Rankin Selberg Convolution Research Articles

Modular symbols for reductive groups and p-adic Rankin–Selberg convolutions over number fields

We give a construction of a wide class of modular symbols attached to reductive groups. As an application we construct a 𝔭-adic distribution interpolating the special values of the twisted Rankin–Selberg L-function attached to cuspidal automorphic representations π and σ of GLn and GLn–1 over a number field k. If π and σ are ordinary at 𝔭, our distribution is bounded and yields analyticity of the associated 𝔭-adic L-function.

The second moment of [formula omitted]L-functions, integrated

We consider the family of Rankin–Selberg convolution L-functions of a fixed SL ( 3 , Z ) Maass form with the family of Hecke–Maass cusp forms on SL ( 2 , Z ) . We estimate the second moment of this family of L-functions with a “long” integration in t-aspect. These L-functions are distinguished by their high degree (12) and large conductors (of size T 12 ).

Computation of the Rankin–Selberg L-functions of Maass wave forms

This note outlines a method for numerically computing the Rankin–Selberg convolutions of Maass wave forms L-functions and reports on the computation of zeros of some of them. Bibliography: 11 titles.

RANKIN’S METHOD AND JACOBI FORMS OF SEVERAL VARIABLES

AbstractFollowing R. A. Rankin’s method, D. Zagier computed the nth Rankin–Cohen bracket of a modular form g of weight k1 with the Eisenstein series of weight k2, computed the inner product of this Rankin–Cohen bracket with a cusp form f of weight k=k1+k2+2n and showed that this inner product gives, up to a constant, the special value of the Rankin–Selberg convolution of f and g. This result was generalized to Jacobi forms of degree 1 by Y. Choie and W. Kohnen. In this paper, we generalize this result to Jacobi forms defined over ℋ×ℂ(g,1).

On Li’s coefficients for the Rankin–Selberg L-functions

We define a generalized Li coefficient for the L-functions attached to the Rankin–Selberg convolution of two cuspidal unitary automorphic representations π and π′ of \(GL_{m}(\mathbb{A}_{F})\) and \(GL_{m^{\prime }}(\mathbb{A}_{F})\) . Using the explicit formula, we obtain an arithmetic representation of the n th Li coefficient \(\lambda _{\pi ,\pi ^{\prime }}(n)\) attached to \(L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime })\) . Then, we deduce a full asymptotic expansion of the archimedean contribution to \(\lambda _{\pi ,\pi ^{\prime }}(n)\) and investigate the contribution of the finite (non-archimedean) term. Under the generalized Riemann hypothesis (GRH) on non-trivial zeros of \(L(s,\pi _{f}\times \widetilde{\pi}_{f}^{\prime })\) , the nth Li coefficient \(\lambda _{\pi ,\pi ^{\prime }}(n)\) is evaluated in a different way and it is shown that GRH implies the bound towards a generalized Ramanujan conjecture for the archimedean Langlands parameters μπ(v,j) of π. Namely, we prove that under GRH for \(L(s,\pi _{f}\times \widetilde{\pi}_{f})\) one has \(|\mathop {\mathrm {Re}}\mu_{\pi}(v,j)|\leq \frac{1}{4}\) for all archimedean places v at which π is unramified and all j=1,…,m.

ANALYTIC CONTINUATION OF NONANALYTIC VECTOR-VALUED EISENSTEIN SERIES

In this paper, we introduce a vector-valued nonanalytic Eisenstein series appearing naturally in the Rankin–Selberg convolution of a vector-valued modular cusp form associated to a monomial representation ρ of SL(2,ℤ). This vector-valued Eisenstein series transforms under a representation χρ associated to ρ. We use a method of Selberg to obtain an analytic continuation of this vector-valued nonanalytic Eisenstein series to the whole complex plane.

Hauteur asymptotique des points de Heegner

AbstractGeometric intuition suggests that the Néron–Tate height of Heegner points on a rational elliptic curveEshould be asymptotically governed by the degree of its modular parametrisation. In this paper, we show that this geometric intuition asymptotically holds on average over a subset of discriminants. We also study the asymptotic behaviour of traces of Heegner points on average over a subset of discriminants and find a difference according to the rank of the elliptic curve. By the Gross–Zagier formulae, such heights are related to the special value at the critical point for either the derivative of the Rankin–Selberg convolution ofEwith a certain weight one theta series attached to the principal ideal class of an imaginary quadratic field or the twistedL-function ofEby a quadratic Dirichlet character. Asymptotic formulae for the first moments associated with theseL-series andL-functions are proved, and experimental results are discussed. The appendix contains some conjectural applications of our results to the problem of the discretisation of odd quadratic twists of elliptic curves.

