Modular Forms and Lfunctions
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Math 820708, 201718, (Fall: Vincent Hall 113) 11:1512:05, MWF
Phenomena, examples, history
This introductory Number Theory course will be accessible to
firstyear and secondyear grad students with a modest background, and
will proceed by extensive examples throughout, as motivation
and explanation for more sophisticated methods and formalism. I will
adapt the content and degree of sophistication to the students who
show up! :)
Course notes will appear here, based in part on earlier notes as
below, with revisions and additions. The general tone and outlook will
be similar to past courses. A tentative partial outline:
 The EulerRiemann zeta function, explicit formulas, Riemann Hypothesis
 Dirichlet's Lfunctions, primes in arithmetic progressions, reciprocity laws
 Simplest cases of Deligne's conjectures on special values of
Lfunctions
 Elliptic functions, elliptic modular forms, Eisenstein series,
theta series
 Application: Kloostermantype equidistribution problems on spheres
 Application: Klingen's evaluation of Lfunctions of totally
real fields
 Hecke operators, Euler products, Mellin transforms, converse theorems
 RankinSelberg Lfunctions
 Meromorphic continuation of Eisenstein series, Hecke summation
 MaassShimura operators and RankinSelberg
 Maass' waveforms, Lfunctions
 Hilbert modular forms, Siegel modular forms
 Decomposition of Eisenstein series
 ...
Math 820708, 201516, (Fall: Vincent Hall 2) 11:1512:05, MWF
Phenomena, examples
This introductory Number Theory course will be accessible to
firstyear and secondyear grad students with a modest background, and
will proceed by extensive examples throughout, as motivation and
explanation for more sophisticated methods and formalism.
I intend (by adapting the content to the population of students
that show up!) that this course be interesting not only to students
in Number Theory or Automorphic Forms, but
also to students whose research areas interact frequently with these
subjects, such as Algebraic Geometry, Mathematical Physics,
Representation Theory, and Combinatorics, among others.
This course will give the phenomenological background to
formalities such as the Langlands program, but I intend to take a broader approach.
Approximate/tentative outline:
[ background on complex analysis ]
 Classical GL(1) stories:
 Euler and zeta
...[ updated
09:09, Sep 14, 2015]
 Estermann phenomenon: obstacles to analytic continuation
...[ updated
09:09, Sep 14, 2015]
 Riemann and zeta
...[ updated
08:09, Sep 24, 2015]
 Guinand and zeta
...[ updated
13:09, Sep 17, 2015]

Keyhole contour and zeta(n)
...[ updated
10:09, Sep 27, 2015]
 Dirichlet's theorem on primes in arithmetic progressions ...[ updated
09:09, Sep 26, 2015]
 Fourier analysis on finite abelian groups ...[ updated
09:09, Sep 26, 2015]
 Somewhat ugly proof of nonvanishing of L(1,chi) ...[ updated
09:09, Sep 26, 2015]
 Special values of L(s,chi) by keyhole contour ...[ updated
10:09, Sep 27, 2015]
 Functional equations of L(s,chi) ...[ updated
11:09, Sep 27, 2015]
 Gauss sums ...[ updated
15:09, Sep 27, 2015]
 Simplest examples of Deligne's conjectures ...[ updated
13:09, Sep 27, 2015]
 Introduction to equidistribution
 Some classical GL(2) stories:

Origins in elliptic functions
...[ updated
11:11, Nov 06, 2015]
 Fundamental domains
...[ updated
11:11, Nov 06, 2015]
 Level one elliptic modular forms,
holomorphic Eisenstein series
...[ updated
11:11, Nov 06, 2015]

