Limit Theorems for Lattices and L-functions

Abstract

This PhD thesis investigates distributional questions related to three types of objects: Unimodular lattices, symplectic lattices, and Hecke L-functions of imaginary quadratic number fields of class number 1. In Paper I, we follow Södergren and examine the asymptotic joint distribution of a collection of random variables arising as geometric attributes of the N = N(n) shortest non-zero lattice vectors (up to sign) in a random unimodular lattice in n-dimensional Euclidean space, as the dimension n tends to infinity: Normalizations of the lengths of these vectors, and normalizations of the angles between them. We prove that under suitable conditions on N, this collection of random variables is asymptotically distributed like the first N arrival times of a Poisson process of intensity 1/2 and a collection of positive standard Gaußians. This generalizes previous work of Södergren. In Paper II, we use methods developed by Björklund and Gorodnik to study the error term in a classical lattice point counting asymptotic due to Schmidt in the context of symplectic lattices and a concrete increasing family of sets in 2n-dimensional Euclidean space. In particular, we show that this error term satisfies a central limit theorem as the volumes of the sets tend to infinity. Moreover, we obtain new Lp bounds on a height function on the space of symplectic lattices originally introduced by Schmidt. In Paper III, we follow Waxman and study a family of L-functions associated to angular Hecke characters on imaginary quadratic number fields of class number 1. We obtain asymptotic expressions for the 1-level density of the low-lying zeros in the family, both unconditionally and conditionally (under the assumption of the Grand Riemann Hypothesis and the Ratios Conjecture). Our results verify the Katz--Sarnak Density Conjecture in a special case for our family of L-functions.