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Vignettes on automorphic and modular forms,representations, L-functions, and number theoryVignettes on automorphic forms,representations, L-functions, and number theory[ambient page updated 27 Sep '08]...[ home ]...[ garrett@math.umn.edu ] ( Also: [ functional analysis ]...[ intro to modular forms ]...[ buildings notes ]...[ number theory ]...[ algebra ]) Talks at ICMS, Edinburgh, August 2008: [Integral moments I]... [ updated 09 Aug '08]Overview of spectral identities for integral moments for GL(n)xGL(n-1) L-functions over number fields. Subconvexity in t-aspect for GL(2) over number fields as evidence for non-triviality of these identities. [Integral moments IIIa]... [ updated 09 Aug '08]General recipe to produce spectrally meaningful integral second moments for all Rankin-Selberg integrals. Examples: GL(n)xGL(n-1) Hecke-type convolutions, GL(n)xGL(n) Rankin-Selberg convolutions, triple-product L-functions, all doubling integrals for classical groups. Heuristic concerning extraction of subconvex bounds. [ characters of principal series ]... [ updated 15 Aug '08] expressed in terms of orbital integrals, without proof that the operators are of trace class. [ Subconvexity bounds for automorphic L-functions for GL(2) over number fields ]... [ updated 30 Mar '08]... [joint with Adrian Diaconu]We prove a subconvex bound in the t-aspect for the standard L-functionattached to a cuspform on GL(2) over a number field. The methodinvolves asymptotics for second integral moments, with a power savingin the error term, for a spectral family of twists by grossencharacters. Some standard integrals for GL(2), with derivations, discussion of normalizations. For example, determination of Whittaker functions. In principle, these computations exist in many places.[archimedean integrals]... [ updated 23 Sep '07][p-adic integrals]... [ updated 01 May '08] [ Elementary asymptotics of integrals]... [ updated 16 Jun '07]... Review of Watson's lemma and Laplace's method, illustrated byobtaining Stirling for gamma, some asymptotics for beta withoutStirling, and asymptotics for Bessel functions. With proofs. [ Integral moments ofautomorphic L-functions ]... [ updated 08 May '08]... [joint with Adrian Diaconu]And-yet-once-more-edited, enhanced/enlarged version of earlier preprint of thesame name (below): integral moments for GL(2) automorphic L-functionsover number fields, by integral representations. This version hastripled in size by comparison to the old one. A recipe is given forproducing spectral identities involving second moments. Appendices prove convergence in detail, evaluate integrals, etc., in response toreferee comments. This version is slightly edited by comparison to theversion on arXiv, too. The original (below) short version may succeedin isolating the really new points better. [to appear inJ. Math. Inst. Jussieu] [ Poles of half-degenerate Eisenstein series]... [ updated 03 Jul '06]... Combine very classical application of Poisson summation withGodement-Jacquet integral representation of L-functions to give a goodestimate on poles of some very special (partly cuspidal-data, partlydegenerate-data) Eisenstein series on GL(n). Treating these as(iterated) residues of cuspidal-data Eisenstein series gives asignificantly worse estimate on poles. This class of Eisenstein seriesis very special, but occurs in applications. [ Discrete decomposition of cuspforms]... [ updated 20 Oct '07]... We prove the Gelfand-Graev-PS theorem: spaces of square-integrablecuspforms on reductive groups (best adelized) decompose discretely,with finite multiplicities, by proving that the operators naturally induced bytest functions on the group are compact . The argumentroughly follows Godement's 1966 Boulder-conference proof, with somecritical details filled in. [ Integral moments ofautomorphic L-functions ]... [ updated 12 Sep '06]... [joint with Adrian Diaconu] Integralmoments for GL(2) automorphic L-functions over number fields, byintegral representations. [ Archimedean zeta integrals for unitary groups]... [ updated 19 May '06]... Evaluation of archimedean zeta integrals arising in decompositionof holomorphic Siegel-type Eisenstein series restricted from larger tosmaller