#### Vol. 8, No. 7, 2014

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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
##### Abstract

The problem of quantum unique ergodicity (QUE) of weight $\frac{1}{2}$ Eisenstein series for ${\Gamma }_{0}\left(4\right)$ 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