Vignettes on automorphic forms,
representations,
Lfunctions, and number theory
[ambient page updated 12 Oct '14]
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[ garrett@math.umn.edu ]
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( Also:
[ functional analysis ]
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[ intro to modular forms ]
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[ representation theory ]
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[ buildings notes ]
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[ number theory ]
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[ algebra ]
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[ complex analysis ]
)
 Some automorphic spectral theory for GL(n,Z):
 [ Discreteness of pseudocuspforms
for GL(n)]
... [ updated 28 Jul '14]...
and more: discreteness of space of L^{2} functions with
all constant terms vanishing above a fixed height, after
LaxPhillips. Thus, this also sets up an extension of Y. Colin de Verdiere's
argument to meromorphic continuation of cuspidaldata Eisenstein
series (as Jacquet's argument in MoeglinWaldspurger).
 Complement: [ Discreteness of
spectrum of Laplacian on compact Riemannian manifolds ]
... [ updated 19 Jun '14]...
Proving that the resolvent is compact selfadjoint, thus allowing
application of the spectral theorem for such operators. Proof
reduces to the case of products of circles, and Sobolev spaces via
Fourier series.
 [Mostcontinuous automorphic spectrum
for GL(n)]
... [ updated 03 Jan '12]...
Example discussion of mostcontinuous automorphic spectrum for
PGL(n,Z), by expressing minimalparabolic
pseudoEisenstein series as integrals of minimalparabolic Eisenstein
series. Plancherel is proven for this part
of the spectrum, modulo fuller spectral decomposition. Supporting material includes
meromorphic continuation of minimalparabolic Eisenstein series,
functional equations, constant terms, review of GL(2), sample
bibliography of prior art.
 [Sporadic isogenies of classical
groups to orthogonal groups]
... [ updated 20 Jul '13]
Elaboration of the wellknown maps
from other classical groups to small orthogonal groups
O(3), O(4), O(5), O(6) over the complex numbers, and also O(p,q) with
p+q≤ 6.
 [ Linear independence of nth roots]
... [ updated 27 Dec '11]...
Surprisingly palatable, relatively elementary argument
for an assertion exemplified by
Q(√2, √3, √5)
= Q(√2+ √3+ √5).
Slightly more generally for an nth root of unity ζ,
with a,b,c relatively prime integers,
Q(ζ)(^{n}√a, ^{n}√b, ^{n}√c)
= Q(ζ)(^{n}√a+^{n}√b+^{n}√c).
Prior art dates at least to Hasse in 1933.
 [Symmetrization maps]
... [ updated 26 Jul '11]...
Small categoricalalgebra exercise characterizing symmetrization maps
on universal enveloping algebras of Lie algebras, thereby proving
independence of coordinates, etc.
 [Pseudocuspforms, pseudoLaplacians ]
... [ updated 12 Aug '11]...
(A few typos fixed...) Pseudocuspforms have constant
term eventually vanishing, and are eventually
eigenfunctions for the Laplacian. Existence of certain
pseudocuspforms is equivalent to vanishing of zeta and
Lfunctions. Unfortunately, not being genuine eigenfunctions, this
does not immediately prove the Riemann
Hypothesis. Nevertheless, circa 1979 there was some excitement due
to confusion about these things. The confusion was dispelled by
Hejhal, who also observed the role of pseudocuspforms, and noted
several serious obstacles. Example computations included.
 [ Explication of ColindeVerdiere's proof
of meromorphic continuation of Eisenstein series]
... [ updated 12 Mar '12]...
A considerable elaboration of ColindeVerdiere's argument, hopefully
useful to people who are not practiced modern analysts. Supporting
material included as appendices. Incidentally proves
discreteness of cuspforms from the same compactness argument
(LaxPhillips' form of a Rellich lemma) used to obtain the
meromorphic continuation. Added March 2012: a few unsurprising but
necessary details about uniform Sobolevnorm estimates on smooth
tails of functions.
 [
(revision) Moments for Lfunctions for GL(r) x
GL(r1)]
... [ updated 26 Mar '11]...
