Double Dirichlet series and quantum unique ergodicity of weight one-half Eisenstein series

Petridis, Yiannis; Raulf, Nicole; Risager, Morten

doi:10.2140/ant.2014.8.1539

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Double Dirichlet series and quantum unique ergodicity of weight one-half Eisenstein series

Yiannis N. Petridis, Nicole Raulf and Morten S. Risager

Vol. 8 (2014), No. 7, 1539–1595

DOI: 10.2140/ant.2014.8.1539

Abstract

The problem of quantum unique ergodicity (QUE) of weight $\frac{1}{2}$ Eisenstein series for $Γ_{0} (4)$ leads to the study of certain double Dirichlet series involving ${GL}_{2}$ automorphic forms and Dirichlet characters. We study the analytic properties of this family of double Dirichlet series (analytic continuation, convexity estimate) and prove that a subconvex estimate implies the QUE result.

Keywords

quantum limits, Eisenstein series, multiple Dirichlet series.

Mathematical Subject Classification 2010

Primary: 58J51

Secondary: 11F68, 11F72

Milestones

Received: 5 October 2012

Revised: 3 August 2014

Accepted: 8 September 2014

Published: 21 October 2014

Authors

Yiannis N. Petridis
Department of Mathematics University College London Gower street London WC1E 6BT United Kingdom

Nicole Raulf
Laboratoire Paul Painlevé U.F.R. de Mathématiques Université Lille 1 Sciences et Technologies 59 655 Villeneuve d’Asqc Cédex France

Morten S. Risager
Department of Mathematical Sciences University of Copenhagen Universitetsparken 5 DK-2100 Copenhagen Ø Denmark

Vol. 8, No. 7, 2014