L-functions
in Number Theory
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L-functions, which are generalizations of the Riemann zeta function and now associated to various objects in Mathematics, contain various types of important information of these objects. Modular forms, on the other hand, were introduced by Hecke to generalize the classical theta series, and they have been playing a central role in modern Number Theory ever since. As Eichler once said, modular form is the fifth operation in Mathematics. In this monograph, we shall see several aspects of L-functions associated to modular forms. The first subject is on an average version of the fourth moment prob...
L-functions, which are generalizations of the Riemann zeta function and now associated to various objects in Mathematics, contain various types of important information of these objects. Modular forms, on the other hand, were introduced by Hecke to generalize the classical theta series, and they have been playing a central role in modern Number Theory ever since. As Eichler once said, modular form is the fifth operation in Mathematics. In this monograph, we shall see several aspects of L-functions associated to modular forms. The first subject is on an average version of the fourth moment problem associated to newforms, where the study of divisor function over quaternion algebras is carried out and a nice identity is derived. We shall also consider the central values of Rankin-Selberg L-functions. At the end, a result on the modularity of some four-dimensional Galois representations is included.
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