[joint with A. Diaconu and D. Goldfeld]
Further details added. Spectral identities for second integral moments of Hecketype
RankinSelberg convolution Lfunctions for Lfunctions
for GL(r) x GL(r1). For all r, the spectral decomposition of
the associated Poincare series involves cuspidal data
only from GL(2).
]
[ updated 26 Mar '11]
 [ product formula for
Delta ]
[ updated 06 Jan '11]
... via a converse theorem, after Weil. Thinking of more general
product expansions of automorphic forms: Borcherds, Zwegers, et alia.
 [ Quadratic reciprocity for global
fields by Poisson summation
]
[ updated 19 Dec '10]
... after Weil, edited, slightly reformatted, from very old version below.
 [ Unbounded operators, Friedrichs
extensions, resolvents
]
[ updated 25 May '14]
 [ Durham, UK, talk on automorphic spectral theory]
... [ updated
30 Sep '10]...
 [ nonexistence of tensor
products of Hilbert spaces]
... [ updated
22 Jul '10]...
One of several points to be made about tensor products of topological
vector spaces: first, tensor products of Hilbert spaces do not
exist, despite a certain cultural mythology. Some further points
about Grothendieck's notion of nuclear spaces and
Schwartz's kernel theorem will be added later.
 [talk in Newark, May 2010]
... [ updated
25 Jun '10]...
Example computation in automorphic spectral theory of
SL(2,Z[i]). Background, details, proofs:
 [Siegel's integral]
... [ updated
02 Mar '10]...
Recollection of a nonobvious computation of a Fourier transform,
dating at least to Siegel in 1939, greatly elaboratedupon in
intervening years, sometimes lost in an ocean of
generalizations.
 Some occasionally obscured small points about IwasawaTate theory
of automorphic Lfunctions for GL(1):
 [ Kummer, Eisenstein, and cyclotomic
Lagrange resolvents]
... [ updated 28 Jul '10]...
Expression of subfields of cyclotomic field by radicals by Lagrange
resolvents: seventh, quintic subfield of eleventh, octic subfield
of seventeenth, and other numerical examples ... by repurposing
Kummer (Stickelberger) estimate on Gauss sums and ideas used by
Eisenstein in his formulation of reciprocity laws.
 [ (revision) Moments for Lfunctions for GL(r) x
GL(r1)]
... [ updated 17 Dec '09]...
[joint with A. Diaconu and
D. Goldfeld]
Updated version: spectral identities involving integral moments of Hecketype
RankinSelberg convolution Lfunctions for Lfunctions
for GL(r) x GL(r1). For all r, the spectral decomposition of
the associated Poincare series involves cuspidal data
only from GL(2).
 [ Asymptotics of integrals of
eigenfunctions ]
... [ updated 24 Nov '09]
... Precise asymptotics for nfold integrals of zonal spherical
functions on SL(2,C), as simplest example of exponential decay.
 [ Peetre's theorem ]
... [ updated 16 Oct '09]
... a linear operator on functions, not
increasing support, is a differential operator.
 [ An iconic error ]
... [ updated 22 Sep '09]
... Debunking the (demonstrably false) claim that
truncated Eisenstein series are eigenfunctions for invariant
operators. This (false) claim continues to be
(needlessly) invoked to give (incorrect) proofs of MaassSelberg
relations. I first heard this claim in 1980, and was amazed to hear
it recently (2009). It's an icon?
 [ Snake lemma, extensions, Gamma function
]
... [ updated 14 Jun '11]
... Illustration of extension and uniqueness of distributions by
simple homological ideas. Easiest example: homogeneous distributions
and Gamma.
 [ shortinterval bounds give
pointwise bounds
]
... [ updated 13 Apr '09]...
... We carry out one version of the obvious argument involving
Cauchy's theorem, convexity bounds, and the functional equation of
Lfunctions, to deduce nontrivial pointwise bounds on Lfunctions
from nontrivial shortinterval bounds. (The far more serious
question of the source of nontrivial shortinterval bounds is not
addressed.)
 [ (updated) Natural boundaries and
the correct notion
of integral moments of Lfunctions
]
... [ updated 23 Sep '09]...
