# Local spectral equidistribution for Siegel modular forms and applications

@article{Kowalski2012LocalSE, title={Local spectral equidistribution for Siegel modular forms and applications}, author={Emmanuel Kowalski and Abhishek Saha and Jacob Tsimerman}, journal={Compositio Mathematica}, year={2012}, volume={148}, pages={335 - 384} }

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 apply results concerning Bessel models and a variant of Petersson’s formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for…

## 39 Citations

Local spectral equidistribution for degree 2 Siegel modular forms in level and weight aspects

- Mathematics
- 2013

We prove an equidistribution statement for the Satake parameters of the local representations attached to Siegel cusp forms of degree $2$ of increasing level and weight, counted with a certain…

Explicit refinements of Böcherer's conjecture for Siegel modular forms of squarefree level

- Mathematics
- 2015

We formulate an explicit refinement of B\"ocherer's conjecture for Siegel modular forms of degree 2 and squarefree level, relating weighted averages of Fourier coefficients with special values of…

On the distribution of Satake parameters for Siegel modular forms

- Mathematics
- 2016

We prove a harmonically weighted equidistribution result for the $p$-th Satake parameters of the family of automorphic cuspidal representations of $\operatorname{PGSp}(2n)$ of fixed weight…

Mathematische Zeitschrift Local and global Maass relations

- Mathematics
- 2017

We characterize the irreducible, admissible, spherical representations of GSp4(F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their…

Local and global Maass relations (expanded version)

- Mathematics
- 2014

We characterize the irreducible, admissible, spherical representations of GSp(4,F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations satisfied by their…

On Bessel models for GSp 4 and Fourier coefficients of Siegel modular forms of degree 2

- Mathematics
- 2016

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…

Local and global Maass relations

- Mathematics
- 2017

We characterize the irreducible, admissible, spherical representations of $$\mathrm{GSp}_4(F)$$GSp4(F) (where F is a p-adic field) that occur in certain CAP representations in terms of relations…

On Fourier coefficients of Siegel modular forms of degree two with respect to congruence subgroups

- Mathematics
- 2011

We prove a formula of Petersson’s type for Fourier coefficients of Siegel cusp forms of degree 2 with respect to congruence subgroups, and as a corollary, show upper bound estimates of individual…

On Fourier coefficients and Hecke eigenvalues of Siegel cusp forms of degree 2

- Mathematics
- 2022

A BSTRACT . We investigate some key analytic properties of Fourier coefﬁcients and Hecke eigenvalues attached to scalar-valued Siegel cusp forms F of degree 2, weight k and level N . First, assuming…

## References

SHOWING 1-10 OF 78 REFERENCES

Ramanujan-type results for Siegel cusp forms of degree 2

- Mathematics
- 2009

A result of Chai-Faltings on Satake parameters of Siegel cusp forms together with the classification of unitary, unramified, irreducible, admissible representations of GSp4 over a p-adic field, imply…

Siegel modular forms and representations

- Mathematics
- 2001

Abstract: This paper explicitly describes the procedure of associating an automorphic representation of PGSp(2n,?) with a Siegel modular form of degree n for the full modular group Γn=Sp(2n,ℤ),…

Explicit formulas for the waldspurger and bessel models

- Mathematics
- 1994

This paper studies certain models of irreducible admissible representations of the split special orthogonal group SO(2n+1) over a nonarchimedean local field. Ifn=1, these models were considered by…

On critical values of L-functions for holomorphic forms on GSp(4) X GL(2)

- Mathematics
- 2009

Let F be a genus two Siegel newform and g a classical newform, both of squarefree levels and of equal weight $ell$. We derive an explicit integral representation for the degree eight L-function L(s,…

An inner product relation on Saito-Kurokawa lifts

- Mathematics
- 2007

Let f be a newform of weight 2k−2 and level M with M an odd square-free integer. Via the Saito-Kurokawa correspondence there is associated to f a Siegel newform Ff of weight k and level M. In this…

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

- Mathematics
- 2009

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…

Endoscopy for GSp(4) and the Cohomology of Siegel Modular Threefolds

- Mathematics
- 2009

An Application of the Hard Lefschetz Theorem.- CAP-Localization.- The Ramanujan Conjecture for Genus two Siegel modular Forms.- Character identities and Galois representations related to the group…

L-functions for holomorphic forms on GSp(4) x GL(2) and their special values

- Mathematics
- 2009

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…

Average Values of Modular L-series Via the Relative Trace Formula

- Mathematics
- 2005

First we reprove, using representation theory and the relative trace formula of Jacquet, an average value result of Duke for modular L-series at the critical center. We also establish a refinement.…

Automorphic Plancherel density theorem

- Mathematics
- 2012

Let F be a totally real field, G a connected reductive group over F, and S a finite set of finite places of F. Assume that G(F ⊗ℚ ℝ) has a discrete series representation. Building upon work of…