Triple Product L Functions and Quantum Chaos on SL(2,C)

We extend the results of Watson, which link quantum unique ergodicity on arithmetic hyperbolic surfaces with subconvexity for the triple product L function, to the case of arithmetic hyperbolic three manifolds. We work with the full unitary dual of SL(2,C), and consider QUE for automorphic forms of arbitrary fixed weight and growing spectral parameter. We obtain our results by constructing microlocal lifts of nonspherical automorphic forms using representation theory, and quantifying the generalised triple product formula of Ichino in the case of complex places.Comment: 28 pages, submitte

Zero repulsion in families of elliptic curve L-functions and an observation of S. J. Miller

We provide a theoretical explanation for an observation of S. J. Miller that if L(s,E) is an elliptic curve L-function for which L(1/2, E) is nonzero, then the lowest lying zero of L(s,E) exhibits a repulsion from the critical point which is not explained by the standard Katz-Sarnak heuristics. We establish a similar result in the case of first-order vanishing.Comment: 9 pages, 2 figure

LpL^p norms of higher rank eigenfunctions and bounds for spherical functions

We prove almost sharp upper bounds for the $L^p$ norms of eigenfunctions of the full ring of invariant differential operators on a compact locally symmetric space, as well as their restrictions to maximal flat subspaces. Our proof combines techniques from semiclassical analysis with harmonic theory on reductive groups, and makes use of new asymptotic bounds for spherical functions that are of independent interest.Comment: 57 pages, 1 figure. Substantially revised, and close to the published versio