[joint with A. Diaconu]
and
D. Goldfeld]
...
Evidence is given that the classical notion of higher
moment of the Riemann zeta function is not correct. We propose
a plausible candidate for a replacement.
 [ Averages of symmetricsquare Lfunctions
and applications]
... [ updated 05 Mar '09]...
[joint with A. Diaconu]
...
Spectral identities involving second moments of symmetricsquare
Lfunctions for SL(2)...
 [ Moments for Lfunctions for GL(r) x GL(r1)]
... [ updated 28 Oct '09]...
[joint with A. Diaconu and
D. Goldfeld]
Spectral identities involving integral moments of Hecketype
RankinSelberg convolution Lfunctions for moments for Lfunctions
for GL(r) x GL(r1). For all r, the spectral decomposition of
the associated Poincare series involves cuspidal data
only from GL(2).
 [Standard periods of Eisenstein series]
... [ updated 07 Sep '09]
Simplest examples of compact periods of Eisenstein series (not
involving cuspidal data).
 Talks at ICMS, Edinburgh, August 2008:
 [Integral moments I]
... [ updated 09 Aug '08]
Overview of spectral identities for integral moments for GL(n)xGL(n1)
Lfunctions over number fields. Subconvexity in taspect for GL(2)
over number fields as evidence for nontriviality of these
identities.
 [Integral moments IIIa]
... [ updated 09 Aug '08]
General recipe to produce spectrally meaningful integral second
moments for all RankinSelberg integrals. Examples: GL(n)xGL(n1)
Hecketype convolutions, GL(n)xGL(n) RankinSelberg convolutions,
tripleproduct Lfunctions, all doubling integrals for classical
groups. Heuristic concerning extraction of subconvex bounds.
 [
characters of principal series
]
... [ updated 14 Jul '11]
in terms of orbital integrals, without proof of trace class.
 [
Subconvexity bounds for automorphic Lfunctions for GL(2) over number fields
]
... [ updated 09 Jul '09]
... [joint with
Adrian Diaconu]
A subconvex bound in the taspect for standard Lfunctions
of GL(2) cuspforms over number fields. The method
involves asymptotics for second integral moments, with a power saving
in the error term, for a spectral family of twists by
grossencharacters.
[to appear in
J. Math. Inst. Jussieu
]
 Some standard i
ntegrals for GL(2), with derivations, discussion
of normalizations. For example, determination of Whittaker
functions. In principle, these computations exist in many places.
 [ Elementary asymptotics of integrals]
... [ updated 29 Dec '12]
... Review of Watson's lemma and Laplace's method, illustrated by
obtaining Stirling for gamma, some asymptotics for beta without
Stirling, and asymptotics for Bessel functions. With proofs.
 [ Integral moments of
automorphic Lfunctions ]
... [ updated 08 May '08]
... [joint with Adrian Diaconu]
Andyetoncemoreedited, enhanced/enlarged version of earlier preprint of the
same name (below): integral moments for GL(2) automorphic Lfunctions
over number fields, by integral representations. This version has
tripled in size by comparison to the old one. A recipe is given for
producing spectral identities involving second moments. Appendices
prove convergence in detail, evaluate integrals, etc., in response to
referee comments. This version is slightly edited by comparison to the
version on arXiv, too. The original (below) short version may succeed
in isolating the really new points better.
Appears in [Volume 8, Issue 02, April 2009, pp 335382,
J. Math. Inst. Jussieu.
]
 [ Poles of halfdegenerate Eisenstein series]
... [ updated 03 Jul '06]
... Combine very classical application of Poisson summation with
GodementJacquet integral representation of Lfunctions to give a good
estimate on poles of some very special (partly cuspidaldata, partly
degeneratedata) Eisenstein series on GL(n). Treating these as
(iterated) residues of cuspidaldata Eisenstein series gives a
significantly worse estimate on poles. This class of Eisenstein series
is very special, but occurs in applications.
