On the cohomology and quantum chaos of the general linear group in two variables.

摘要

The subject of this thesis is the cohomology and quantum unique ergodicity of various arithmetic manifolds arising from quaternion algebras over a number field.;Our first main theorem bounds the number of cohomological forms of fixed level and growing weight when the associated locally symmetric space is a hyperbolic 3-manifold. To state it in a special case, let Gamma ⊂ SL(2, C ) be a congruence lattice and Ed the restriction of the representation Symd ⊗ Symd of SL(2, C ) to Gamma. Classes in H1(Gamma, Ed) correspond to cohomological forms on Gamma SL(2, C ), and we are able to improve the trivial bound dim H 1(Gamma, Ed) d 2 for the dimension of these groups by a power to d2-delta. Our proof involves choosing an auxiliary prime p and applying a theorem of Calegari and Emerton on Fp cohomology growth in the level aspect, which we transfer to the weight aspect by a reduction mod p argument. We also prove that cohomological forms on a quaternion algebra over any number field must have the same weights as a form base changed from a totally real subfield.;Our first result on QUE deals with Hecke-Maass eigenforms of large eigenvalue on arithmetic quotients of SL(2, C ). We construct representation theoretic microlocal lifts for every element of the unitary dual of SL(2, C ) following Silberman and Venkatesh, and show that QUE for these lifts is implied by a subconvex bound for a triple product L-function.;Our second main theorem establishes QUE for cohomological forms on GL2 over an arbitrary number field. Assuming Ramanujan, we show that the mass of cohomological forms of fixed level and growing weight becomes equidistributed, generalising work of Holowinsky and Soundararajan. In particular, our theorem is unconditional over totally real and imaginary quadratic fields. We use Holowinsky and Soundararajan's methods, applying Soundararajan's weak subconvexity to certain triple product L-functions, and adapting Holowinsky's sieve method to the more complicated structure of the cusp in the presence of units. In the totally real case, our result implies that the zero divisors of holomorphic Hecke eigenforms of large weight become equidistributed, generalising a result of Rudnick.