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

Following the prequel work \cite{VO3}, we prove a generalization of "Mazur's conjecture" for $L$-functions of elliptic curves in abelian extensions of imaginary quadratic fields, including the assertion that the Mordell-Weil rank of an…

We prove a weak version of Beilinson's conjecture for non-critical values of $L$-functions for the Rankin-Selberg product of two modular forms.

We develop two structure theorems for vector valued Siegel modular forms for Igusa's subgroup \Gamma_2[2,4], the multiplier system induced by the theta constants and the representation Sym^2. In the proof, we identify some of these modular…

Assuming the Generalized Riemann Hypothesis, we provide explicit upper bounds for moduli of $\log{\mathcal{L}(s)}$ and $\mathcal{L}'(s)/\mathcal{L}(s)$ in the neighbourhood of the 1-line when $\mathcal{L}(s)$ are the Riemann, Dirichlet and…

Recently, Allen, Grove, Long, and Tu proposed an explicit Hypergeometric-Modularity method which gives a concrete link between certain hypergeometric objects and modular forms. The theory is exemplified by a collection of 199 weight 3…

We calculate certain "wide moments" of central values of Rankin--Selberg $L$-functions $L(\pi\otimes \Omega, 1/2)$ where $\pi$ is a cuspidal automorphic representation of $\mathrm{GL}_2$ over $\mathbb{Q}$ and $\Omega$ is a Hecke character…

Many important analytic statements about automorphic forms, such as the analytic continuation of certain L-functions, rely on the well-known rapid decay of K-finite cusp forms on Siegel sets. We extend this here to prove a more general…

Let $X$ be an orbisurface, meaning a compact hyperbolic Riemann surface possibly with a finite number of elliptic points, and let $X_1$ denote its unit tangent bundle. We consider the twisted Selberg zeta function $Z(s;\rho)$ associated to…

We obtain the basic analytic properties, i.e. meromorphic continuation, polar structure and bounds for the order of growth, of all the nonlinear twists with exponents $\leq 1/d$ of the L-functions of any degree $d \geq 1$ in the extended…

We derive integral representations for the Rankin-Selberg L-functions on GL(3) x GL(1) and GL(3) x GL(2) by a process of unipotent averaging at archimedean places. A key feature of our result is that it allows one to fix the choice of test…

The author reviews results and conjectures of Selberg on a class of Dirichlet series functions which share properties with the Riemann zeta function, and he relates this work to the theory of Artin L-functions.

In this article we show analytic properties of certain Rankin-Selberg type Dirichlet series for holomorphic Jacobi cusp forms of integral weight and of half-integral weight. The numerators of these Dirichlet series are the inner products of…

This paper gives a simple method for constructing vector-valued Siegel modular forms from scalar-valued ones. The method is efficient in producing the siblings of Delta, the smallest weight cusp forms that appear in low degrees. It also…

The theta correspondence has been an important tool in studying cycles in locally symmetric spaces of orthogonal type. We generalize the Kudla-Millson relation between intersection numbers of cycles and Fourier coefficients of Siegel…

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…

In this paper we prove that the $p$-adic $L$-function that interpolates the Rankin-Selberg product of a general weight two modular form which is unramified and non-ordinary at $p$, and an ordinary CM form of higher weight contains the…

This article studies the finite--slope analogue of Loeffler's conjectural framework for Rankin--Selberg $p$-adic $L$-functions in universal deformation families. Starting from residual representations $\bar\rho_1,\bar\rho_2$ of tame…

We establish standard zero-free regions with no exceptional Landau-Siegel zeros for Rankin-Selberg $L$-functions and triple product $L$-functions in several new families for which modularity is not yet known.

The purpose of this paper is to prove that a primitive Hilbert cusp form $\mathbf{g}$ is uniquely determined by the central values of the Rankin-Selberg $L$-functions $L(\mathbf{f}\otimes\mathbf{g}, \frac{1}{2})$, where $\mathbf{f}$ runs…

The modular transformation behavior of theta series for indefinite quadratic forms is well understood in the case of elliptic modular forms due to Vign\'eras, who deduced that solving a differential equation of second order serves as a…