Some New Aspects of Mass Equidistribution

Abstract

This thesis presents two new results concerning the limiting behavior of families of automorphic forms. First, let F be a totally real number field of degree n, and let f traverse a sequence of nondihedral holomorphic eigencuspforms on GL(2,F) of trivial central character and full level. We show that the mass of $f$ equidistributes on the Hilbert modular variety as any one of the weight components of f tends to infinity. Our result answers affirmatively a natural analogue of a conjecture of Rudnick and Sarnak (1994). Our proof generalizes the argument of Holowinsky-Soundararajan (2008) who established the case F = Q. The essential difficulty in doing so is to adapt Holowinsky's bounds for the Weyl periods of the equidistribution problem in terms of manageable shifted convolution sums of Fourier coefficients to the case of a number field with nontrivial unit group. Second, let f traverse a sequence of classical holomorphic newforms of fixed weight and increasing squarefree level q tending to infinity. We prove that the pushforward of the mass of f to the modular curve of level 1 equidistributes with respect to the Poincare measure. Our result answers affirmatively the squarefree level case of a conjecture spelled out by Kowalski, Michel, and VanderKam (2002) in the spirit of a conjecture of Rudnick and Sarnak (1994). Our proof follows the strategy of Holowinsky and Soundararajan (2008) who showed that newforms of level 1 and large weight have equidistributed mass. The new ingredients required to treat forms of fixed weight and large level are an adaptation of Holowinsky's reduction of the problem to one of bounding shifted sums of Fourier coefficients, a refinement of his bounds for shifted sums, an evaluation of the p-adic integral needed to extend Watson's formula to the case of three newforms where the level of one divides but need not equal the common squarefree level of the other two, and some additional technical work in the problematic case that the level has many small prime factors.