Modular Forms and L-functions, Math
8207-8208
A course in modern number theory and harmonic analysis
[ambient page updated 20:03, Mar 31, 2020]
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[ garrett@math.umn.edu
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( See also:
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[ functional analysis ]
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[ intro to modular forms ]
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[ representation theory ]
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[ Lie theory, symmetric spaces ]
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[ number theory ]
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[ complex analysis ]
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[ real analysis ]
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[ homological algebra ]
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My book Modern Analysis of
Automorphic Forms by Example (also: Cambridge University
Press, Cambridge Studies in Advanced Mathematics, vols 173, 174
). The contract allows CUP to sell physical copies, and (by
design) allows me to put this PDF on-line, under the reasonable
condition that I ask that people only download copies for
personal use (or, conceivably, just link to this page), and that
I link to CUP's pages for this book:
CUP's page for volume one ,
CUP's page for volume two
2019-2020
Fall 2019: Vincent Hall 2,
11:15-12:05, MWF
Phenomena, examples, history
This introductory Number Theory course will be accessible to
first-year and second-year 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.
Students are encouraged to look at other sources, and, especially,
to ask questions in class. The chief (only?) benefit of live lectures,
as opposed to textbooks or notes, is the possibility of asking
questions.
- Classical ideas for Euler-Riemann zeta
- Euler and zeta
[ updated 12:09, Sep 15, 2019]
- Estermann
phenomenon [updated 17:11, Nov 11, 2021]
- Riemann and zeta
[updated 18:09, Sep 14, 2019]
- Guinand and
zeta [updated 13:09, Sep 15, 2019]
-
Background
- Fourier series
[updated 14:09, Sep 15, 2019]
- The Gamma
function [updated 14:09, Sep 15, 2019]
-
Hadamard products [updated 14:09, Sep 15, 2019]
-
Phragmen-Lindelof [updated 14:09, Sep 15, 2019]
-
asymptotics of integrals [updated 14:09, Sep 15, 2019]
-
counting zeros [updated 11:02, Feb 23, 2023]
-
Jensen's formula [updated 07:09, Sep 16, 2019]
- Dirichlet L-functions, primes in arithmetic progressions,
reciprocity laws
- Dirichlet
L-functions, primes in arithmetic progressions [
updated 14:09, Sep 21, 2019]
- First
proof of non-vanishing on Re(s)=1 [ updated
14:09, Sep 21, 2019]
- Fourier
analysis on finite abelian groups [ updated
14:09, Sep 21, 2019]
- Special
values [ updated 14:09, Sep 21, 2019]
-
Functional equations [ updated 14:09, Sep 21, 2019]
- Gauss sums
[ updated 14:09, Sep 21, 2019]
- Dedekind zeta functions, Hecke L-functions
-
Definition of Dedekind zetas
- Quadratic Hilbert symbols and
quadratic reciprocity
- Factorization of zeta functions of
quadratic extensions
- Reciprocity for cyclotomic extensions,
and factorization of zeta functions
- Hecke's proofs of
analytic continuation
- Hecke L-functions: examples
-
Preview of Iwasawa-Tate
- Artin L-functions: definitions, partial results, and
conjectures
- Deligne's conjectures on special values of L-functions
- Elliptic functions, elliptic modular forms, Eisenstein
series, theta series
- Klingen's theorem on special values via
Hilbert-Blumenthal modular Eisenstein series
- Kloosterman-type equidistribution problems on
spheres
- Iwasawa-Tate theory for GL(1): (see second part of My
algebraic number theory notes
- Global zeta
integrals as Euler products (of local zeta integrals)
- Local
zeta integrals: local functional equation, meromorphic
continuation
- Global functional equation, meromorphic
continuation
- Details:
- p-adic and adelic Fourier
transforms
- Convergence of half-zeta integrals
- Adelic
Poisson summation
- Duality (A/k)^=k
- Self-duality of A,
R, C, Qp
- Hecke operators, Euler products, standard L-functions
attached to modular forms
- Rankin-Selberg L-functions
- Meromorphic continuation of Eisenstein series, Hecke
summation
- Maass' waveforms
- Hilbert modular forms, Siegel
modular forms
- ...
- Heisenberg groups, Segal-Shale-Weil, theta correspondences,
Siegel-Weil theorem
- representations of finite
abelian groups [ updated 10:03, Mar 12, 2020]
- representations of
finite dihedral groups [ updated
10:03, Mar 12, 2020]
- generalities
on representations of finite groups [ updated
10:03, Mar 12, 2020]
- more
generalities on representations of finite groups
[ updated 10:03, Mar 12, 2020]
- representations of
finite GL2 [ updated 10:03, Mar 12, 2020]
- finite
Heisenberg groups and Segal-Shale-Weil [
updated 10:03, Mar 12, 2020]
- Stone-vonNeumann
theorem over local fields [ updated
17:03, Mar 23, 2020]
- Lie algebra
models of Segal-Shale-Weil/oscillator representation of
SL2 [ updated 13:03, Mar 31, 2020]
- Local SL(2) x O(2)
[ updated 13:03, Mar 31, 2020]
- Reciprocity for
global quadratic norm residue symbols [ updated 16:02, Feb 26, 2023]
- ...
2017-18
Phenomena, examples, history
This introductory Number Theory course will be accessible to
first-year and second-year 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.
Tentative partial course outline:
- The Euler-Riemann zeta function, explicit formulas, Riemann Hypothesis
- Dirichlet's L-functions, primes in arithmetic progressions, reciprocity laws
- Simplest cases of Deligne's conjectures on special values of
L-functions
- Elliptic functions, elliptic modular forms, Eisenstein series,
theta series
- Application: Kloosterman-type equidistribution problems on spheres
- Application: Klingen's evaluation of L-functions of totally
real fields
- Hecke operators, Euler products, Mellin transforms, converse theorems
- Rankin-Selberg L-functions
- Meromorphic continuation of Eisenstein series, Hecke summation
- Maass-Shimura operators and Rankin-Selberg
- Maass' waveforms, L-functions
- Hilbert modular forms, Siegel modular forms
- Decomposition of Eisenstein series
- ...
