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Hybrid bounds for automorphic forms on ellipsoids over number fields
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 12 / Issue 4 / October 2013
- Published online by Cambridge University Press:
- 20 December 2012, pp. 727-758
- Print publication:
- October 2013
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- Article
- Export citation
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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 } $.
