Weil Representation and Waldspurger Formula

2012 ◽  
Author(s):  
Xinyi Yuan ◽  
Shou-Wu Zhang ◽  
Wei Zhang
Keyword(s):  
Integral Representation ◽  
Eisenstein Series ◽  
Local Field ◽  
Integral Formula ◽  
Theta Functions ◽  
Direct Consequence ◽  
Weil Representation ◽  
Computational Results ◽  
Weil Representations ◽  
Central Value

This chapter reviews some basic results on Weil representations, theta liftings and Eisenstein series. In particular, it introduces a proof of the Waldspurger formula. The theory of Weil representation is applied to an integral representation of the Rankin–Selberg L-function and to a proof of Waldspurger's central value formula. The chapter mostly follows Waldspurger's treatment with some modifications including Kudla's construction of incoherent Eisenstein series. It first describes the classical theory of Weil representation for an orthogonal space over a local field before discussing theta functions, the Siegel–Weil formula, and normalized local Shimizu lifting. The main result is an integral formula for the L-series using a kernel function. The Waldspurger formula is a direct consequence of the Siegel–Weil formula. After presenting the proof of Waldspurger formula, the chapter lists some computational results on three types of incoherent Eisenstein series.

scholarly journals Euler factors determine local Weil representations

2016 ◽  
Vol 2016 (717) ◽  
Author(s):  
Tim Dokchitser ◽  
Vladimir Dokchitser
Keyword(s):  
Local Field ◽  
Weil Representation ◽  
Weil Representations

AbstractWe show that a Frobenius-semisimple Weil representation over a local field

scholarly journals An integral representation, complete monotonicity, and inequalities of the Catalan numbers

Filomat ◽  
2018 ◽  
Vol 32 (2) ◽  
pp. 575-587 ◽  
Author(s):  
Feng Qi ◽  
Xiao-Ting Shi ◽  
Fang-Fang Liu
Keyword(s):  
Integral Representation ◽  
Generating Function ◽  
Integral Formula ◽  
Cauchy Integral Formula ◽  
Catalan Numbers ◽  
Complete Monotonicity ◽  
Cauchy Integral ◽  
Complex Functions

In the paper, by virtue of the Cauchy integral formula in the theory of complex functions, the authors establish an integral representation for the generating function of the Catalan numbers in combinatorics. From this, the authors derive an alternative integral representation, complete monotonicity, determinantal and product inequalities for the Catalan numbers.

scholarly journals Geometric Weil representation: local field case

2009 ◽  
Vol 145 (1) ◽  
pp. 56-88 ◽  
Author(s):  
Vincent Lafforgue ◽  
Sergey Lysenko
Keyword(s):  
Local Field ◽  
Dual Pair ◽  
Weil Representation ◽  
Closed Field ◽  
Perverse Sheaves ◽  
Algebraically Closed Field ◽  
Field Case ◽  
Metaplectic Group ◽  
Geometric Langlands ◽  
Langlands Functoriality

AbstractLet k be an algebraically closed field of characteristic greater than 2, and let F=k((t)) and G=𝕊p2d. In this paper we propose a geometric analog of the Weil representation of the metaplectic group $\widetilde G(F)$. This is a category of certain perverse sheaves on some stack, on which $\widetilde G(F)$ acts by functors. This construction will be used by Lysenko (in [Geometric theta-lifting for the dual pair S𝕆2m, 𝕊p2n, math.RT/0701170] and subsequent publications) for the proof of the geometric Langlands functoriality for some dual reductive pairs.

scholarly journals Eisenstein Series Identities Involving the Borweins' Cubic Theta Functions

2012 ◽  
Vol 2012 ◽  
pp. 1-14 ◽  
Author(s):  
Ernest X. W. Xia ◽  
Olivia X. M. Yao
Keyword(s):  
Eisenstein Series ◽  
Theta Functions ◽  
Elliptic Functions ◽  
The Other ◽  
Fundamental Theory ◽  
Convolution Sums

Based on the theories of Ramanujan's elliptic functions and the (p,k)-parametrization of theta functions due to Alaca et al. (2006, 2007, 2006) we derive certain Eisenstein series identities involving the Borweins' cubic theta functions with the help of the computer. Some of these identities were proved by Liu based on the fundamental theory of elliptic functions and some of them may be new. One side of each identity involves Eisenstein series, the other products of the Borweins' cubic theta functions. As applications, we evaluate some convolution sums. These evaluations are different from the formulas given by Alaca et al.

DIFFERENTIAL EQUATIONS FOR CUBIC THETA FUNCTIONS

2011 ◽  
Vol 07 (07) ◽  
pp. 1945-1957 ◽  
Author(s):  
TIM HUBER
Keyword(s):  
Differential Equations ◽  
Eisenstein Series ◽  
Theta Function ◽  
Modular Group ◽  
Short Proof ◽  
Nonlinear Differential Equations ◽  
Theta Functions ◽  
Coupled Systems ◽  
Function Identity ◽  
Theta Function Identity

We show that the cubic theta functions satisfy two distinct coupled systems of nonlinear differential equations. The resulting relations are analogous to Ramanujan's differential equations for Eisenstein series on the full modular group. We deduce the cubic analogs presented here from trigonometric series identities arising in Ramanujan's original paper on Eisenstein series. Several consequences of these differential equations are established, including a short proof of a famous cubic theta function identity derived by J. M. Borwein and P. B. Borwein.

scholarly journals Relation between Borweins’ Cubic Theta Functions and Ramanujan’s Eisenstein Series

2021 ◽  
Vol 2021 ◽  
pp. 1-6
Author(s):  
B. R. Srivatsa Kumar ◽  
Shruthi ◽  
D. Anu Radha
Keyword(s):  
Eisenstein Series ◽  
Geometric Mean ◽  
Theta Functions ◽  
Two Dimensional

Two-dimensional theta functions were found by the Borwein brothers to work on Gauss and Legendre’s arithmetic-geometric mean iteration. In this paper, some new Eisenstein series identities are obtained by using ( p , k )-parametrization in terms of Borweins’ theta functions.

scholarly journals The Jacobi form D_L, theta functions, and Eisenstein series

2016 ◽  
Vol 28 (1) ◽  
pp. 75-88
Author(s):  
Abdelmejid Bayad ◽  
Gilles Robert
Keyword(s):  
Eisenstein Series ◽  
Theta Functions ◽  
Jacobi Form

scholarly journals Vector-valued Eisenstein series of small weight

2019 ◽  
Vol 15 (02) ◽  
pp. 265-287 ◽  
Author(s):  
Brandon Williams
Keyword(s):  
Eisenstein Series ◽  
Weil Representation ◽  
Small Weight ◽  
Vector Valued

We study the (mock) Eisenstein series [Formula: see text] of weight [Formula: see text] for the Weil representation on an even lattice, defined as the result of Bruinier and Kuss’s coefficient formula for the Eisenstein series naively evaluated at [Formula: see text]. We describe the transformation law of [Formula: see text] in general. Most of this paper is dedicated to collecting examples where the coefficients of [Formula: see text] contain interesting arithmetic information. Finally, we make a few remarks about the case [Formula: see text].

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