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Algebraic Number Theory 20112012
201112 Notes
[updated
Friday, 09Jan2015 17:41:03 CST]
... 209 pages: overheads reformatted as normal text, repetitions
eliminated, some examples and details added.
See also
modular forms notes from 20056 and 201011 and 201314 (current).
201112 Overheads in reverse chrono order:

05042012
Sketch of main points of homological formulation of classfield theory,
with some explanations: Tate cohomology of groups, etc,
[updated
Saturday, 05May2012 11:06:24 CDT]

05022012
Classfield theory, (co)homological formulation, cont'd:
Herbrand quotient, example: local cyclic norm index equality.
[updated
Thursday, 03May2012 14:04:26 CDT]

04302012
Classfield theory, (co)homological formulation, cont'd:
Herbrand quotient, homology long exact sequences, Euler
characteristics, example: local cyclic norm index equality.
[updated
Tuesday, 01May2012 07:39:57 CDT]

04272012
Classfield theory, (co)homological formulation, cont'd:
Herbrand quotient, homology long exact sequences, Euler
characteristics, snake lemma, example: meromorphic continuation of
Gamma (!)
[updated
Thursday, 26Apr2012 20:13:49 CDT]

04252012
Classfield theory, (co)homological formulation, cont'd:
homology long exact sequences, Euler characteristics, snake lemma, examples
[updated
Thursday, 26Apr2012 07:33:43 CDT]

04232012
Classfield theory, (co)homological formulation, cont'd:
recollection of how to understand holes in higherdimensional
spheres by homology long exact sequences, intro to group (co)homology
[updated
Tuesday, 24Apr2012 16:31:49 CDT]

04202012
Classfield theory, cont'd: assembly of local to global
statement. Grounding for post1940 (co)homological formulation:
recollection of how to understand holes in higherdimensional
spheres by homology long exact sequences.
[updated
Friday, 20Apr2012 16:12:26 CDT]

04182012
Classfield theory, cont'd: assembly of local to global
statement. Recollection in quadratic case that
k^{x}invariance of idelic norm residue symbol yields quadratic
reciprocity.
Grounding for post1940 (co)homological formulation.
[updated
Friday, 20Apr2012 16:04:24 CDT]

04162012
Classfield theory, cont'd: idelic statement of global classfield
theory, review of Kummer theory and independence of roots,
cyclotomic extensions of Q, Hilbert's theorem 90
[updated
Monday, 16Apr2012 15:58:22 CDT]

04132012
Classfield theory, cont'd: idelic statement of global classfield
theory, review of Kummer theory and independence of roots
[updated
Monday, 16Apr2012 15:58:06 CDT]

04112012
Classfield theory, cont'd: review of extensions of
unramified extensions of local fields, quadratic extensions of local
fields, Kummer theory
[updated
Monday, 16Apr2012 15:57:43 CDT]

04092012
Classfield theory, cont'd: review of extensions of finite fields,
unramified extensions of local fields
[updated
Monday, 09Apr2012 17:37:38 CDT]

04062012
Classfield theory: basic overview
[updated
Saturday, 07Apr2012 17:28:20 CDT]

04042012
Completion of proof of Hecke's identity... introduction of
intrinsic action on representation spaces of Casimir operator, from
center of universal enveloping algebra. Direct computation that
Casimir acts as spherical Laplacian.
[updated
Tuesday, 03Apr2012 14:00:28 CDT]

04022012
Completion of proof of Hecke's identity... via some representation
theory of orthogonal groups
[updated
Monday, 02Apr2012 20:21:21 CDT]

03302012
Interlude: Hecke's identity for harmonic polynomial multiples of
Gaussians, cont'd... some representation theory of orthogonal groups
[updated
Saturday, 31Mar2012 14:54:08 CDT]

03282012
Interlude: Hecke's identity for harmonic polynomial multiples of
Gaussians, cont'd
[updated
Wednesday, 28Mar2012 16:44:22 CDT]

03262012
Interlude: Hecke's identity for harmonic polynomial multiples of Gaussians
[updated
Tuesday, 27Mar2012 10:47:24 CDT]

03232012
Details supporting IwasawaTate:
unramified and ramified archimedean local zeta integrals, convergence
of local zetas, Fourier transforms of ramified archimedean
integrals, Hecke's identity
[updated
Saturday, 24Mar2012 14:50:42 CDT]

03212012
Details supporting IwasawaTate:
Convergence of global halfzeta integrals, Hecke's identity, harmonic
polynomials, harmonic analysis on spheres.
[updated
Tuesday, 20Mar2012 14:42:53 CDT]

03192012
Details supporting IwasawaTate: good finiteprime local zeta integrals,
unramified and ramified archimedean local zeta integrals
[updated
Tuesday, 20Mar2012 09:22:43 CDT]
 Products of entire functions, especially (s1)zeta(s):

03092012
Beginning followup details supporting IwasawaTate: elementary
global integrals, good finiteprime local integrals, local
functional equation
[updated
Friday, 09Mar2012 16:46:20 CST]

