rallis, Piatetski-Shapiro, and also Böcherer, have obtained the integral representation in special cases. We will modify their methods to suit our level of generality. We will illustrate an application of the integral representation to algebraicity of special values of the .-function in the genus 2 case.

汇总

Examples,ito–Kurokawa lifts. These are concrete examples of cuspidal Siegel modular forms constructed from elliptic cusp forms. Finally, we consider Siegel modular forms with level . in the genus . case, corresponding to the standard congruence subgroups.

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Hecke Theory and ,-Functions, consisting of simultaneous eigenforms of the Hecke algebra. We explicate the relation between the Hecke eigenvalues and the Fourier coefficients of the modular forms. For genus greater than 1, this relation is very complicated. Finally, we introduce the two .-functions associated with Hecke eigenfo

obtuse

Cuspidal Automorphic Representations Corresponding to Siegel Modular Forms, There are several steps for achieving this— construct a function . on . corresponding to ., understand the properties it inherits from ., and study the local components of the representation generated by .. The main reference for this chapter is the article [6] by Asgari and Schmidt. We suggest the

DEMUR

Integral Representation of the Standard ,-Function,ters. The goal is to assume as little as possible about the Siegel modular forms. We will allow vector-valued modular forms with respect to any congruence subgroup. Rallis, Piatetski-Shapiro, and also Böcherer, have obtained the integral representation in special cases. We will modify their methods

TIGER

Book 2019ace, and the approach via representation theory on the symplectic group. By illustrating the interconnections shared by the two, this book fills an important gap in the existing literature on modular forms. It begins by establishing the basics of the classical theory of Siegel modular forms, and the

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