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EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE

Part of: Zeta and $L$-functions: analytic theory Partial differential equations on manifolds; differential operators Discontinuous groups and automorphic forms Forms and linear algebraic groups

Published online by Cambridge University Press:  02 April 2018

Valentin Blomer
Valentin Blomer*
Affiliation:
Mathematisches Institut, Bunsenstr. 3-5, 37073 Göttingen, Germany (vblomer@math.uni-goettingen.de)
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Abstract

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.

Keywords

Epstein zeta-functionquadratic formssubconvexitysup-normsSarnak’s purity conjecturemultiple Dirichlet series

MSC classification

Primary: 11M36: Selberg zeta functions and regularized determinants; applications to spectral theory, Dirichlet series, Eisenstein series, etc. Explicit formulas 11E45: Analytic theory (Epstein zeta functions; relations with automorphic forms and functions)
Secondary: 11F68: Dirichlet series in several complex variables associated to automorphic forms; Weyl group multiple Dirichlet series 58J50: Spectral problems; spectral geometry; scattering theory
Type
Research Article
Information
Journal of the Institute of Mathematics of Jussieu , Volume 19 , Issue 2 , March 2020 , pp. 581 - 596
Copyright
© Cambridge University Press 2018

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Footnotes

The author was supported in part by SNF-DFG grant BL 915/2 and NSF grant 1128155 while enjoying the hospitality of the Institute for Advanced Study. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notation herein.

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