Product expansion of Ramanujan's Delta (equivalently, Dedekind's eta)
...[ updated
14:11, Nov 17, 2013]
 Klingen's theorem on special values of zeta functions of totally real number fields
 Equidistribution example
 Classical Lfunctions for GL(2)
 Hecke operators and Euler products
 simplest RankinSelberg Lfunctions and meromorphic continuation and functional equation of relevant Eisenstein series
[ updated
17:07, Jul 09, 2010]
 Waveforms
 Pretrace formulas and spectral theory of automorphic forms/functions
 HilbertBlumenthal modular forms
 Siegel modular forms
 Transition to GL_{2}(A) and GL_{n}(A)
 Theta lifts/correspondences, SegalShaleWeil representations:
 HeckeMaass correspondence: SL(2) x SO(2)
 DoiNaganuma correspondence: SL(2) x SO(4)
 Yoshida correspondence: Sp(4) x SO(2,2)
 Comments on: ShintaniNiwa, SaitoKurakawa correspondences
Questions? Send me email! :)
(200506, 201011, and 201314 notes lower on the page)
Math 820708, 201314, (Spring: 206 Vincent Hall) 11:1512:05, MWF
This course introduces many phenomena that
led to much contemporary research, including the Langlands program
and much more. Little prior acquaintance with higherlevel
prerequisites is assumed. Rather, we will give examples that
led to formation of many contemporary concepts and abstractions in
number theory, complex analysis, Lie theory, harmonic analysis,
representation theory, and algebraic geometry.
Units are listed in reverse chronological order. Notes will be linkedto as
we go, somewhat in advance of progress inclass. If you must print
notes, please don't do so until just before reading, because many updates
will occur.
See
also
Number theory notes 201112 for related discussions
 ...
 14 Spectral theory of waveforms
 13
 12 Automorphic forms on adele groups, Hecke operators
 11 Levelone
waveforms, Heegner points, geodesic periods of Eisenstein
series, nonvanishing... ...[ updated
14:11, Nov 11, 2014]
 10 Sums of squares, harmonic theta series, equidistribution
problems on spheres ...[ updated
16:12, Dec 15, 2013]
 09 Harmonic analysis on
spheres
...[ updated
18:12, Dec 21, 2014]
 08
Levelone holomorphic elliptic modular forms
...[ updated
11:11, Nov 17, 2013]
 07 Fundamental domain for
SL(2,Z) on the upper halfplane
...[ updated
08:10, Oct 21, 2013]
 06
Geometry of homogeneous spaces: spheres, projective spaces, hyperbolic
spaces
...[ updated
17:10, Oct 09, 2013]
 05
Dirichlet and Lfunctions: equidistribution of primes modulo N
...[ updated
11:09, Sep 24, 2013]
 04 Fourier series:
Dirichlet approximation, Kronecker approximation, Weyl
equidistribution
...[ updated
13:10, Oct 14, 2015]
 03 From trigonometric functions to elliptic functions to elliptic modular forms
...[ updated
11:09, Sep 24, 2013]
 02 Riemann and
zeta: the explicit formula
...[ updated
08:08, Aug 30, 2013]
 Supplements:

Poisson summation
and convergence of Fourier series
...[ updated
19:08, Aug 29, 2013]

Hadamard products
...[ updated
19:08, Aug 29, 2013]

counting
zeros of zeta in the critical strip
...[ updated
19:08, Aug 29, 2013]

asymptotics
of integrals, including the Gamma function
...[ updated
09:08, Aug 30, 2013]

PhragmenLindelof theorem
...[ updated
09:08, Aug 30, 2013]

Estermann
phenomenon: nonexistence of meromorphic continuations for some
natural Dirichlet series
...[ updated
13:09, Sep 12, 2013]

Keyhole contour and zeta(n)
...[ updated
10:09, Sep 27, 2015]
 01 Euler and the
zeta function
...[ updated
13:08, Aug 22, 2013]
 00 Quick review
of some basic complex analysis
...[ updated [IOU pictures, though!]
15:09, Sep 05, 2013]
Our course will include much supporting material, beyond the strict
topics of the title. Samples of other sources about modular forms
themselves are below. Siegel's notes give numbertheoretic
applications of Hilbert modular forms.
Math 820708, 201011, 209 Vincent Hall, 2:303:20, MWF
Office hours: MWF 1:252:15 or by appointment, email anytime
An introduction to number theory, zeta functions and
Lfunctions, and the role of modular and automorphic forms
Notes and exercises (reverse chrono order)
 ... [ Transition exercise
on Eisenstein series ]
...[ updated
11:01, Jan 07, 2012]...
Rewriting GL(2) Eisenstein series as functions on adele groups, to
illustrate the appearance of GL_{2}(Q_{p}), to
see Bruhatcell decomposition of constant term with Euler
product of bigcell summand, and to compute Hecke eigenvalues
locally.
 ... [ Traces, Cauchy's identity,
Schur functions]
...[ updated
16:06, Jun 28, 2011]... A
representationtheory identity giving the spherical GL(n) case of the
local RankinSelberg integral.

