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[Lie theory], ...
Algebraic Number Theory 2011-2012
MWF 11:15-12:05, Vincent 02
See also
modular forms notes from 2005-6 and 2010-11.
Additional references:
- H. Cohen, Number Theory: Vol I, Tools and Diophantine
Equations, Vol. II, Analytic and Modern Tools
- S. Lang, Algebraic Number Theory
- F. Lemmermeyer Reciprocity Laws, from Euler to
Eisenstein
- F. Lemmermeyer, "Harbingers of Artin's Reciprocity Law":
- I. The Continuing Story of Auxiliary Primes, arXiv:1109:1228
- II. Irreducibility of Cyclotomic Polynomials, arXiv:1201:5776
- III. Gauss' Lemma and Artin's Transfer, arXiv:1202.5772
- IV. Bernstein's Reciprocity Law, arXiv:1202.5774
- E. Artin, J. Tate, Class Field Theory (!?)
- N. Childress Class Field Theory (!?)
- J. Neukirch, big book, and little book, Class Field
Theory
- E. Artin, J. Tate Class Field Theory
- J. Tate Global Class Field Theory , in
Cassels-Frohlich 1967.
- C. Weibel, An intro to Homological Algebra
2011-12 Overheads in reverse chrono order:
-
05-04-2012
Sketch of main points of homological formulation of classfield theory,
with some explanations: Tate cohomology of groups, etc,
[updated
Saturday, 05-May-2012 11:06:24 CDT]
-
05-02-2012
Classfield theory, (co)homological formulation, cont'd:
Herbrand quotient, example: local cyclic norm index equality.
[updated
Thursday, 03-May-2012 14:04:26 CDT]
-
04-30-2012
Classfield theory, (co)homological formulation, cont'd:
Herbrand quotient, homology long exact sequences, Euler
characteristics, example: local cyclic norm index equality.
[updated
Tuesday, 01-May-2012 07:39:57 CDT]
-
04-27-2012
Classfield theory, (co)homological formulation, cont'd:
Herbrand quotient, homology long exact sequences, Euler
characteristics, snake lemma, example: meromorphic continuation of
Gamma (!)
[updated
Thursday, 26-Apr-2012 20:13:49 CDT]
-
04-25-2012
Classfield theory, (co)homological formulation, cont'd:
homology long exact sequences, Euler characteristics, snake lemma, examples
[updated
Thursday, 26-Apr-2012 07:33:43 CDT]
-
04-23-2012
Classfield theory, (co)homological formulation, cont'd:
recollection of how to understand holes in higher-dimensional
spheres by homology long exact sequences, intro to group (co-)homology
[updated
Tuesday, 24-Apr-2012 16:31:49 CDT]
-
04-20-2012
Classfield theory, cont'd: assembly of local to global
statement. Grounding for post-1940 (co)homological formulation:
recollection of how to understand holes in higher-dimensional
spheres by homology long exact sequences.
[updated
Friday, 20-Apr-2012 16:12:26 CDT]
-
04-18-2012
Classfield theory, cont'd: assembly of local to global
statement. Recollection in quadratic case that
kx-invariance of idelic norm residue symbol yields quadratic
reciprocity.
Grounding for post-1940 (co)homological formulation.
