Moments of Eisenstein Series on Convex Co-Compact Hyperbolic Manifolds

Abstract

ACKNOWLEDGEMENTSI would like to thank my supervisor, Dmitry Jakobson, for the opportunity of working under his guidance and for his encouragement and support throughout all the stages of this work. I am also grateful to him for suggesting the problem addressed in this thesis.It is a pleasure for me to thank Colin Guillarmou, to whom I am especially grateful for many helpful discussions, which contributed significantly to my general understanding of the field of research to which this work belongs. I must also thank him for his exceptional hospitality during my stay atÉcole Normale Supérieure de Paris.I would also like to thank Frédéric Naud for discussions and comments that were particularly helpful in the latter phase of the research. ii ABSTRACTWe investigate the high-energy limits of the moments of Eisenstein series, when these functions are considered as real random variables over a compact subset of a convex cocompact hyperbolic manifold. In the first part, under a restriction on the Hausdorff dimension of the limit set of the fundamental group of the manifold, we prove a general formula describing all the moments of the Eisenstein series at high-energy. In particular, we show that all the odd-order moments vanish. In the second part, we study the rate of convergence of the moments in the high-energy limit. In the case of the odd-order moments, we prove that the rate of convergence is at least polynomial. As for the even-order moments, following an approach based on the work of Guillarmou and Naud [16] concerning the equidistribution of Eisenstein series, we find a polynomial error term for the fourth moment under the additional hypothesis that the manifold in question is a surface.iii