On characteristic twists of multiple Dirichlet series associated to Siegel cusp forms

We define a twisted two complex variables Rankin-Selberg convolution of Siegel cusp forms of degree 2. We find its group of functional equations and prove its analytic continuation to $${\mathbb{C}^2}$$ . As an application we obtain a non-vanishing result for special values of the Fourier Jacobi coefficients. We also prove the analytic properties for the characteristic twists of convolutions of Jacobi cusp forms.

The Rankin-Selberg convolution for real analytic Cohen's Eisenstein series of half-integral weight

We give a meromorphic continuation and a functional equation for the Rankin�Selberg convolution of certain real analytic Eisenstein series of half-integral weight. Our result and method have several applications to the Koecher�Maass series associated with the real analytic Siegel�Eisenstein series.

On a vanishing conjecture appearing in the geometric Langlands correspondence

0.1. This paper should be regarded as a sequel to [7]. There it was shown that the geometric Langlands conjecture for GLn follows from a certain vanishing conjecture. The goal of the present paper is to prove this vanishing conjecture. Let X be a smooth projective curve over a ground field k. Let E be an m-dimensional local system on X, and let Bunm be the moduli stack of rank m vector bundles on X. The geometric Langlands conjecture says that to E we can associate a perverse sheaf FE on Bunm, which is a Hecke eigensheaf with respect to E. The vanishing conjecture of [7] says that for all integers n < m, a certain functor AvE , depending on E and a parameter d ∈ Z+, which maps the category D(Bunn) to itself, vanishes identically, when d is large enough. The fact that the vanishing conjecture implies the geometric Langlands conjecture may be regarded as a geometric version of the converse theorem. Moreover, as will be explained in the sequel, the vanishing of the functor AvE is analogous to the condition that the Rankin-Selberg convolution of E, viewed as an m-dimensional Galois representation, and an automorphic form on GLn with n < m is well-behaved. Both the geometric Langlands conjecture and the vanishing conjecture can be formulated in any of the sheaf-theoretic situations, e.g., Q -adic sheaves (when char(k) = ), D-modules (when char(k) = 0), and sheaves with coefficients in a finite field F (again, when char(k) = ). When the ground field is the finite field Fq and we are working with -adic coefficients, it was shown in [7] that the vanishing conjecture can be deduced from Lafforgue’s theorem that establishes the full Langlands correspondence for global fields of positive characteristic; cf. [9].

Mean square values of Hecke L–series formed with r–th order characters

Let L be a number field containing the r–th roots of unity. Starting with the Rankin-Selberg convolution of a metaplectic Eisenstein series on the r–fold cover of GL(2) with itself, we construct a Dirichlet series defined over L whose coefficients involve the r–th order twists of a fixed Hecke L–function. We then observe that a group of functional equations can be naturally associated with this construction. Combining this with the convexity theorem for holomorphic functions of several complex variables, we show that this object, as a function of two complex variables, admits meromorphic continuation to ℂ2. As an application, we obtain asymptotic formulae for mean square values of the r–th order twists of an arbitrary Hecke L–function defined over L.

On a Rankin-Selberg convolution of two variables for Siegel modular forms

On a Rankin-Selberg convolution of two variables for Siegel modular forms

Period relations and p-adic measures

Let (π,σ) be a pair of cuspidal automorphic representations of GL n × GL n −1 over the adele ring of ℚ having non-vanishing cohomology with constant coefficients. The p-adic distribution interpolating the critical values of the twisted corresponding Rankin–Selberg convolution is shown to be p-adically bounded thus leading to an associated p-adic L-function.

The local Rankin-Selberg convolution for GL( n ): divisibility of the conductor

Let F be a non-Archimedean local field and \(n\geq 2\) an integer. Let \(\pi, \pi'\) be irreducible supercuspidal representations of GL\(_n(F)\) with \(\pi\not\cong\pi'\). One knows that there exists an irreducible supercuspidal representation \(\rho\) of GL\(_m(F)\), with \(m < n\), such that the local constants (in the sense of Jacquet, Piatetskii-Shapiro and Shalika) \(\varepsilon(\pi\times\rho,s,\psi),\varepsilon(\pi'\times\rho,s,\psi)\) are distinct. In this paper, we show that, when \(\pi'\) is an unramified twist \(\chi\pi\) of \(\pi\), one may here takem dividingn and \(\leq n/\ell\), for a prime divisor \(\ell\) ofn depending on \(\pi\) and the order of \(\chi\): in particular, \(m\leq n/\ell_0\), where \(\ell_0\) is the least prime divisor of \(n\). This follows from a result giving control of certain divisibility properties of the conductor of a pair of supercuspidal representations.

