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Local spectral equidistribution for Siegel modular forms and applications

Part of: Discontinuous groups and automorphic forms

Published online by Cambridge University Press:  21 February 2012

Emmanuel Kowalski ,
Abhishek Saha  and
Jacob Tsimerman
Emmanuel Kowalski
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, 8092 Zürich, Switzerland (email: kowalski@math.ethz.ch)
Abhishek Saha
Affiliation:
ETH Zürich – D-MATH, Rämistrasse 101, 8092 Zürich, Switzerland (email: abhishek.saha@math.ethz.ch)
Jacob Tsimerman
Affiliation:
Mathematics Department, Princeton University, Fine Hall, Washington Road, Princeton, NJ 08540, USA (email: jtsimerm@math.princeton.edu)
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Abstract

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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 restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic nature of Fourier coefficients of Siegel cusp forms.

Keywords

Siegel cusp formsfamilies of L-functionslocal spectral equidistributionL-functionsBöcherer’s conjecturelow-lying zeros

MSC classification

Secondary: 11F30: Fourier coefficients of automorphic forms 11F46: Siegel modular groups; Siegel and Hilbert-Siegel modular and automorphic forms 11F66: Langlands $L$-functions; one variable Dirichlet series and functional equations 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols 11F70: Representation-theoretic methods; automorphic representations over local and global fields
Type
Research Article
Information
Compositio Mathematica , Volume 148 , Issue 2 , March 2012 , pp. 335 - 384
Copyright
Copyright © Foundation Compositio Mathematica 2012

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