 [ Discrete decomposition of cuspforms]
... [ updated 05 Jun '09]
... We prove the GelfandGraevPS theorem: spaces of squareintegrable
cuspforms on reductive groups (best adelized) decompose discretely,
with finite multiplicities, by proving that the operators naturally induced by
test functions on the group are compact . The argument
roughly follows Godement's 1966 Boulderconference proof, with some
critical details filled in.
 [ Integral moments of
automorphic Lfunctions ]
... [ updated 12 Sep '06]
... [joint with
Adrian Diaconu] Integral
moments for GL(2) automorphic Lfunctions over number fields, by
integral representations.
 [ Archimedean zeta integrals for unitary groups]
... [ updated 19 May '06]
... Evaluation of archimedean zeta integrals arising in decomposition
of holomorphic Siegeltype Eisenstein series restricted from larger to
smaller unitary groups. [This paper itself does not explain the larger
context. A version of it will appear in the AIM conference volume on
Eisenstein series, likely as a toolong appendix to a paper of
Michael Harris' ,
namely
"A simple proof of rationality of SiegelWeil 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/Gamma
is the group homology of Gamma, under some mild conditions on X (such
as that X be a ball). We recap work of Hopf, Hurewicz,
EilenbergMacLane: homology of spaces with vanishing higher
homotopy is determined by the first homotopy group,
giving a functor on groups. Then recall Dold's
result that removes the requirement that the spaces be
CWcomplexes. Historical notes.
 [ Lie algebra sl(2)
version of SegalShaleWeil (oscillator) representation ]
... [ updated 12 May '14]
... Setting up Lie algebra action of sl(2) on Schwartz
functions: archimedean case of Weil representation. The benefits of
looking at the Lie algebra rather than Lie group action are compelling
in this example. Amusing connection to spherical
harmonics... Invocation of subrepresentation theorem to study
irreducible quotients in positivedefinite case.
 [Buildings, Bruhat decompositions, etc.]
... [ updated 02 Jul '05]
... Development of basic theory of buildings, Bruhat
decompositions, aiming especially at simple discussion of
IwahoriHecke algebra and BorelMatsumoto theorem. Graphics
meant to suggest proof techniques. (Small novelty is the realization
that one need not presume the combinatorial group theory of Coxeter
groups.)
 [Kernels of intertwinings for SL(2,R)]
... [ updated 03 Jan '09]
... Computing natural intertwining operators among
unramified principal series for SL(2,R). Meromorphic
continuation in terms of the gamma
function. Holomorphic discrete series (summed with antiholomorphic)
detected. (Meromorphically continued) intertwining operators extend to
smooth vectors.
 [
Convergent SiegelWeil
]
... [ updated 09 Aug '05]
... Proof of SiegelWeil in the farconvergent range, by inequalities
separating principal series (Satake) parameters of Eisenstein series
and cuspforms. May be viewed as an updated version of an argument of
Andrianov from 1979.
 [Archimedean zeta integrals]
... [ updated 16 Mar '05]
... Overheads for Bowling Green, KY, talk on Archimedean zeta
integrals, and qualitative rationality arguments.
 [ Artin Lfunctions ]
... [ updated 01 Mar '05]
... Definition of Artin Lfunctions, brief comments on Artin's
conjecture 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 on
groups. Banach space representations of real reductive groups are of
moderate growth. Further, the Frechet spaces of smooth vectors are of
moderate growth.
 [
Inducing cuspidals from compactopens
]
... [ updated 19 Feb '05]
... (Jacquet 1970) the (smooth) induced module of cuspidal from
compactopen is admissible and supercuspidal.
 [
Some facts about discrete series ]
... [ updated 19 Feb '05]
... Recollection of some basic facts on discrete series of real
reductive groups, with table showing which classical groups do and
don't have discrete series, holomorphic discrete series, and
quaternionic discrete series. Bibliographical pointers.
 [
A possibly amusing little stunt involving traces: ]
... [ updated 17 Apr '14]
... Computing zeta(2) as the trace of an integral operator
on [a,b] solving u''= f with boundary conditions
u(a)=u(b)=0. Traces of the iterates of this kernel evaluate
zeta(2k).
 [ The zeroth Fourier
coefficient lies in the field generated by the higher coefficients
]
...