03072012
IwasawaTate executed for Dedekind zetas, and then the general
case. Comments on needed supporting material.
[updated
Friday, 09Jan2015 17:38:09 CST]

03052012
IwasawaTate executed for Dirichlet Lfunctions, for Dedekind zetas
[updated
Monday, 05Mar2012 15:24:42 CST]

03022012
IwasawaTate executed for EulerRiemann
zeta, for Dirichlet Lfunctions
[updated
Thursday, 01Mar2012 11:25:29 CST]

Nonoverhead 02292012
First pass at IwasawaTate: the argument executed for EulerRiemann
zeta, for Dirichlet Lfunctions, for Dedekind zetas, for
grossencharacter Lfunctions.
... some details postponed.
[updated
Monday, 06Jun2011 17:30:55 CDT]
 02272012
Toward IwasawaTate: compact operators, HilbertSchmidt integral operators.
...
[updated
Friday, 24Feb2012 15:52:05 CST]
 02242012
Toward IwasawaTate: decomposition of L^{2} of compact
abelian groups by compact operators: HilbertSchmidt integral operators.
...
[updated
Friday, 24Feb2012 14:50:42 CST]
 02222012
Toward IwasawaTate: Schwartz spaces, Fourier inversion,
decomposition of L^{2} of compact abelian groups, by compact
operators. Toward adelic Poisson summation.
...
[updated
Thursday, 23Feb2012 09:33:59 CST]
 02202012
Toward IwasawaTate: padic Fourier transform, Fourier inversion.
...
[updated
Monday, 20Feb2012 16:42:19 CST]
 02172012
Toward IwasawaTate: Fourier transform, Fourier inversion, on
archimedean and padic completions
...
[updated
Sunday, 19Feb2012 15:50:59 CST]
 02152012
Toward IwasawaTate: unitary duals, characters, Fourier inversion
...
[updated
Thursday, 16Feb2012 10:01:36 CST]
 02132012
Toward IwasawaTate's modernization of Hecke's treatment of
Lfunctions: unitary duals
of Q_{p}, A,
and A/Q.
...
[updated
Monday, 13Feb2012 15:40:48 CST]
 02102012
General argument for uniqueness of invariant functionals. Introduction
to IwasawaTate's modernization of Hecke's extensions of Riemann's
argument for meromorphic continuation and functional equation.
...
[updated
Friday, 10Feb2012 15:37:11 CST]
 02082012
measure and integration on Q_{p} and A. General
uniqueness proof.
...
[updated
Thursday, 09Feb2012 18:06:24 CST]
 02062012
volumes of arithmetic quotients, special values of Lfunctions
...
[updated
Tuesday, 07Feb2012 08:48:01 CST]
 [hiatus]
 01232012
measures on quotients, volume of SL(n,Z)\SL(n,R)
...
[updated Sunday, 22Jan2012 20:49:30 CST]
 01202012
Background for Fujisaki's lemma (and corollaries on class groups,
units): invariant measures on quotients, iterated integrals, comparison to
Minkowski's classical results, external characterization of
invariant measures/integrals.
...
[updated Friday, 20Jan2012 13:13:54 CST]
 01182012
(Recap of Fujisaki's lemma, ideal class groups as idele class groups,
units theorem.) Invariant measures on quotients, iterated integrals.
...
[updated Thursday, 19Jan2012 14:31:52 CST]
 12142011
Interlude/preview... toward IwasawaTate theory: selfduality
of Q_{p} and A, mutual duality
of A/k and k, compactopen
topology, nosmallsubgroups, ...
[updated Tuesday, 13Dec2011 19:35:15 CST]
 12122011
Generalized ideal class groups are images of the idele class group,
classification of closed subgroups of R^{n},
other supporting stuff...
[updated Monday, 12Dec2011 18:16:25 CST]
 12092011
Proof of Fujisaki's compactness lemma, measuretheory pigeonhole
principle, finiteness of class number, Dirichlet
Units Theorem, extensions
[updated Saturday, 10Dec2011 12:03:20 CST]
 12072011
Proof of Fujisaki's compactness lemma, finiteness of class number, Dirichlet
Units Theorem, extensions
[updated Wednesday, 07Dec2011 14:15:45 CST]
 12052011
Fujisaki's compactness lemma, finiteness of class number, Dirichlet
Units Theorem
[updated Monday, 05Dec2011 12:25:09 CST]
 12022011
Ostrowski's theorem, Approximation Theorem, adelic solenoid A/k.
[updated Friday, 02Dec2011 12:19:57 CST]
 11302011
Product formula, Ostrowski's theorem, Approximation Theorem
[updated Wednesday, 30Nov2011 18:11:20 CST]
 11282011
... completions, finitedimensional topological vector
spaces, product formula
[updated Tuesday, 29Nov2011 11:34:52 CST]
 11232011
... completions, finitedimensional topological vector
spaces, product formula
[updated Saturday, 26Nov2011 09:31:26 CST]
 11212011
Absolute values, completions, finitedimensional topological vector spaces
