Related papers: On the Rankin-Selberg method for vector valued Sie…

A well-known principle states that a congruence between objects should give rise to a corresponding congruence between the special values of $L$-functions attached to these objects. We computationally investigate this principle for…

We formulate and prove the analogue of the Ramanujan Conjectures for modular forms of half-integral weight subject to some ramification restriction in the setting of a polynomial ring over a finite field. This is applied to give an…

We introduce formulas for the logarithms of Drinfeld modules using a framework recently developed by the second author. We write the logarithm function as the evaluation under a motivic map of a product of rigid analytic trivializations of…

For $g=8,12,16$ and $24$, there is a nonzero alternating $g$-multilinear form on the ${\rm Leech}$ lattice, unique up to a scalar, which is invariant by the orthogonal group of ${\rm Leech}$. The harmonic Siegel theta series built from…

We proof a Garrett-B\"ocherer decomposition of a vector valued Siegel Eisenstein series $E_{l,0}^2$ of genus 2 transforming with the Weil representation of $\text{Sp}_2(\mathbb{Z})$ on the group ring $\mathbb{C}[(L'/L)^2]$. We show that the…

In this paper, some asymptotic formulas are proved for the harmonic mollified second moment of a family of Rankin-Selberg L-functions. One of the main new input is a substantial improvement of the admissible length of the mollifier which is…

\begin{abstract} In this article, we will get non-trivial estimates for the central values of degree six Rankin-Selberg $L$-functions $L(1/2+it, \pi \times f)$ associated with a ${GL(3)}$ form $\pi$ and a ${GL(2)} $ form $f$ using the delta…

In one of his final research papers, Alan Turing introduced a method to certify the completeness of a purported list of zeros of the Riemann zeta-function. In this paper we consider Turing's method in the analogous setting of Selberg…

Let $p$ be an odd prime, $ f$ be a $ p $-ordinary newform of weight $ k $ and $ h $ be a normalized cuspidal $ p $-ordinary Hecke eigenform of weight $ l < k$. In this article, we study the $p$-adic $ L $-function and $ p^{\infty} $-Selmer…

The object of this article is to construct certain classes of arithmetically significant, holomorphic Siegel cusp forms F of genus 2, which are neither of Saito-Kurokawa type, in which case the degree 4 spinor L-function L(s, F) is…

We describe a numerical algorithm for evaluating the numbers of roots minus the number of poles contained in a region based on the argument principle with the function of interest being written as a Mellin transformation of a usually…

Zagier proved that the traces of singular moduli, i.e., the sums of the values of the classical j-invariant over quadratic irrationalities, are the Fourier coefficients of a modular form of weight 3/2 with poles at the cusps. Using the…

In this paper, we prove some divisibility results for the Fourier coefficients of reduced modular forms of sign vectors. More precisely, we generalize a divisibility result of Siegel on constant terms when the weight is non-positive, which…

By the unfolding method, Rankin-Selberg L-functions for ${\rm GL}(n)\times{\rm GL}(m)$ can be expressed in terms of period integrals. These period integrals actually define invariant forms on tensor products of the relevant automorphic…

We prove an exact formula for the second moment of Rankin-Selberg $L$-functions $L(1/2,f \times g)$ twisted by $\lambda_f(p)$, where $g$ is a fixed holomorphic cusp form and $f$ is summed over automorphic forms of a given level $q$. The…

We investigate the twisting of motivic $L$-functions by a family of multiplicative characters $\psi$, defined on prime ideals $\mathfrak{p}$ via $\psi(\mathfrak{p})=\alpha^{N(\mathfrak{p})}$ for a fixed $\alpha \in \mathbb{C}$. One can…

We extend Venkatesh's proof of the converse theorem for classical holomorphic modular forms to arbitrary level and character. The method of proof, via the Petersson trace formula, allows us to treat arbitrary degree 2 gamma factors of…

Given a Hilbert cuspidal newform $g$ we construct a family of modular forms of half-integral weight whose Fourier coefficients give the central values of the twisted $L$-series of $g$ by fundamental discriminants. The family is parametrized…

We compute the second moment of a certain family of Rankin-Selberg $L$-functions L(f x g, 1/2) where f and g are Hecke-Maass cusp forms on GL(n). Our bound is as strong as the Lindel\"of hypothesis on average, and recovers individually the…

We apply the technique of localization for vertex algebras to the Segal-Sugawara construction of an ``internal'' action of the Virasoro algebra on affine Kac-Moody algebras. The result is a lifting of twisted differential operators from the…