[ updated 19 Feb '05]
...
This principle was used by Klingen c. 1960 in application to pullbacks
of Hilbert modular Eisenstein series to elliptic modular forms, to
analyze the constant terms (certain values of Lfunctions). It is very
easy to give incorrect proofs of this.
 [ Godement's criterion for convergence of
Eisenstein series ]
... [ updated 18 Aug '08]
.. Provides the little bit of adelic reductiontheoretic background (with
proofs, following Godement) to give an adelic version of Borel's 1966
account of Godement's criterion for convergence of very simple
Siegeltype (degenerate) Eisenstein series.
 [ holomorphic discrete series ]
... [ updated 19 Feb '05]
... Proving (for Sp(n,R) and U(p,q)) the apparently apocryphal result
that for sufficiently high lowest Ktype rho the universal
lowestKtype (g,K)module with lowest Ktype rho is
irreducible. This implies that rho determines the isomorphism class of
the (g,K)module, and also that the whole representation space is the
tensor product of the enveloping algebra U(p+) with rho. The latter
freeness property is essential in a treatment of MaassShimura
operators.
 [ von Neumann
density theorem ]
... [ updated 20 Feb '05]
... A onepage proof, not entangled with anything else.
 [ Unitary representations of
topological groups ]
... [ updated 28 Jul '14]
... Basics, emphasizing discrete series,
compact quotients, integrationtheory methods.
 [ GL(2) over a finite field ]
... [ updated 19 Apr '09]
... Very simple illustration of
irreducibility of principal series, Jacquet modules, uniqueness of
Whittaker models, MackeyBruhat orbit decomposition, GelfandGraev
involution method. (dumb errors fixed Jan 2004, but SL(2) part not
done. Toy Weil/oscillator representation stuff to be added.)
 [ Extended automorphic forms ]
... [ updated 19 Feb '05]
... after Casselman and Zagier, MaassSelberg for SL(2,Z) as
illustration.
 [ Classical groups and classical
domains ]
... [ updated 22 Sep '10]
... classical cones, classical groups over R and C,
HarishChandra and Borel realizations of bounded symmetric domains.
 [ Basic RankinSelberg method
]
... [ updated 09 Jul '10]
... The classical simplest possible example, obtaining the
tensor product Lfunction for two holomorphic cuspforms for SL(2,Z),
discussing also the Mellintransform trick to see the meromorphic
continuation of the relevant Eisenstein series.
 [ Representations with Iwahorifixed
vectors ]
... [ updated 19 Feb '05]
... BorelMatsumoto theorem and applications to
irreducibility of unramified principal series and degenerate principal
series representations of reductive padic groups. [
Note: It seems to me now that the generic algebras
business is needless, and amounts to taking the long way around. This
will be written up in a different style...(11 June 2005)]
 [ Jacquet theory ]
... [ updated 05 Dec '05]
... Standard basic features of representation theory of padic reductive groups:
exactness of Jacquet module functors, Jacquet's lemmas, admissibility
and finitegeneration of Jacquet modules of admissible
finitelygenerated smooth representations.
 [ Slightly nontrivial examples of
MaassSelberg relations ]
... [ updated 19 Feb '05]
... Inner products of truncated
Eisenstein series attached to spherical cuspidaldata on maximal
proper parabolics in GL(n), with standard corollaries about possible
poles, squareintegrability of residues.
 [ Simplest example of MaassSelberg relations
]
... [ updated 19 Feb '05]
... The absolutely simplest case:
spherical Eisenstein series for SL(2,Z), of course, assuming basic results from the theory
of the constant term, paying attention to the proper notion of
truncation. Standard corollaries about possible poles,
squareintegrability of residues, in this simple case.
 [ Eisenstein series bibliography ]
... [ updated 01 May '06]
... concerning analytical properties of Eisenstein series, constant terms,
RankinSelberg and LanglandsShahidi integral representations of
Lfunctions, related representation theory of reductive Lie and padic
groups, etc.
 [ Volumes of SL(n,Z) and Sp(n,Z) ]
... [ updated 20 Apr '14]
... following Siegel et alia. Essentially elementary argument using
Poisson summation.
