Related papers: Non annulation des fonctions L automorphes au poin…

We use relative trace formula to prove a non-vanishing result and a subconvexity result for the twisted base change $L$-functions associated to Hilbert modular forms whose local components at ramified places are some supercuspidal…

In this paper, we show the nonvanishing of some Hecke characters on cyclotomic fields. The main ingredient of this paper is a computation of eigenfunctions and the action of Weil representation at some primes including the primes above $2$.…

Let $M_k^\sharp(N)$ be the space of weight $k$, level $N$ weakly holomorphic modular forms with poles only at the cusp at $\infty$. We explicitly construct a canonical basis for $M_k^\sharp(N)$ for $N\in\{8,9,16,25\}$, and show that many of…

We present a proof of Selberg's Central Limit Theorem for automorphic $L$-functions of degree 2 using Radziwi\l\l\space and Soundararajan's method. Additionally, we prove the independence of the automorphic $L$-functions associated with the…

We prove three main results: all Langlands-Shahidi automorphic $L$-functions over function fields are rational; after twists by highly ramified characters our automorphic $L$-functions become polynomials; and, if $\pi$ is a globally generic…

We prove an asymptotic formula for the second moment of automorphic L-functions of even weight and prime power level. The error term is estimated uniformly in all parameters: level, weight, shift and twist.

In the early 1980s, Rohrlich began a study of canonical Hecke characters, which are closely related to the simplest examples of CM elliptic curves. He and Montgomery showed the non-vanishing of the central value when the L-function has an…

We give a geometric criterion for Dirichlet $L$-functions associated to cyclic characters over the rational function field $\mathbb{F}_q(t)$ to vanish at the central point $s=1/2$. The idea is based on the observation that vanishing at the…

Given elliptic modular forms f and g satisfying certain conditions on their weights and levels, we prove (a quantitative version of the statement) that there exist infinitely many imaginary quadratic fields K and characters chi of the ideal…

We describe algorithms for computing central values of twists of $L$-functions associated to Hilbert modular forms, carry out such computations for a number of examples, and compare the results of these computations to some heuristics and…

In this article, we study the density conjecture of Katz and Sarnak for $L$-functions of ad\'elic Hilbert modular forms and their convolutions. In particular, under the generalised Riemann hypothesis, we establish several instances…

The Katz-Sarnak philosophy predicts that the behavior of zeros near the central point in families of $L$-functions agrees with that of eigenvalues near 1 of random matrix ensembles. Under GRH, Iwaniec, Luo and Sarnak showed agreement in the…

We prove two principal results. Firstly, we characterise Maass forms in terms of functional equations for Dirichlet series twisted by primitive characters. The key point is that the twists are allowed to be meromorphic. This weakened…

Let $K$ be an imaginary quadratic field. In this article, we study the eigenvariety for $\mathrm{GL}_2/K$, proving an \'etaleness result for the weight map at non-critical classical points and a smoothness result at base-change classical…

We prove the exceptional zero conjecture for the symmetric powers of CM cuspidal eigenforms at ordinary primes. In other words, we determine the trivial zeroes of the associated p-adic L-functions, compute the L-invariants, and show that…

We prove the following statement about any Siegel modular form $F$ of degree $n$ and arbitrary odd level $N$ on the group $\Gamma_{0}^{(n)}(N)$. Let $A(F,T)$ denote the Fourier coefficients of $F$ and write $T=(T(i,j))$. Suppose that $F$…

The aim of this paper is to prove inequalities towards instances of the Bloch-Kato conjecture for Hilbert modular forms of parallel weight two, when the order of vanishing of the $L$-function at the central point is zero or one. We achieve…

It has been conjectured that varieties of general type do not admit nowhere vanishing holomorphic one-forms. We confirm this conjecture for smooth minimal varieties and for varieties whose Albanese variety is simple.

In this paper, we extend previous results to prove that generalized modular forms with rational Fourier expansions whose divisors are supported only at the cusps and certain other points in the upper half plane are actually classical…

Let $\lambda$ be a self-dual Hecke character over an imaginary quadratic field $K$ of infinity type $(1,0)$. Let $\ell$ and $p$ be primes which are coprime to $6N_{K/\mathbb{Q}}({\mathrm cond}(\lambda))$. We determine the $\ell$-adic…