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EPSTEIN ZETA-FUNCTIONS, SUBCONVEXITY, AND THE PURITY CONJECTURE

Published online by Cambridge University Press:  02 April 2018

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

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.

Type
Research Article
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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