# Steinberg representation of GSp(4): Bessel models and integral representation of L-functions

@article{Pitale2009SteinbergRO,
title={Steinberg representation of GSp(4): Bessel models and integral representation of L-functions},
author={Ameya Pitale},
journal={Pacific Journal of Mathematics},
year={2009},
volume={250},
pages={365-406}
}
• Ameya Pitale
• Published 23 September 2009
• Mathematics
• Pacific Journal of Mathematics
We obtain explicit formulas for the test vector in the Bessel model, and derive the criteria for existence and uniqueness of Bessel models for the unramified quadratic twists of the Steinberg representation π of GSp 4 (F), where F is a nonarchimedean local field of characteristic zero. We also give precise criteria for the Iwahori spherical vector in π to be a test vector. We apply the formulas for the test vector to obtain an integral representation of the local L-function of π, twisted by any…
On Bessel models for GSp 4 and Fourier coefficients of Siegel modular forms of degree 2
In this work, we make a detailed study of the Fourier coefficients of cuspidal Siegel modular forms of degree 2. We derive a very general relation between the Fourier coefficients that extends
Some results on Bessel functionals for GSp(4)
• Mathematics
• 2014
We prove that every irreducible, admissible representation of GSp(4,F), where F is a non-archimedean local field of characteristic zero, admits a Bessel functional, provided the representation is not
Test vectors and central L-values for GL(2) (Automorphic Forms and Related Zeta Functions)
• Mathematics
• 2015
We determine local test vectors for Waldspurger functionals for GL2, in the case where both the representation of GL2 and the character of the degree two extension are ramied, with certain
Local spectral equidistribution for Siegel modular forms and applications
• Mathematics
Compositio Mathematica
• 2012
Abstract We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight k, subject to a specific weighting which allows us to
Restrictions of Eisenstein series and Rankin-Selberg convolution
• Mathematics
• 2017
In a 2005 paper, Yang constructed families of Hilbert Eisenstein series, which when restricted to the diagonal are conjectured to span the underlying space of elliptic modular forms. One approach to
Special values of L-functions for Saito–Kurokawa lifts
• Mathematics
Mathematical Proceedings of the Cambridge Philosophical Society
• 2013
Abstract In this paper we obtain special value results for L-functions associated to classical and paramodular Saito–Kurokawa lifts. In particular, we consider standard L-functions associated to
On ratios of Petersson norms for Yoshida lifts
We prove an algebraicity property for a certain ratio of Petersson norms associated to a Siegel cusp form of degree 2 (and arbitrary level) whose adelization generates a weak endoscopic lift. As a
Transfer of Siegel Cusp Forms of Degree 2
• Mathematics
• 2011
Introduction Notation Distinguished vectors in local representations Global L-functions for GSp? X GL? The pullback formula Holomorphy of global L-functions for GSp? X GL? Applications Bibliography
N T ] 1 4 A pr 2 01 3 ON RATIOS OF PETERSSON NORMS FOR YOSHIDA
• 2014

## References

SHOWING 1-10 OF 26 REFERENCES
L-functions for holomorphic forms on GSp(4) x GL(2) and their special values
We provide an explicit integral representation for L-functions of pairs (F, g), where F is a holomorphic genus two Siegel newform and g a holomorphic elliptic newform, both of square-free levels and
Bessel models for lowest weight representations of GSp(4,R)
• Mathematics
• 2008
We prove uniqueness and give precise criteria for existence of split and non-split Bessel models for a class of lowest and highest weight representations of the groups GSp(4,R) and Sp(4,R) including
Integral representation for L-functions for GSp(4) x GL(2), II
• Mathematics
• 2009
Based on Furusawa's theory, we present an integral representation for the L-function L(s,\pi \times \tau), where \pi is a cuspidal automorphic representation on GSp(4) related to a holomorphic Siegel
On L-functions for GSp(4) X GL(2) and their special values.
The purpose of this paper is to present a new integral representation of a degree eight L-function for GSp(4) x GL(2). In principle our method always works when both forms correspond to holomorphic
Bessel Models for Lowest Weight Representations of
• Mathematics
• 2009
We prove uniqueness and give precise criteria for existence of split and nonsplit Bessel models for a class of lowest and highest weight representations of the groups and including all holomorphic
On the arithmetic of Siegel-Hilbert cuspforms: Petersson inner products and Fourier coefficients
The latter is a sort of 'L-indistinguishability' result concerning the (presumably transcendental) Petersson norms-squared of Hecke eigenfunctions. The general assertion is essential in discussion of
Critical values of L-functions on GSp2 × Gl2
• Mathematics
• 2006
This paper deals with Deligne's conjecture on the critical values of L-functions. Let ZG⊗h(s) denote the tensor product L-function attached to a Siegel modular form G of weight k and an elliptic cusp
Iwahori-spherical representations of GSp(4) and Siegel modular forms of degree 2 with square-free level
A theory of local old- and newforms for representations of GSp(4) over a p-adic field with Iwahori-invariant vectors is developed. The results are applied to Siegel modular forms of degree 2 with
Bessel models for GSp(4)
• Mathematics
• 2007
Abstract Methods of theta correspondence are used to analyze local and global Bessel models for GSp4 proving a conjecture of Gross and Prasad which describes these models in terms of local epsilon