 [ Hartogs' theorem]
... [ updated 19 Feb '05]
... that separate analyticity implies joint analyticity. Used in
reduction of the nonmaximal parabolic case to the maximal parabolic
case in treatment of Eisenstein series, and in the proof (for the
SelbergBernstein argument for meromorphic continuation) that a
composition of weakly holomorphic morphisms of topological
vectorspaces maps is again weakly holomorphic.
 [ Algebras and involutions ]
... [ updated 19 Feb '05]
... Especially over local fields. Crossed product, cyclic algebra
constructions. Local splitting almost everywhere. Classifies
involutions over local fields.
 [ Jacobi product formula ]
... [ updated 23 Aug '01]
 [
Weak smoothness implies strong smoothness ]
... [ updated 21 Nov '06]
... for functions with values in quasicomplete locally convex topological
vectorspaces.
 [ Uniqueness of invariant
distributions ]
... [ updated 03 Aug '05]
... on Lie groups, totally disconnected groups, adele groups, etc.
 [ Very easy nonunitarizability
criterion for principal series ]
... [ updated 24 Apr '05]
... for padic 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 April
2001 ]
... [ updated 19 Feb '05]
... Remarks on spectral decompositions,
intro to meromorphic continuation. Maybe expanded later.
 [ Expanded version of Tel Aviv talk
]
... [ updated 21 Jul '14]
... Meromorphic continuation of cuspidaldata Eisenstein series
for maximal proper parabolics in GL(n).
 [ Meromorphic continuation
of Eisenstein series ]
... [ updated 21 Jul '14]
... after BernsteinSelberg. Revised setup
and treatment of SL(2,Z).
 Complementary material for meromorphic continuation:
 [`Easy' proof of SiegelWeil]
... [ updated 06 Mar '14]
... 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 conference
1996. Gives multiplicityone sufficient conditions for factorization
of global integrals into Euler product and period .
 [ The GelfandKazhdan criterion
]
... [ updated 19 Feb '05]
... for multiplicityfreeness
 [ Bernstein's Rationality
Lemma ]
... [ 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 TamagawaMostow and Godement). Other basic stuff about affine
heights and Minkowski reduction in modern setting.
 [ Fujisaki's lemma: ]
... [ updated 16 Jan '08]
... Compactness of
arithmetic quotients of division algebras (after Weil)
 [ Satake parameters versus principal
series ]
... [ updated 19 Feb '05]
... comparison of Satake transform with principal series
parameters. Obvious in hindsight.
 [ Factoring unitary representations
over primes ]
... [ updated 19 Feb '05]
... about admissibility, type I groups,
liminality/CCR property, etc.
 [ What are automorphic forms and
Lfunctions? ]
... [ updated 19 Feb '05]
... informal historical introduction
 [ Primer of unramified principal series
]
... [ updated 19 Feb '05]
... explaining the "facts" of Casselman's 1980 Compositio paper
in elementary terms for a few classical groups

[ Admissibility of irreducibles of
reductive groups ]
... [ updated 15 Jan '09]
... proofs in real case, Bernstein's proof for
supercuspidals in padic case.
 [ Smooth representations of totally
disconnected groups ]
... [ updated 12 Mar '12]
... introductory notes
 [ Injectivity of supercuspidals ]
... [ updated 19 Feb '05]
 [ von Neumann algebras:
terminology ]
... [ updated 19 Feb '05]
 [ Quadratic reciprocity over global fields
]
... [ updated 17 Sep '10]
... using Poisson summation, Fourier transforms of distributions

[ Newton polygons ]
...
[ updated 19 Feb '05]
 [ Bernstein's analytic
continuation of complex powers ]
... [ updated 03 Apr '11]
... a slight rewrite of the original article
© 19962013, Creative Commons license,
This work
by Paul Garrett is licensed
under a Creative
Commons Attribution 3.0
Unported License.
...
[ garrett@math.umn.edu ]
[this page is http://www.math.umn.edu/~garrett/m/v/index.html]
The University of Minnesota explicitly requires that I
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