Modular Forms and L-functions
[ambient page updated 11:04, Apr 14, 2012]
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(2005-06 notes lower on the page)
Math 8207-08, 2010-11, 209 Vincent Hall, 2:30-3:20, MWF
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 ]
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
- ... [ Traces, Cauchy's identity,
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]
14:06, Jun 06, 2011]...
- ... [ Weil's proof of
product expansion of Delta
] ... with appendix giving Siegel's proof
continuation and functional equation for the simplest Eisenstein
series, for SL(2,Z), by Poisson summation
18:06, Jun 07, 2011]...
cont'n and functional equation for GL(2) Eisenstein
series, over number fields, with general data, by Poisson summation
19:06, Jun 06, 2011]...
- [ Continuous automorphic
spectrum for SL(2,Z)]
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
- [ spheres and
hyperbolic spaces ]
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
- [ Exercise 16:]
Due approximately Fri, April 15, 2011.
- Superpositions of eigenfunctions, intertwinings, asymptotic behavior
- [asymptotics at regular
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
10:08, Aug 13, 2011]... Examples:
rotationally invariant eigenfunctions of Laplacian in
Euclidean spaces at infinity; Whittaker/Bessel functions at
- [exceptional regular
16:05, May 14, 2011]... Exceptional
regular singular points. These occur in some important
examples, and it is essential to understand the second
Harmonic analysis on spheres I, invariant Laplacian,
spherical harmonics ]
17:02, Feb 28, 2011]
Harmonic analysis on spheres II, Sobolev inequalities
] ... [ updated 16:02, Feb 27, 2011]
Intrinsic characterization of Laplacians, Casimir elements ]
07:11, Nov 11, 2011]
- [ Exercise 15:
] Due approximately Fri, Mar 11, 2011.
Iwasawa-Tate: first pass
17:06, Jun 06, 2011]...
- 08 Analytic
continuations and functional equations
[ updated 12:06, Jun 22, 2012] examples:
Dirichlet L-functions, Dedekind zeta for Gaussian integers,
Hecke grossencharacters for Gaussian
[ 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
- 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
[ updated 11:10, Oct 02, 2010]
... with thanks to Hadamard, von Mangoldt... and a host of others...
- 01 Introduction to
[ 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
- Typical sources: not required, but interesting supplements,
contrasting viewpoints, styles, contexts, goals
- H. Cohen, Number Theory Vol I: Tools and Diophantine
Equations; Vol II: Analytic and Modern Tools
- Serre Course in arithmetic
- Iwaniec Spectral methods in automorphic forms
- Bump Automorphic forms and representations
- Borel Automorphic forms on SL(2,R)
- Garrett Holomorphic Hilbert modular forms
- Bernstein, Gelbart, editors, An introduction to the
- Kowalski-Iwaniec Analytic Number Theory
Notes from 2005-06
Notes (reverse chronological order):
- [ 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
-  (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.
-  (DRAFT)
... [ updated 13:11, Nov 01, 2010]
Beginning of study of eigenfunctions for the invariant Laplacian on
the upper half-plane. Introduction of (non-holomorphic) Eisenstein
-  (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.
-  (DRAFT)
Functions on spheres]
... [ updated 15:10, Oct 10, 2010]
Harmonic polynomials, Fourier-Laplace series, Sobolev spaces, on
spheres. Duals, distributions (generalized functions).
-  (Functions on the line)
- [exercises 10]...
[ updated 13:12, Dec 11, 2005]
... Easy exercises about distributions, Fourier transforms, tempered
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).
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.
-  (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.
... [ 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.
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]...
basic classical viewpoint on p-adic numbers
- [exercises 04]...
[ updated 14:10, Oct 30, 2005]
metrics, completeness, more colimits, more automorphism of solenoids
-  [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.
-  [Bigger diagrams, more
... [ 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.
-  [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.
-  [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?
© 1996-2012, Creative Commons license,
by Paul Garrett is licensed
under a Creative
Commons Attribution 3.0
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