unitary groups. [This paper itself does not explain the largercontext. A version of it will appear in the AIM conference volume onEisenstein series, likely as a too-long appendix to a paper of Michael Harris' , namely"A simple proof of rationality of Siegel-Weil Eisenstein series"[http://www.math.jussieu.fr/~harris/SW.pdf]which does explain something of the context. ] [Geometric homology versus group homology]... [ updated 14 Sep '05]... We want to prove that the singular homology of quotients X/Gammais the group homology of Gamma, under some mild conditions on X (suchas that X be a ball). We recap work of Hopf, Hurewicz,Eilenberg-MacLane: homology of spaces with vanishing higherhomotopy is determined by the first homotopy group, giving a functor on groups. Then recall Dold'sresult that removes the requirement that the spaces beCW-complexes. Historical notes. [ Lie algebra sl(2)version of Segal-Shale-Weil (oscillator) representation ]... [ updated 07 Jan '06]... Setting up Lie algebra action of sl(2) on Schwartzfunctions: archimedean case of Weil representation. The benefits oflooking at the Lie algebra rather than Lie group action are compellingin this example. Amusing connection to sphericalharmonics... Invocation of subrepresentation theorem to studyirreducible quotients in positive-definite case. [Buildings, Bruhat decompositions, etc.]... [ updated 02 Jul '05]... [DRAFT] Development of basic theory of buildings, Bruhatdecompositions, aiming especially at simple discussion ofIwahori-Hecke algebra and Borel-Matsumoto theorem. Graphicsmeant to suggest proof techniques. (Small novelty is the realizationthat one need not presume the combinatorial group theory of Coxetergroups.) [Kernels of intertwinings for SL(2,R)]... [ updated 02 Jul '05]... Computing natural intertwining operators amongunramified principal series for SL(2,R). Meromorphiccontinuation in terms of the gammafunction. Holomorphic discrete series (summed with antiholomorphic)detected. (Meromorphically continued) intertwining operators extend tosmooth vectors. [ Convergent Siegel-Weil [draft] ]... [ updated 09 Aug '05]... Proof of Siegel-Weil in the far-convergent range, by inequalitiesseparating principal series (Satake) parameters of Eisenstein seriesand cuspforms. May be viewed as an updated version of an argument ofAndrianov from 1979. [Archimedean zeta integrals]... [ updated 16 Mar '05]... Overheads for Bowling Green, KY, talk on Archimedean zetaintegrals, and qualitative rationality arguments. [ Artin L-functions ]... [ updated 01 Mar '05]... Definition of Artin L-functions, brief comments on Artin'sconjecture on analytic continuations, Brauer's result on meromorphy,Langlands' reformulation. [ Moderate growth representations ]... [ updated 19 Feb '05]... Amplification of part of paper of N. Wallach from 1982: Norms ongroups. Banach space representations of real reductive groups are ofmoderate growth. Further, the Frechet spaces of smooth vectors are ofmoderate growth. [ Inducing cuspidals from compact-opens ]... [ updated 19 Feb '05]... (Jacquet 1970) the (smooth) induced module of cuspidal fromcompact-open is admissible and supercuspidal. [ Some facts about discrete series ]... [ updated 19 Feb '05]... Recollection of some basic facts on discrete series of realreductive groups, with table showing which classical groups do anddon't have discrete series, holomorphic discrete series, andquaternionic discrete series. Bibliographical pointers. [ A possibly amusing little stunt involving traces: ]... [ updated 19 Feb '05]... Computing zeta(2) as the trace of an integral operatoron [a,b] solving u''= f with boundary conditionsu(a)=u(b)=0. Traces of the iterates of this kernel evaluatezeta(2k). [ The zeroth Fouriercoefficient lies in the field generated by the higher coefficients]... [ updated 19 Feb '05]... This principle was used by Klingen c. 1960 in application to pullbacksof Hilbert modular Eisenstein series to elliptic modular forms, toanalyze the constant terms (certain values of L-functions). It is veryeasy to give incorrect proofs of this. [ Godement's criterion for convergence ofEisenstein series ]... [ updated 18 Aug '08].. Provides the little bit of adelic reduction-theoretic background (withproofs, following