Book ChapterDOI
The Rankin–Selberg Method: A Survey
Daniel Bump
- pp 49-109
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TLDR
A survey of the Rankin-Selberg method for modular forms can be found in this article, where Rankin and Selberg introduced a new tool into the study of cusp forms, which is known as the rankin-selberg method and gave the functional equation of a new kind of Euler product, and new estimates for the Fourier coefficients.Abstract:
Publisher Summary This chapter presents a survey of the Rankin-Selberg method. The earliest applications of the Rankin-Selberg method were to the estimation of the Fourier coefficients of modular forms. The Rankin-Selberg method gives an improvement over the “trivial estimate”.. Hecke introduced operators on cusp forms for each p that play a role similar to the Laplacian. These operators are self-adjoint and mutually commutative, hence might be simultaneously diagonalized. Rankin and, independently at around the same time, Selberg introduced a new tool into the study of cusp forms, which is known as the Rankin-Selberg method. This gives the functional equation of a new kind of Euler product, and gives new estimates for the Fourier coefficients.read more
Citations
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Book ChapterDOI
Perspectives on the Analytic Theory of L-Functions
Henryk Iwaniec,Peter Sarnak +1 more
TL;DR: In this paper, the authors introduce L-functions and describe the central problems connected with them, as well as some problems that can be resolved by finessing these conjectures.
Journal ArticleDOI
Directed polymers and the quantum Toda lattice
TL;DR: In this article, the authors characterized the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice, via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion.
Journal ArticleDOI
Nonvanishing theorems for L-functions of modular forms and their derivatives.
Book
Local newforms for GSp(4)
Brooks Roberts,Ralf Schmidt +1 more
TL;DR: In this article, the main theorem of representation theory and representation theory is presented. But they do not consider non-supercuspidal representations. And they ignore Hecke Operators.
Journal ArticleDOI
Directed polymers and the quantum Toda lattice
TL;DR: In this article, the authors characterized the partition function of a Brownian directed polymer model in terms of a diffusion process associated with the quantum Toda lattice, via a multidimensional generalization of a theorem of Matsumoto and Yor concerning exponential functionals of Brownian motion.
References
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Journal ArticleDOI
Harmonic Analysis and Discontinuous Groups in Weakly Symmetric Riemannian Spaces With Applications to Dirichlet Series
TL;DR: The trace-formula as discussed by the authors is a general relation which can be considered as a generalization of the so-called Poisson summation formula (in one or more dimensions) and is used in many of these works.
Journal ArticleDOI
On Modular Forms of Half Integral Weight
TL;DR: In this article, the connection of modular forms with zeta functions was clarified, and a more affirmative aspect of the subject was revealed, which might have given a rather negative and somewhat misleading impression that one would not be able to do much except in some special cases.
Book ChapterDOI
Modular Forms of Half Integral Weight
TL;DR: The Dedekind eta function and the theta function θ as discussed by the authors satisfy the automorphic factor (cz + d)k/2 with a positive odd integer k. (For some practical reasons, we take \( e^{2\pi in^2 z} \) instead of the usual \( e^{\pi in √ 2 z} in the definition of θ.)