... [ intro to FourierWhittaker
expansions of cuspforms on GL(n), and some Lfunctions]
...[ updated
14:06, Jun 06, 2011]...
 ... [ Weil's proof of
product expansion of Delta
] ... with appendix giving Siegel's proof
 ...
 19
 meromorphic
continuation and functional equation for the simplest Eisenstein
series, for SL(2,Z), by Poisson summation
...[ updated
18:06, Jun 07, 2011]...
 mero
cont'n and functional equation for GL(2) Eisenstein
series, over number fields, with general data, by Poisson summation
...[ updated
19:06, Jun 06, 2011]...
 18
 [ Continuous automorphic
spectrum for SL(2,Z)]
...[ updated
14:05, May 05, 2011]... Orthogonal
complement to cuspforms is spanned by pseudoEisenstein series
with testfunction data. PseudoEisenstein series are
integrals of Eisenstein series. Half of Plancherel theorem for
continuous spectrum.
 17
 [ spheres and
hyperbolic spaces ]
...[ updated
09:04, Apr 16, 2011]... Action
of orthogonal and unitary groups on spheres, general linear
groups on projective spaces, O(n,1) and U(n,1) on real and
complex hyperbolic spaces
 16
 [ Exercise 16:]
Due approximately Fri, April 15, 2011.
 Superpositions of eigenfunctions, intertwinings, asymptotic behavior
 [asymptotics at regular
singular points]
...[ updated
12:05, May 07, 2011]... Examples for SL(2,R):
translationequivariant eigenfunctions (Whittaker/Bessel functions) at 0; dilationequivariant
eigenfunctions at infinity
 [asymptotics at irregular
singular points]
...[ updated
14:07, Jul 06, 2013]... Examples:
rotationally invariant eigenfunctions of Laplacian in
Euclidean spaces at infinity; Whittaker/Bessel functions at
infinity
 [exceptional regular
singular points]
...[ updated
16:05, May 14, 2011]... Exceptional
regular singular points. These occur in some important
examples, and it is essential to understand the second
solution
 15
 [
Harmonic analysis on spheres I, invariant Laplacian,
spherical harmonics ]
...[ updated
17:02, Feb 28, 2011]
 [
Harmonic analysis on spheres II, Sobolev inequalities
] ... [ updated 16:02, Feb 27, 2011]
 [
Intrinsic characterization of Laplacians, Casimir elements ]
...[ updated
07:11, Nov 11, 2011]
 [ Exercise 15:
] Due approximately Fri, Mar 11, 2011.
 14
 13
 12
 11
IwasawaTate: first pass
...[ updated
17:06, Jun 06, 2011]...
 10
 09
 08 Analytic
continuations and functional equations
[ updated 12:06, Jun 22, 2012] examples:
Dirichlet Lfunctions, Dedekind zeta for Gaussian integers,
Hecke grossencharacters for Gaussian
integers...
 07

