Rankin–Selberg

Rankin–Selberg
Name	Rankin–Selberg
Field	Analytic number theory
Introduced	1939
Contributors	Rankin; Selberg

Rankin–Selberg is a method in analytic number theory connecting Fourier coefficients of modular forms to Dirichlet series and L-functions via integral transforms. It links ideas from the theory of Modular forms, Hecke operators, and Eisenstein series to produce analytic continuations and functional equations for convolution L-functions. The approach has influenced work in Automorphic forms, the Langlands program, and investigations by mathematicians at institutions such as Institute for Advanced Study, Princeton University, and École Normale Supérieure.

Introduction

The method originated in the study of Modular forms for SL(2,Z) and variations such as Congruence subgroups, where one pairs cusp forms with Eisenstein series to form integrals that represent Dirichlet series. Important figures include Atle Selberg, R. A. Rankin, Goro Shimura, Herbert S. Wilf, and Jacquet Langlands. The technique interacts with structures like Dirichlet characters, Maass forms, and Petersson inner product, and it informs results on Ramanujan–Petersson conjecture instances and bounds for Fourier coefficients.

Historical development

Early instances appeared in work of R. A. Rankin and Atle Selberg in the 1930s and 1940s, building on classical analyses of Ramanujan tau function and Hecke eigenform theory. Subsequent extensions were developed by Goro Shimura, Hans Maass, and Ilya Piatetski-Shapiro, connecting to the theory of Eisenstein series and integral representations studied at University of Cambridge and Columbia University. Later contributors include Jacques Tits, Robert Langlands, Friedrich Hirzebruch, Stephen Gelbart, and Dorian Goldfeld, who broadened the scope to adelic and representation-theoretic settings at institutions like Harvard University and Yale University.

Rankin–Selberg convolution and L-functions

The convolution produces L-functions combining coefficients from two automorphic entities such as a Holomorphic modular form and a Maass form or two Hecke eigenforms. Classical examples include the convolution L-function for a pair of cusp forms, generalizing the Dirichlet L-series and relating to the Dedekind zeta function in special cases. The resulting objects are central to the Langlands functoriality conjectures and tie into the Arthur–Selberg trace formula, Godement–Jacquet constructions, and work of Frey and Faltings on arithmetic applications.

Method of integral representations

The technique evaluates integrals over fundamental domains of groups like SL(2,R) or adelic quotients, pairing modular forms with Eisenstein series or Poincaré series to obtain Mellin transforms that equal Dirichlet series. Key tools include the Petersson formula, unfolding arguments used by Rankin and Selberg, and the use of intertwining operators from Representation theory of GL(n) groups. Adelic formulations were developed by Jacquet, Piatetski-Shapiro, and Shalika, while analytic preparations often invoke estimates from Iwaniec and Kowalski.

Applications in automorphic forms and number theory

Applications include subconvexity bounds for L-functions, nonvanishing results, and reciprocity laws predicted by Langlands reciprocity. The method underpins proofs of special value formulas connecting to Deligne conjecture variants, results on equidistribution like those of Duke, and relations to periods appearing in work of Waldspurger and Gan Gross Prasad. It has been applied to study zeros of zeta and L-functions related to Riemann zeta function, to bound Fourier coefficients influencing Sato–Tate conjecture investigations, and to arithmetic statistics pursued at Princeton University and University of Chicago.

Analytic properties and functional equations

Rankin–Selberg integrals yield meromorphic continuation and functional equations of convolution L-functions by relating integrals to Eisenstein series functional equations originally studied by Selberg. Analytic continuation arguments exploit the analytic properties of Eisenstein series from Langlands and spectral decompositions using the Selberg trace formula. Critical tools include gamma factors appearing in functional equations as in the Godement–Jacquet theory and local factors described by Tate's thesis and the local Langlands correspondence established by researchers like Harris, Taylor, and Henniart.

Generalizations and modern developments

Modern generalizations extend to higher rank groups such as GL(n), symmetric power L-functions, and Rankin–Selberg style integrals for groups like GSp(4) and SO(n). Researchers including Cogdell, Piatetski-Shapiro, Soudry, Bump, Friedberg, Harris, Li, Blomer, and Michel have developed integral representations, converse theorems, and subconvexity results. The landscape now blends analytic number theory, representation theory, and arithmetic geometry, interacting with programs at Institute for Advanced Study, Princeton University, Massachusetts Institute of Technology, and Stanford University.

Category:Analytic number theory