Bounds for traces of Hecke operators and applications to modular and elliptic curves over a finite field

We give an upper bound for the trace of a Hecke operator acting on the space of holomorphic cusp forms with respect to certain congruence subgroups. Such an estimate has applications to the analytic theory of elliptic curves over a finite field, going beyond the Riemann hypothesis over finite fields. As the main tool to prove our bound on traces of Hecke operators, we develop a Petersson formula for newforms for general nebentype characters.Comment: Final version; to appear in Algebra & Number Theor

Moments of L'(1/2) in the Family of Quadratic Twists

We prove the asymptotic formulae for several moments of derivatives of GL(2) L-functions over quadratic twists. The family of L-functions we consider has root number fixed to -1 and odd orthogonal symmetry. Assuming GRH we prove the asymptotic formulae for (1) the second moment with one secondary term, (2) the moment of two distinct modular forms f and g and (3) the first moment with controlled weight and level dependence. We also include some immediate corollaries to elliptic curves via the modularity theorem and the work of Gross and Zagier.Comment: Many minor typos fixed and improvements in the writing made in this revision. To appear in IMR

A Twisted Motohashi Formula and Weyl-Subconvexity for LL-functions of Weight Two Cusp Forms

We derive a Motohashi-type formula for the cubic moment of central values of $L$-functions of level $q$ cusp forms twisted by quadratic characters of conductor $q$, previously studied by Conrey and Iwaniec and Young. Corollaries of this formula include Weyl-subconvex bounds for $L$-functions of weight two cusp forms twisted by quadratic characters, and estimates towards the Ramanujan-Petersson conjecture for Fourier coefficients of weight 3/2 cusp forms.Comment: One reference fixe

The Weyl law for algebraic tori

We give an asymptotic evaluation for the number of automorphic characters of an algebraic torus $T$ with bounded analytic conductor. The analytic conductor which we use is defined via the local Langlands correspondence for tori by choosing a finite dimensional complex algebraic representation of the $L$-group of $T$. Our results therefore fit into a general framework of counting automorphic representations on reductive groups by analytic conductor

Moments of Automorphic L-functions and Related Problems

We present in this dissertation several theorems on the subject of moments of automorphic L-functions. In chapter 1 we give an overview of this area of research and summarize our results. In chapter 2 we give asymptotic main term estimates for several different momentsof central values of L-functions of a fixed GL2 holomorphic cusp form f twisted by quadratic characters. When the sign of the functional equation of the twist L(s, f ⊗ χ_d) is −1, the central value vanishes and one instead studies the derivative L′(1/2, f ⊗ χ_d). We prove two theorems in the root number −1 case which are completely out of reach when the root number is +1. In chapter 3 we turn to an average of GL2 objects. We study the family of cuspforms of level q^2 which are given by f ⊗ χ, where f is a modular form of prime level q and χ is the quadratic character modulo q. We prove a precise asymptotic estimate uniform in shifts for the second moment with the purpose of understanding the off-diagonal main terms which arise in this family. In chapter 4 we prove an precise asymptotic estimate for averages of shifted con-volution sums of Fourier coefficients of full-level GL2 cusp forms over shifts. We find that there is a transition region which occurs when the square of the average overshifts is proportional to the length of the shifted sum. The asymptotic in this rangedepends very delicately on the constant of proportionality: its second derivative seems to be a continuous but nowhere differentiable function. We relate this phenomenon to periods of automorphic forms, multiple Dirichlet series, automorphic distributions, and moments of Rankin-Selberg L-functions

A Recursion Formula for Moments of Derivatives of Random Matrix Polynomials

We give asymptotic formulae for random matrix averages of derivatives of characteristic polynomials over the groups USp(2N), SO(2N) and O^-(2N). These averages are used to predict the asymptotic formulae for moments of derivatives of L-functions which arise in number theory. Each formula gives the leading constant of the asymptotic in terms of determinants of hypergeometric functions. We find a differential recurrence relation between these determinants which allows the rapid computation of the (k+1)-st constant in terms of the k-th and (k-1)-st. This recurrence is reminiscent of a Toda lattice equation arising in the theory of \tau-functions associated with Painlev\'e differential equations

Oscillatory integrals with uniformity in parameters

We prove a sharp asymptotic formula for certain oscillatory integrals that may be approached using the stationary phase method. The estimates are uniform in terms of auxiliary parameters, which is crucial for application in analytic number theory.Comment: Final version. To appear in Journal de Th\'eorie des Nombres de Bordeaux. Portions of this work originally appeared in arXiv:1608.06854 (Petrow-Young) and arXiv:1701.07507 (Kiral-Young). arXiv admin note: text overlap with arXiv:1701.0750

The volumes of Miyauchi subgroups

Miyauchi described the $L$ and $\varepsilon$-factors attached to generic representations of the unramified unitary group of rank three in terms of local newforms defined by a sequence of subgroups. We calculate the volumes of these Miyauchi groups.Comment: v2: typos corrected and some more direct citations adde