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Hybrid bounds for automorphic forms on ellipsoids over number fields

Part of: Discontinuous groups and automorphic forms Forms and linear algebraic groups Algebraic number theory: global fields Partial differential equations on manifolds; differential operators

Published online by Cambridge University Press:  20 December 2012

Valentin Blomer  and
Philippe Michel
Valentin Blomer
Affiliation:
Mathematisches Institut, Universität Göttingen, Bunsenstr. 3-5, 37073 Göttingen, Germany ([email protected])
Philippe Michel
Affiliation:
EPFL/SB/IMB/TAN, Station 8, CH-1015 Lausanne, Switzerland ([email protected])
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Abstract

We prove upper bounds for Hecke–Laplace eigenfunctions on certain Riemannian manifolds $X$ of arithmetic type, uniformly in the eigenvalue and the volume of the manifold. The manifolds under consideration are $d$-fold products of $2$-spheres or $3$-spheres, realized as adelic quotients of quaternion algebras over totally real number fields. In the volume aspect we prove a (‘Weyl-type’) saving of $\mathrm{vol} \hspace{0.167em} (X)^{- 1/ 6+ \varepsilon } $.

Keywords

definite quaternion algebrastrace formulasup-normshybrid boundsnorm formsspherical harmonics

MSC classification

Secondary: 11R52: Quaternion and other division algebras 11F70: Representation-theoretic methods; automorphic representations over local and global fields 11F72: Spectral theory; Selberg trace formula 11E20: General ternary and quaternary quadratic forms; forms of more than two variables 58J50: Spectral problems; spectral geometry; scattering theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 12 , Issue 4 , October 2013 , pp. 727 - 758
Copyright
©Cambridge University Press 2012 

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