On the meromorphic continuation of Eisenstein series
Joseph Bernstein, Erez Lapid
Abstract
Eisenstein series are ubiquitous in the theory of automorphic forms. The traditional proofs of the meromorphic continuation of Eisenstein series, due to Selberg and Langlands, start with cuspidal Eisenstein series as a special case, and deduce the general case from spectral theory. We present a “soft” proof which relies only on rudimentary Fredholm theory (needed only in the number field case). It is valid for Eisenstein series induced from an arbitrary automorphic form. The proof relies on the principle of meromorphic continuation. It is close in spirit to Selberg’s later proofs.
Topics & Concepts
Meromorphic functionEisenstein seriesMathematicsAutomorphic formLanglands–Shahidi methodMathematical proofAnalytic continuationPure mathematicsAutomorphic L-functionSeries (stratigraphy)Artin L-functionContinuationAlgebra over a fieldModular formMathematical analysisGeometryPaleontologyProgramming languageConductorBiologyComputer scienceAdvanced Algebra and GeometryAnalytic Number Theory ResearchAdvanced Mathematical Identities