Epstein zeta-functions, subconvexity, and the purity conjecture

TL;DR

This paper establishes subconvexity bounds for Epstein zeta functions associated with k-ary quadratic forms, revealing that Sarnak's purity conjecture does not hold for Eisenstein series when k is odd.

Contribution

It provides the first subconvexity bounds for general Epstein zeta functions and determines the exact sup-norm exponent for Eisenstein series on GL(k).

Findings

Subconvexity bounds are proved for Epstein zeta functions.

The exact sup-norm exponent for Eisenstein series on GL(k) is (k-2)/8.

Sarnak's purity conjecture fails for Eisenstein series when k is odd.

Abstract

Subconvexity bounds are proved for general Epstein zeta functions of k-ary quadratic forms. This is related to sup-norm bounds for Eisenstein series on GL(k), and the exact sup-norm exponent is determined to be (k-2)/8 for k >= 2. In particular, if $k$ is odd, this exponent is not in Z/4, which shows that Sarnak's purity conjecture does not hold for Eisenstein series.