Related papers: Explicit mean value theorems for toric periods and…

Partial zeta functions of algebraic varieties over finite fields generalize the classical zeta function by allowing each variable to be defined over a possibly different extension field of a fixed finite field. Due to this extra variation…

This thesis contributes to the analytic theory of automorphic L-functions. We prove an approximate functional equation for the central value of the L-series attached to an irreducible cuspidal automorphic representation of GL(m) over a…

Let $F$ be the function field of an elliptic curve $X$ over $\F_q$. In this paper, we calculate explicit formulas for unramified Hecke operators acting on automorphic forms over $F$. We determine these formulas in the language of the graph…

Let $f$ be a fixed (holomorphic or Maass) modular cusp form, with $L$-function $L(f,s)$. We describe an algorithm that computes the value $L(f,1/2+ iT)$ to any specified precision in time $O(1+|T|^{7/8})$.

Let $E/F$ be a CM extension of number fields, and let $H < G$ be a unitary Gan--Gross--Prasad pair defined with respect to $E/F$ that is compact at infinity. We consider a family $\mathcal{F}$ of automorphic representations of $G \times H$…

In this paper, we prove the following asymptotic formula for the spectral cubic moment of central $L$-values: $$ \sum_{t_f \leqslant T} \frac {2 L \big( \tfrac 1 2 , f \big)^3} {L(1, \mathrm{Sym}^2 f)} + \frac {2} {\pi} \int_{0}^{T} \frac…

In the paper we prove an explicit formula for the central values of certain Rankin L-functions. These L-functions are L-functions of Hilbert newforms over a totally real field F, twisted by unitary Hecke characters of a totally imaginary…

Let $Q(x)$ be a quadratic form over $\mathbb{R}^n$. The Epstein zeta function associated to $Q(x)$ is a well known function in number theory. We generalize the construction of the Epstein zeta function to a class of function $\phi(x)$…

We prove that the space of cuspidal quaternionic modular forms on the groups of type $F_4$ and $E_n$ have a purely algebraic characterization. This characterization involves Fourier coefficients and Fourier-Jacobi expansions of the cuspidal…

We prove a strong form of the trivial zero conjecture at the central point for the $p$-adic $L$-function of a non-critically refined self-dual cohomological cuspidal automorphic representation of $\mathrm{GL}_2$ over a totally real field,…

Let $L/F$ be a quadratic extension of totally real number fields. For any prime $p$ unramified in $L$, we construct a $p$-adic $L$-function interpolating the central values of the twisted triple product $L$-functions attached to a…

We prove a highly uniform version of the prime number theorem for a certain class of $L$-functions. The range of $x$ depends polynomially on the analytic conductor, and the error term is expressed in terms of an optimization problem…

Given $\mathbf{F}$ a number field with ring of integers $\mathcal{O}_{\mathbf{F}}$ and $\mathfrak{p},\mathfrak{q}$ two squarefree and coprime ideals of $\mathcal{O}_{\mathbf{F}}$, we prove a reciprocity relation for the first moment of the…

We prove Lindel\"of-on-average upper bounds on the cubic moment of central values of $L$-functions over certain families of $\operatorname{PGL}_2/\mathbb{Q}$ automorphic representations $\pi$ given by specifying the local representation…

We consider the group $\mathrm{Aut}(T)$ of isometries of a semi-homogeneous tree $T=T_{q_+,q_-}$ with valencies $q_+ +1$ and $q_- +1$ and its two orbits $V_+$, $V_-$ respectively. We make use of the action of $\mathrm{Aut} (T)$ to equip the…

We study Fourier coefficients of $GL_n(\A)$-automorphic functions $\phi$, for $\A$ being the adele group of a number field $\kkk$. Let FC be an abbreviation for such a Fourier coefficient (and FCs for plural). Roughly speaking, in the…

Following a strategy suggested by Michel--Venkatesh, we study the cubic moment of automorphic $L$-functions on $\operatorname{PGL}_2$ using regularized diagonal periods of products of Eisenstein series. Our main innovation is to produce…

We discuss computation of the special values of partial zeta functions associated to totally real number fields. The main tool is the \emph{Eisenstein cocycle} $\Psi $, a group cocycle for $GL_{n} (\Z )$; the special values are computed as…

Xian-Jin Li gave a criterion for the Riemann hypothesis in terms of the positivity of a set of coefficients lambda_n, indexed by the integers. We define similar coefficients attached to principal automorphic L-functions over GL(N). We…

In this note we study the symmetric powers of strongly modular icosahedral representations $\rho$ of ${\rm Gal} (\bar{F}/F)$, $F$ a number field, and their twisted $L$--functions. We prove that for such $\rho$, there exists a cuspidal…