Godement) to give an adelic version of Borel's 1966account of Godement's criterion for convergence of very simpleSiegel-type (degenerate) Eisenstein series. [ holomorphic discrete series ]... [ updated 19 Feb '05]... Proving (for Sp(n,R) and U(p,q)) the apparently apocryphal resultthat for sufficiently high lowest K-type rho the universallowest-K-type (g,K)-module with lowest K-type rho isirreducible. This implies that rho determines the isomorphism class ofthe (g,K)-module, and also that the whole representation space is thetensor product of the enveloping algebra U(p+) with rho. The latterfreeness property is essential in a treatment of Maass-Shimuraoperators. [ von Neumanndensity theorem ] ... [ updated 20 Feb '05]... A one-page proof, not entangled with anything else. [ Unitary representations oftopological groups ] ... [ updated 13 Feb '08]... Basics, emphasizing discrete series,compact quotients, integration-theory methods. [ GL(2) over a finite field ] ... [ updated 20 Dec '07]... Very simple illustration ofirreducibility of principal series, Jacquet modules, uniqueness ofWhittaker models, Mackey-Bruhat orbit decomposition, Gelfand-Graevinvolution method. (dumb errors fixed Jan 2004, but SL(2) part notdone. Toy Weil/oscillator representation stuff to be added.) [ Extended automorphic forms ] ... [ updated 19 Feb '05]... after Casselman and Zagier, Maass-Selberg for SL(2,Z) asillustration. [ Classical groups and classicaldomains ] ... [ updated 24 Apr '05]... classical cones, classical groups over R and C,Harish-Chandra and Borel realizations of bounded symmetric domains. [ Basic Rankin-Selberg method] ... [ updated 31 May '05]... The classical simplest possible example, obtaining thetensor product L-function for two holomorphic cuspforms for SL(2,Z),discussing also the Mellin-transform trick to see the meromorphiccontinuation of the relevant Eisenstein series. [ Representations with Iwahori-fixedvectors ] ... [ updated 19 Feb '05]... Borel-Matsumoto theorem and applications toirreducibility of unramified principal series and degenerate principalseries representations of reductive p-adic groups. [Note: It seems to me now that the generic algebrasbusiness is needless, and amounts to taking the long way around. Thiswill be written up in a different style...(11 June 2005)] [ Jacquet theory ] ... [ updated 05 Dec '05]... Standard basic features of representation theory of p-adic reductive groups:exactness of Jacquet module functors, Jacquet's lemmas, admissibilityand finite-generation of Jacquet modules of admissiblefinitely-generated smooth representations. [ Slightly non-trivial examples ofMaass-Selberg relations ] ... [ updated 19 Feb '05]... Inner products of truncatedEisenstein series attached to spherical cuspidal-data on maximalproper parabolics in GL(n), with standard corollaries about possiblepoles, square-integrability of residues. [ Simplest example of Maass-Selberg relations] ... [ updated 19 Feb '05]... The absolutely simplest case:spherical Eisenstein series for SL(2,Z), of course, assuming basic results from the theoryof the constant term, paying attention to the proper notion oftruncation. Standard corollaries about possible poles,square-integrability of residues, in this simple case. [ Eisenstein series bibliography ]... [ updated 01 May '06]... concerning analytical properties of Eisenstein series, constant terms,Rankin-Selberg and Langlands-Shahidi integral representations ofL-functions, related representation theory of reductive Lie and p-adicgroups, etc. [ Volumes of SL(n,Z) and Sp(n,Z) ]... [ updated 19 Feb '05]... following Siegel et alia. Essentially elementary argument usingPoisson summation. [ Hartogs' theorem] ... [ updated 19 Feb '05]... that separate analyticity implies joint analyticity. Used inreduction of the non-maximal parabolic case to the maximal paraboliccase in treatment of Eisenstein series, and in the proof (for the Selberg-Bernstein argument for meromorphic continuation) that acomposition of weakly holomorphic morphisms of topologicalvectorspaces maps is again weakly holomorphic. [ Algebras and involutions ]... [ updated 19 Feb '05]... Especially over local fields. Crossed product, cyclic algebraconstructions. Local splitting almost