Rankin selberg functional equation pdf

We know from the derivation of this formula that the integral and the sum must converge for. Except for the notable case of n 2, r, this lfunction is not known to correspond to an automorphic form. Analytic theory of lfunctions for gln proceedings of the school on automorphic forms, lfunctions, and number theory, the. We discuss the nonvanishing of the family of central values l1 2,f. The paper gives complete proofs of the properties of the rankinselberg integrals for the group gln.

The global integral admits a factorization into an euler product of local factors, called the local rankinselberg integrals. A note on the rankinselberg method rims, kyoto university. Vanderkam to all of c and a functional equation of the form l. It is well known that the central critical value l12. One approach to obtain the meromorphic continuation and functional equation would be the theory of rankinselberg integrals following the work of jacquet et al. We would like to show something similar for the rankin lseries.

Functional equation for padic rankinselberg lfunctions authors. Hybrid level aspect subconvexity for gl2 x gl1 rankin. To begin, we want to know if there is a formula describing an, where lf g,s. I certify that i have read this dissertation and that, in my opinion, it is fully adequate in scope and quality as a dissertation for the degree of doctor of philosophy. Rankinselberg integrals following the work of jacquet et al.

The centralvalue oftherankinselberg lfunctions arxiv. Selberg proves the functional equation and then states essentially the same the rankinselberg method. Root number of the rankinselberg convolution of two newforms. In this paper we calculate the asymptotics of various moments of the central values of rankinselberg convolution lfunctions of large level, thus generalizing the results and methods of dukefriedlanderiwaniec and the authors. Applying the rankinselberg identity 3 to the tinvariant function f.

Unlike rankin, seiberg treated the functional equations as the main point of the paper. On the nonvanishing of the central value of the rankinselberg lfunctions david ginzburg, dihua jiang, and stephen rallis dedicated to ilya i. We brie y describe the steps of the method, following cogdell cog06. The weight aspect alia hamieh and naomi tanabe abstract. Introduction and first notations polar divisor considerations proof of the main theorem acknowledgements.

Pdf we obtain the approximate functional equation for the rankinselberg zetafunction on the 12line. Equation 3 then gives the corresponding properties for the mellin transform of. Basic rankinselberg july 9, 2010 be the usual zeta function s x n 1 1 ns along with its gamma factor. Kazim buyukboduk, antonio lei submitted on 3 jul 2017 v1, last revised 25 oct 2017 this version, v2. The purpose of this paper is to prove that a primitive hilbert cusp form g is uniquely determined by the central values of the rankinselberg lfunctions lf g. In the diagonal zero frequency we save qt2k which is su cient if k t12. The subconvexity problem for rankinselberg lfunctions. Cohomological construction of the distribution 20 7. The rankinselberg method for studying langlands automorphic lfunctions is to nd integral representations for these functions. In proceedings of the session in analytic number theory and diophantine equations bonner mathematische schriften 360, university of bonn bonn, 2003, 19.

Abstract we prove a voronoi formula for coecients of a large class of lfunctions including maass cusp forms, rankinselberg convolutions, and certain isobaric sums. July 9, 2010 basic rankinselberg university of minnesota. We prove a functional equation for the threevariable p adic lfunction attached to the rankinselberg convolution of. To show is bounded in vertical strips and to find the location of the poles one could try using the constant terms of eisenstein series following the approach in 11. Introduction the theory of padic rankinselberg lfunctions for hida families. Each of these includes a discussion of the rankinselberg convolution and zeta function. The subconvexity problem for rankinselberg lfunctions 187 remark 1.

Rankinselberg lfunctions in the level aspect people. Piatetskishapiro with admiration on the occasion of his 75th birthday 1. A survey 55 consequences for the fourier coefficients without proof. The centralvalue oftherankinselberg lfunctions xiaoqingli abstract let f be a maass form for sl3,z which is. Functional equation for padic rankinselberg lfunctions by kazim buyukboduk and antonio lei download pdf 237 kb. Heegner points and nonvanishing of rankinselberg lfunctions. Proceedings of the xxii journees arithmetiques, lille, 2001. The functional equation of the eisenstein series is. We present the simplest possible example of the rankinselberg method, namely for a. Let f and g be two primitive holomorphic or maass cusp forms of arbitrary level, character and infinity parameter by which we mean the weight in the holomorphic case and the spectral parameter in the maass case. On the gcd of local rankinselberg integrals for even. In sections 5 and 6 we compute the first moment in two.

The formula for general n looks complicated a priori, so let us start by calculating ap, for p a prime number. In order to overcome this problem, one resorts to a trick attributed in 3 to goldfeld 5 which amounts to opening up the kloosterman sums and the jbessel functions, and then extracting. Hidden functional equations for rankinselberg transforms. Special value formulae of rankinselberg lfunctions penn math. Nontriviality of rankinselberg lfunctions and cm points. Our proof is based on the functional equations of lfunctions twisted by dirichlet characters and does not directly depend on automorphy. Nonvanishing of the central value of the rankinselberg lfunctions dihua jiang school of mathematics, university of minnesota. Introduction let fbe a primitive holomorphic cusp form of level pand. Next we recall the definition of the rankinselberg convolution l function and its functional equation. Heegner points and nonvanishing of rankinselberg lfunctions philippe michel and akshay venkatesh abstract. Hence it has wider application than previous proofs. Archimedean rankinselberg integrals columbia university. We discuss the nonvanishing of central values l1 2,f.

Pdf we obtain the approximate functional equation for the rankin selberg zeta function in the critical strip and, in particular, on the critical line. Sketch of proof i now we apply cauchy inequality, and our job reduces to saving t2 plus extra in x n. Find, read and cite all the research you need on researchgate. M 1, ma squarefree positive integer, and p a prime number. The universality of symmetric power lfunctions and their rankinselberg lfunctions li, hongze and wu, jie, journal of the mathematical society of japan, 2007. Real zeros and size of rankinselberg lfunctions in the level aspect ricotta, g. Special values of lfunctions, rankinselberg convolution, subconvexity, method 2010 mathematics subject classi cation. Rankinselberg lfunctions associated to the rankinselberg product of a coleman family and a cm hida family. Let g be a fixed modular form of full level, and let fj, k be a basis of holomorphic cuspidal newforms of even weight k, fixed level and fixed primitive nebentypus. Although its standard analytic properties analytic continuation, functional equation have been known for a while from the work of rankin, selberg and others, see j, js. Jacquet, which has analytic continuation and satisfies the functional equation.