Local geometrized Rankin-Selberg method for GL( n) and its application

We propose a geometric interpretation of the classical Rankin-Selberg method for GL( n) in the framework of the geometric Langlands program. Our main result is local. For n = 2 we apply it to prove a global result that gives a geometric version of the computation of the scalar product of two normalized cusp Hecke eigenforms as a residue of the Rankin-Selberg convolution.

On standard L-functions attached to automorphic forms on definite orthogonal groups

Abstract.We show an explicit functional equation of the standard L-function associated with an automorphic form on a definite orthogonal group over a totally real algebraic number field. This is a completion and a generalization of our previous paper, in which we constructed standard L-functions by using Rankin-Selberg convolution and the theory of Shintani functions under certain technical conditions. In this article we remove these conditions. Furthermore we show that the L-function of f has a pole at s = m/2 if and only if f is a constant function.

Distinguished representations of metaplectic groups

We propose a conjecture that automorphic distinguished representations on n -fold covering groups of GL ( nr ) are parametrized by automorphic cuspidal representations of GL ( r ). We also give a local result on unramified distinguished representations; noncuspidal automorphic distinguished representations are constructed by Eisenstein series; and some types of Rankin-Selberg convolutions of a metaplectic form against a distinguished form are shown to have Euler products.

Local Rankin-Selberg convolutions for 𝐺𝐿_{𝑛}: Explicit conductor formula

Let F F be a non-Archimedean local field and n 1 n_{1} , n 2 n_{2} positive integers. For i = 1 , 2 i=1,2 , let G i = G L n i ( F ) G_{i}=\mathrm {GL}_{n_{i}}(F) and let π i \pi _{i} be an irreducible supercuspidal representation of G i G_{i} . Jacquet, Piatetskii-Shapiro and Shalika have defined a local constant ε ( π 1 × π 2 , s , ψ ) \varepsilon (\pi _{1}\times \pi _{2},s,\psi ) to the π i \pi _{i} and an additive character ψ \psi of F F . This object is of central importance in the study of the local Langlands conjecture. It takes the form ε ( π 1 × π 2 , s , ψ ) = q − f s ε ( π 1 × π 2 , 0 , ψ ) , \begin{equation*}\varepsilon (\pi _{1}\times \pi _{2},s,\psi ) = q^{-fs}\varepsilon (\pi _{1} \times \pi _{2},0,\psi ), \end{equation*} where f = f ( π 1 × π 2 , ψ ) f=f(\pi _{1}\times \pi _{2},\psi ) is an integer. The irreducible supercuspidal representations of G = G L n ( F ) G=\mathrm {GL}_{n}(F) have been described explicitly by Bushnell and Kutzko, via induction from open, compact mod centre, subgroups of G G . This paper gives an explicit formula for f ( π 1 × π 2 , ψ ) f(\pi _{1} \times \pi _{2},\psi ) in terms of the inducing data for the π i \pi _{i} . It uses, on the one hand, the alternative approach to the local constant due to Shahidi, and, on the other, the general theory of types along with powerful existence theorems for types in G L ( n ) \mathrm {GL}(n) , developed by Bushnell and Kutzko.

On the gamma factors attached to representations ofU(2,1) over ap-adic field

In this paper we complete the local non-archimedean theory of Rankin-Selberg convolutions forU(2,1) that was suggested by S. Gelbart and I. Piatetski-Shapiro in [G,PS,1]. In addition, we prove two fundamental properties of the gamma factors (Theorem 6.3) which allow us to give a new proof of a strong multiplicity one theorem forU(2,1).

Distribution of lattice points on surfaces of second order

Let f1(x1,..., xl1) and f2(y1,...,yl2) be positive definite primitive quadratic forms in l1 and l2 variables, respectively. We obtain new results in the well-known problem on the number of lattice points on the cone f1(x1,...,xl1)=f2(y1,...,yl2), in the domain f1(x1,...,xl1)≦N for N»∞. Our technical tool is the Rankin-Selberg convolution. In several special cases the results can be sharpened by other methods. In addition, new facts concerning the uniform distribution of lattice points on ellipsoids in l variables, l odd, l≧5 are obtained.