everywhere. Classifiesinvolutions over local fields. [ Jacobi product formula ]... [ updated 23 Aug '01] [ Weak smoothness implies strong smoothness ] ... [ updated 21 Nov '06]... for functions with values in quasi-complete locally convex topologicalvectorspaces. [ Uniqueness of invariantdistributions ] ... [ updated 03 Aug '05]... on Lie groups, totally disconnected groups, adele groups, etc. [ Very easy non-unitarizabilitycriterion for principal series ] ... [ updated 24 Apr '05]... for p-adic GL(n), and more generally [ Prime Number Theorem, Etcetera ] ... [ updated 19 Feb '05]... Further simplification of D.J. Newman's simplified method,applied to general Euler products. [ Purdue talk, 20 April2001 ] ... [ updated 19 Feb '05]... Remarks on spectral decompositions,intro to meromorphic continuation. Maybe expanded later. [ Expanded version of Tel Aviv talk] ... [ updated 19 Feb '05]... Meromorphic continuation of cuspidal-data Eisenstein seriesfor maximal proper parabolics in GL(n). Draft. [ Meromorphic continuationof Eisenstein series ] ... [ updated 19 Feb '05]... after Bernstein-Selberg. Revised setupand treatment of SL(2,Z). Draft. Complementary material for meromorphic continuation:[ Compactness of some operators ] ... Technical enhancement of the classical result for integraloperators on cuspforms [ Theory of the Constant Term ] ... [ updated 15 Dec '06]... Basic estimates, rapid decay, for GL(n) ... [ Spectral Theory for SL(2,Z) ] ... [ updated 23 Jun '08]... Continuous spectrum ... simplest example. [Vector-valued integrals ] ... [ updated 24 Jul '08]... Proof that certain `weak' (Gelfand-Pettis) integrals exist, and that `weakly holomorphic' implies `holomorphic' [`Easy' proof of Siegel-Weil]... [ updated 09 Aug '05]... in the region of convergence, for SL(2) [Euler Factorizations of Global Integrals ]... [ updated 19 Feb '05]... my paper from Proc. Symp. AMS 66, from Texas conference1996. Gives multiplicity-one sufficient conditions for factorizationof global integrals into Euler product and period . [ The Gelfand-Kazhdan criterion]... [ updated 19 Feb '05]... for multiplicity-free-ness [ Bernstein's RationalityLemma ] ... [ updated 19 Feb '05]... algebraic version of his "continuation principle" [ Reduction theory ] ... [ updated 18 Aug '08]...compactness of arithmetic quotients of anisotropic orthogonal groups(after Tamagawa-Mostow and Godement). Other basic stuff about affineheights and Minkowski reduction in modern setting. [ Fujisaki's lemma: ] ... [ updated 16 Jan '08]... Compactness ofarithmetic quotients of division algebras (after Weil) [ Satake parameters versus principalseries ] ... [ updated 19 Feb '05]... comparison of Satake transform with principal seriesparameters. Obvious in hindsight. [ Factoring unitary representationsover primes ] ... [ updated 19 Feb '05]... about admissibility, type I groups,liminality/CCR property, etc. [ What are automorphic forms andL-functions? ] ... [ updated 19 Feb '05]... informal historical introduction [ Primer of unramified principal series] ... [ updated 19 Feb '05]... explaining the "facts" of Casselman's 1980 Compositio paperin elementary terms for a few classical groups [ Admissibility of irreducibles ofreductive groups ] ... [ updated 19 Feb '05]... proofs in real case, Bernstein's proof forsupercuspidals in p-adic case. [ Smooth representations of totallydisconnected groups ] ... [ updated 08 Jul '05]... introductory notes [ Injectivity of supercuspidals ]... [ updated 19 Feb '05] [ von Neumann algebras:terminology ]... [ updated 19 Feb '05] [ Quadratic reciprocity over global fields] ... [ updated 19 Feb '05]... using Poisson summation, Fourier transforms of distributions[ Newton polygons ]... [ updated 19 Feb '05] [ Bernstein's analyticcontinuation of complex powers ] ... [ updated 19 Feb '05]... a slight rewrite of the original article© 1996-2008, Paul Garrett ...[ garrett@math.umn.edu ][this page is http://www.math.umn.edu/~garrett/m/v/?] The University of Minnesota explicitly requires that Istate that "The views and opinions expressed in this page arestrictly those of the page author. The contents of this page have notbeen reviewed or approved by the University of Minnesota." |
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