2015-16
Phenomena, examples
This introductory Number Theory course will be accessible to
first-year and second-year 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 non-vanishing 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 L-functions for GL(2)
- Hecke operators and Euler products
- simplest Rankin-Selberg L-functions and meromorphic continuation and functional equation of relevant Eisenstein series
[ updated
17:07, Jul 09, 2010]
- Waveforms
- Pre-trace formulas and spectral theory of automorphic forms/functions
- Hilbert-Blumenthal modular forms
- Siegel modular forms
- Transition to GL2(A) and GLn(A)
- Theta lifts/correspondences, Segal-Shale-Weil representations:
- Hecke-Maass correspondence: SL(2) x SO(2)
- Doi-Naganuma correspondence: SL(2) x SO(4)
- Yoshida correspondence: Sp(4) x SO(2,2)
- Comments on: Shintani-Niwa, Saito-Kurakawa correspondences
2013-14
This course introduces many phenomena that
led to much contemporary research, including the Langlands program
and much more. Little prior acquaintance with higher-level
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 linked-to as
we go, somewhat in advance of progress in-class. If you must print
notes, please don't do so until just before reading, because many updates
will occur.
See
also
Number theory notes 2011-12 for related discussions
- ...
- 14 Spectral theory of waveforms
- 13
- 12 Automorphic forms on adele groups, Hecke operators
- 11 Level-one
waveforms, Heegner points, geodesic periods of Eisenstein
series, non-vanishing... ...[ updated
15:09, Sep 07, 2023]
- 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
Level-one holomorphic elliptic modular forms
...[ updated
11:11, Nov 17, 2013]
- 07 Fundamental domain for
SL(2,Z) on the upper half-plane
...[ 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 L-functions: 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]
-
Phragmen-Lindelof theorem
...[ updated
09:08, Aug 30, 2013]
-
Estermann
phenomenon: non-existence 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 number-theoretic
applications of Hilbert modular forms.
2010-11
Office hours: MWF 1:25-2:15 or by appointment, email anytime
An introduction to number theory, zeta functions and
L-functions, 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 GL2(Qp), to
see Bruhat-cell decomposition of constant term with Euler
product of big-cell summand, and to compute Hecke eigenvalues
locally.
- ... [ Traces, Cauchy's identity,
Schur functions]
...[ updated
16:06, Jun 28, 2011]... A
representation-theory identity giving the spherical GL(n) case of the
local Rankin-Selberg integral.
-
... [ intro to Fourier-Whittaker
expansions of cuspforms on GL(n), and some L-functions]
...[ 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 pseudo-Eisenstein series
with test-function data. Pseudo-Eisenstein 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):
translation-equivariant eigenfunctions (Whittaker/Bessel functions) at 0; dilation-equivariant
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
Iwasawa-Tate: first pass
...[ updated
16:05, May 23, 2020]...
- 10
- 09
- 08 Analytic
continuations and functional equations
[ updated 11:10, Oct 24, 2018] examples:
Dirichlet L-functions, Dedekind zeta for Gaussian integers,
Hecke grossencharacters for Gaussian
integers...
- 07
-
[ Exercise 07:
Euclidean-ness ] (and starred exercises about topological
subgroups of Rn, 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 non-vanishing 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) non-vanishing.
- 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
L-functions. Automorphic/modular forms: holomorphic Eisenstein
series, theta series, waveform Eisenstein series
2005-06
- [ 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 finite-dimensional
complex vectorspaces.
Notes (reverse chronological order):
- [14] (DRAFT)
[
Dirichlet series from automorphic forms]
... [ updated 11:10, Oct 23, 2018]
...
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 11:10, Oct 23, 2018]
...
Beginning of study of eigenfunctions for the invariant Laplacian on
the upper half-plane. Introduction of (non-holomorphic) 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, Fourier-Laplace 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 k-fold continuously differentiable
functions. Hilbert-space 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, n-balls]
... [ updated 17:09, Sep 25, 2010]
...
Spheres with rotation groups acting, projective spaces with
projectivized linear actions, translation to linear-fractional
transformations [sic], groups acting transitively on complex n-balls.
- [07] (DRAFT)
[
Modular curves, raindrops through kaleidoscopes]
... [ updated 19:10, Oct 09, 2011]
...
Modular curves formed as quotients of the upper half-plane. Limits of
quotients by p-power congruence subgroups, action of
SL(2,Zp) and SL(2,Qp),
or GL(2,Zp) and GL(2,Qp) on the upper-and-lower
half-planes. Non-abelian 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 p-adic numbers]
... [ updated 10:09, Sep 14, 2010]
...
Hensel's lemma, classical metric definition of p-adic numbers, p-adic
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 p-adic numbers
- [exercises 04]...
[ updated 14:10, Oct 30, 2005]
...
metrics, completeness, more colimits, more automorphism of solenoids
- [04] [The ur-solenoid and adeles]
... [ updated 16:09, Sep 11, 2010]
...
More solenoids, with automorphism groups factoring over primes,
leading to the universal or ur-solenoid, which also introduces the
adeles, as a colimit, much as Qp is a
colimit of p-nZp. 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
2-solenoid. 2-adic numbers as colimit. Slightly
broader discussion of colimits, strict colimits.
- [exercises 02]...
[ updated 17:10, Oct 03, 2005]
...
Further mapping-property 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 p-adic
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 mapping-property 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@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."