[updated Sunday, 20Nov2011 11:41:54 CST]
 some class number data
class numbers of the first few hundred complex quadratic fields
[updated Monday, 21Nov2011 07:32:40 CST]
 11182011
class number formula for complex quadratic fields
[updated Friday, 18Nov2011 12:39:10 CST]
 11162011
... Dedekind zeta functions, class numbers, residues of Epstein zetas,
class number formula for complex quadratic fields
[updated Thursday, 17Nov2011 10:18:40 CST]
 11142011
... ramification degrees, residue class field degrees, ideal norms,
Dedekind zeta functions
[updated Tuesday, 15Nov2011 10:22:59 CST]
 11112011
... basic big theorem characterizing Dedekind domains, cont'd + sequel
[updated Friday, 11Nov2011 14:03:09 CST]
 11092011
... basic big theorem on factorization in Dedekind domains
[updated Thursday, 10Nov2011 10:13:56 CST]
 11072011
... more: Galois theory of primes lying over, intro to Dedekind rings.
[updated Monday, 07Nov2011 12:20:49 CST]
 11042011
...better localization... Galois theory of primes lying over, intro to
Dedekind rings.
[updated Saturday, 05Nov2011 10:07:11 CDT]
 11022011
A better version of localization.
[updated Thursday, 03Nov2011 10:26:23 CDT]
 10312011
Primes lying over, Galois action. Review of localization.
[updated Saturday, 29Oct2011 11:29:10 CDT]
 10282011
Brief recollection of quadratic normresidue, Hilbert symbols, reciprocity laws. Primes
lying over, Galois action.
[updated Saturday, 29Oct2011 11:29:10 CDT]
 10262011
Quadratic normresidue, Hilbert symbols, reciprocity laws. Primes
lying over, Galois action.
[updated Thursday, 27Oct2011 12:41:38 CDT]
 10242011
function fields over finite fields ... quadratic normresidue, Hilbert
symbols, reciprocity laws
[updated Sunday, 23Oct2011 12:21:32 CDT]
 10212011
Cont'd: function fields... especially over finite
fields... ArtinSchreier extensions... quadratic norm/Hilbert
reciprocity laws
[updated Sunday, 23Oct2011 11:24:42 CDT]
 10192011
Cont'd: function fields, Galois groups of algebraic closures,
comments on Galois groups and their representations, the usefulness of
repn theory, ... and then focus on function fields over finite fields.
[updated Friday, 21Oct2011 07:59:15 CDT]
 10172011
Cont'd: function fields, especially extensions
of C(X) and C((X)). Formal Puiseux
expansions, Newton polygons.
[updated Tuesday, 18Oct2011 11:44:09 CDT]
 10142011
function fields, especially extensions
of C(X) and C((X)). Formal Puiseux
expansions, Newton polygons.
[updated Saturday, 15Oct2011 13:52:27 CDT]
 10122011
function fields
[updated Thursday, 13Oct2011 09:36:36 CDT]
 10102011
commutative algebra: integral extensions, Noetherianness, function
field case
[updated Monday, 10Oct2011 12:20:01 CDT]
 10072011
commutative algebra: integral extensions, Noetherianness
[updated Friday, 07Oct2011 12:56:29 CDT]
 10052011
commutative algebra: integral extensions, algebraic integers
[updated Friday, 07Oct2011 13:02:16 CDT]
 10032011
some commutative algebra: integral extensions, algebraic integers
[updated Sunday, 09Oct2011 16:55:37 CDT]
 09302011
padic numbers, projective limits, colimits, adeles
[updated Friday, 30Sep2011 14:44:47 CDT]
 09282011
more on Hensel's Lemma, padic numbers, projective limits
[updated Thursday, 29Sep2011 09:06:27 CDT]
 09262011
more on Hensel's Lemma, padic numbers
[updated Sunday, 25Sep2011 17:04:29 CDT]
 09232011
Hensel's Lemma, padic numbers
[updated Sunday, 25Sep2011 11:07:41 CDT]
 09212011
Cont'd: more factoring Dedekind zeta functions
[updated Wednesday, 21Sep2011 12:21:22 CDT]
 09192011
Cont'd: factoring Dedekind zeta functions into Dirichlet Lfunctions
[updated Tuesday, 20Sep2011 07:49:20 CDT]
 09162011
factorization of some
Dedekind zeta functions into Dirichlet Lfunctions
[updated Saturday, 17Sep2011 15:06:55 CDT]
 09142011
Quadratic Reciprocity over Q, by Gauss sums
[updated Wednesday, 14Sep2011 14:15:36 CDT]
 09122011
meromorphic continuation and functional equation of zeta, Poisson
summation and functional equation of theta, integral representation
of zeta in terms of theta
[updated Sunday, 11Sep2011 18:00:38 CDT]
 09092011
Riemann's explicit formula
[updated Sunday, 11Sep2011 18:17:10 CDT]
Older notes
Unless explicitly noted otherwise, everything here, work
by Paul Garrett, is licensed
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