2 results
EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE
- Part of
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- Journal:
- Journal of the Institute of Mathematics of Jussieu / Volume 19 / Issue 2 / March 2020
- Published online by Cambridge University Press:
- 02 April 2018, pp. 581-596
- Print publication:
- March 2020
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- Article
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Subconvexity bounds on the critical line are proved for general Epstein zeta-functions of
$k$-ary quadratic forms. This is related to sup-norm bounds for unitary Eisenstein series on 
$\text{GL}(k)$ associated with the maximal parabolic of type 
$(k-1,1)$, and the exact sup-norm exponent is determined to be 
$(k-2)/8$ for 
$k\geqslant 4$. In particular, if 
$k$ is odd, this exponent is not in 
$\frac{1}{4}\mathbb{Z}$, which is relevant in the context of Sarnak’s purity conjecture and shows that it can in general not directly be generalized to Eisenstein series.
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
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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 } $.
