The Quantum Unique Ergodicity and the L4-norm Problems for Newform Eisenstein Series
Abstract
This thesis deals with two closely related problems about Eisenstein series on varying levels, both of which stem from the Random Wave Conjecture.
The first problem is quantum unique ergodicity for Eisenstein series in the level aspect. With a fixed nice test function, we see equidistribution as the level grows. A new feature for the level aspect is a term of the logarithmic derivative of the Dirichlet L-function, which connects quantum unique ergodicity and Siegel zeroes. Going one step further, we let the test function change with the growth of level in the manner analogous to the recently known results on quantum unique ergodicity on shrinking sets, and surprisingly, we observe some distorting behavior.
The second problem is bounding the regularized L4-norm for newform Eisenstein series. We manage to express the fourth moment as an average of L-functions.
Citation
Pan, Jiakun (2020). The Quantum Unique Ergodicity and the L4-norm Problems for Newform Eisenstein Series. Doctoral dissertation, Texas A&M University. Available electronically from https : / /hdl .handle .net /1969 .1 /192783.