[ Exercise 07:
Euclideanness ] (and starred exercises about topological
subgroups of R^{n}, application to units)
Due approximately Wed, Nov 03, 2010.
 06 Factorization of
zeta functions of number fields
[ updated 12:01, Jan 11, 2011] examples: reciprocity laws, application to nonvanishing of
L(1,chi)
 05 Dirichlet's theorem: primes
in arithmetic progressions
[ updated 09:04, Apr 12, 2011] with background on characters, Landau's lemma, L(1,chi) nonvanishing.
 04 Fourier analysis on
finite abelian groups
[ updated 07:04, Apr 01, 2012]
 03 Fourier expansions of polynomials
and applications to values of zeta, Bernoulli polynomials
[ updated 12:09, Sep 13, 2011]
 02 Riemann's explicit
formula
[ updated 11:10, Oct 02, 2010]
... with thanks to Hadamard, von Mangoldt... and a host of others...
 01 Introduction to
phenomena:
[ updated 12:03, Mar 01, 2011] examples. Distribution of primes, zeta and
Lfunctions. Automorphic/modular forms: holomorphic Eisenstein
series, theta series, waveform Eisenstein series
Notes from 200506
 [ Fourier analysis on finite
abelian groups ]
... [ updated 17:10, Oct 17, 2007]
... Decomposition of the regular representation of a finite abelian
group, that is, acting on functions on itself, under
translation. Assumes only spectral theory on finitedimensional
complex vectorspaces.
Notes (reverse chronological order):
 [14] (DRAFT)
[
Dirichlet series from automorphic forms]
... [ updated 07:09, Sep 17, 2010]
...
Beginning of study of Dirichlet series with meromorphic continuation
and functional equation obtained from automorphic forms, both
holomorphic ones and waveforms.
 [13] (DRAFT)
[
toward waveforms]
... [ updated 13:11, Nov 01, 2010]
...
Beginning of study of eigenfunctions for the invariant Laplacian on
the upper halfplane. Introduction of (nonholomorphic) Eisenstein
series, cuspforms.
 [12] (DRAFT)
[
Invariant differential operators]
... [ updated 14:10, Oct 28, 2010]
...
More intrinsic discussion of differential operators related to group
actions. Introduction of Casimir operator in the
universal enveloping algebra attached to a Lie
algebra, etc. No assumption of prior acquaintance with Lie
algebras or Lie groups.
 [11] (DRAFT)
[
Functions on spheres]
... [ updated 15:10, Oct 10, 2010]
...
Harmonic polynomials, FourierLaplace series, Sobolev spaces, on
spheres. Duals, distributions (generalized functions).
 [10] (Functions on the line)
 [exercises 10]...
[ updated 13:12, Dec 11, 2005]
... Easy exercises about distributions, Fourier transforms, tempered
distributions
 [09]
[
Functions on circles, Fourier series, Sobolev spaces]
... [ updated 07:04, Apr 26, 2012]
...
Natural function spaces of kfold continuously differentiable
functions. Hilbertspace theory of Fourier series. Sobolev's
comparison of natural function spaces with certain Hilbert
spaces. Duals, distributions (generalized functions).
 [08]
[
Homogeneous spaces: spheres, projective spaces, nballs]
... [ updated 17:09, Sep 25, 2010]
...
Spheres with rotation groups acting, projective spaces with
projectivized linear actions, translation to linearfractional
transformations [sic], groups acting transitively on complex nballs.
 [07] (DRAFT)
[
Modular curves, raindrops through kaleidoscopes]
... [ updated 19:10, Oct 09, 2011]
...
Modular curves formed as quotients of the upper halfplane. Limits of
quotients by ppower congruence subgroups, action of
SL(2,Z_{p}) and SL(2,Q_{p}),
or GL(2,Z_{p}) and GL(2,Q_{p}) on the upperandlower
halfplanes. Nonabelian analogue of
solenoids. In a picture, the simplest modular curve looks like a
raindrop, and the projective limit is something seen through a
kaleidoscope. Rudimentary pictures eventually.
 [06]
[Historical origins]
... [ updated 10:10, Oct 08, 2011]
...
Bits of history, especially to clarify etymology: integrals for
arc length of ellipses, elliptic integrals, elliptic functions,
lattices/modules, modular forms. (Perhaps the traditional pictures
will be inserted at some later point.)
 [exercises 06]...
[ updated 13:11, Nov 19, 2005]
...
Constructions of periodic functions, orbits on projective spaces, some
counting issues, other oddments.
 [05]
[Comparison with
classical presentations of padic numbers]
... [ updated 10:09, Sep 14, 2010]
...
Hensel's lemma, classical metric definition of padic numbers, padic
exponential and logarithm (developing formal power series as useful
device), comparison with projective limit definition. Similar
comparison of definitions of adeles.
 [exercises 05]...
[pdf]
...
basic classical viewpoint on padic numbers
 [exercises 04]...
[ updated 14:10, Oct 30, 2005]
...
metrics, completeness, more colimits, more automorphism of solenoids
 [04] [The ursolenoid and adeles]
... [ updated 16:09, Sep 11, 2010]
...
More solenoids, with automorphism groups factoring over primes,
leading to the universal or ursolenoid, which also introduces the
adeles, as a colimit, much as Q_{p} is a
colimit of p^{n}Z_{p}. Incidental
very general results about limits and products commuting, isomorphism of
cofinal (directed) limits.
 [exercises 03]...
[ updated 14:10, Oct 30, 2005]
...
Some topology, some commutation of operators, some galois theory.
 [03] [Bigger diagrams, more
automorphisms, colimits]
... [ updated 19:09, Sep 19, 2011]
...Bigger automorphism groups visible via bigger diagram for
2solenoid. 2adic numbers as colimit. Slightly
broader discussion of colimits, strict colimits.
 [exercises 02]...
[ updated 17:10, Oct 03, 2005]
...
Further mappingproperty exercises.
 [02] [Solenoids]
... [ updated 15:09, Sep 11, 2010]
...
Initial fragment of discussion of projective limits
illustrated by solenoids (after Eilenberg). This is the beginning of a
story that will show how padic
groups, adele groups, etc. arise naturally as automorphisms of
families of more primitive, simpler objects. Review
of fundamentals regarding topological groups.
 [exercises 01]...
...
Some basic mappingproperty exercises.
 [01] [Review example: product topology]
...
[ updated 12:01, Jan 06, 2006]
...
Review example: characterization of objects by (universal) mapping
properties, the product topology. Why is the product topology so coarse?
Unless explicitly noted otherwise, everything here, work
by Paul Garrett, is licensed
under a Creative
Commons Attribution 3.0
Unported License.
...
[ garrett@math.umn.edu ]
The University of Minnesota explicitly requires that I
state that "The views and opinions expressed in this page are
strictly those of the page author. The contents of this page have not
been reviewed or approved by the University of Minnesota."