[updated
Friday, 20-Apr-2012 16:04:24 CDT]
-
04-16-2012
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, 16-Apr-2012 15:58:22 CDT]
-
04-13-2012
Classfield theory, cont'd: idelic statement of global classfield
theory, review of Kummer theory and independence of roots
[updated
Monday, 16-Apr-2012 15:58:06 CDT]
-
04-11-2012
Classfield theory, cont'd: review of extensions of
unramified extensions of local fields, quadratic extensions of local
fields, Kummer theory
[updated
Monday, 16-Apr-2012 15:57:43 CDT]
-
04-09-2012
Classfield theory, cont'd: review of extensions of finite fields,
unramified extensions of local fields
[updated
Monday, 09-Apr-2012 17:37:38 CDT]
-
04-06-2012
Classfield theory: basic overview
[updated
Saturday, 07-Apr-2012 17:28:20 CDT]
-
04-04-2012
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, 03-Apr-2012 14:00:28 CDT]
-
04-02-2012
Completion of proof of Hecke's identity... via some representation
theory of orthogonal groups
[updated
Monday, 02-Apr-2012 20:21:21 CDT]
-
03-30-2012
Interlude: Hecke's identity for harmonic polynomial multiples of
Gaussians, cont'd... some representation theory of orthogonal groups
[updated
Saturday, 31-Mar-2012 14:54:08 CDT]
-
03-28-2012
Interlude: Hecke's identity for harmonic polynomial multiples of
Gaussians, cont'd
[updated
Wednesday, 28-Mar-2012 16:44:22 CDT]
-
03-26-2012
Interlude: Hecke's identity for harmonic polynomial multiples of Gaussians
[updated
Tuesday, 27-Mar-2012 10:47:24 CDT]
-
03-23-2012
Details supporting Iwasawa-Tate:
unramified and ramified archimedean local zeta integrals, convergence
of local zetas, Fourier transforms of ramified archimedean
integrals, Hecke's identity
[updated
Saturday, 24-Mar-2012 14:50:42 CDT]
-
03-21-2012
Details supporting Iwasawa-Tate:
Convergence of global half-zeta integrals, Hecke's identity, harmonic
polynomials, harmonic analysis on spheres.
[updated
Tuesday, 20-Mar-2012 14:42:53 CDT]
-
03-19-2012
Details supporting Iwasawa-Tate: good finite-prime local zeta integrals,
unramified and ramified archimedean local zeta integrals
[updated
Tuesday, 20-Mar-2012 09:22:43 CDT]
- Products of entire functions, especially (s-1)zeta(s):
-
03-09-2012
Beginning follow-up details supporting Iwasawa-Tate: elementary
global integrals, good finite-prime local integrals, local
functional equation
[updated
Friday, 09-Mar-2012 16:46:20 CST]
-
03-07-2012
Iwasawa-Tate executed for Dedekind zetas, and then the general
case. Comments on needed supporting material.
[updated
Wednesday, 07-Mar-2012 15:13:22 CST]
-
03-05-2012
Iwasawa-Tate executed for Dirichlet L-functions, for Dedekind zetas
[updated
Monday, 05-Mar-2012 15:24:42 CST]
-
03-02-2012
Iwasawa-Tate executed for Euler-Riemann
zeta, for Dirichlet L-functions
[updated
Thursday, 01-Mar-2012 11:25:29 CST]
-
Non-overhead 02-29-2012
First pass at Iwasawa-Tate: the argument executed for Euler-Riemann
zeta, for Dirichlet L-functions, for Dedekind zetas, for
grossencharacter L-functions.
... some details postponed.
[updated
Monday, 06-Jun-2011 17:30:55 CDT]
- 02-27-2012
Toward Iwasawa-Tate: compact operators, Hilbert-Schmidt integral operators.
...
[updated
Friday, 24-Feb-2012 15:52:05 CST]
- 02-24-2012
Toward Iwasawa-Tate: decomposition of L2 of compact
abelian groups by compact operators: Hilbert-Schmidt integral operators.
...
[updated
Friday, 24-Feb-2012 14:50:42 CST]
- 02-22-2012
Toward Iwasawa-Tate: Schwartz spaces, Fourier inversion,
decomposition of L2 of compact abelian groups, by compact
operators. Toward adelic Poisson summation.
...
[updated
Thursday, 23-Feb-2012 09:33:59 CST]
- 02-20-2012
Toward Iwasawa-Tate: p-adic Fourier transform, Fourier inversion.
...
[updated
Monday, 20-Feb-2012 16:42:19 CST]
- 02-17-2012
Toward Iwasawa-Tate: Fourier transform, Fourier inversion, on
archimedean and p-adic completions
...
[updated
Sunday, 19-Feb-2012 15:50:59 CST]
- 02-15-2012
Toward Iwasawa-Tate: unitary duals, characters, Fourier inversion
...
[updated
Thursday, 16-Feb-2012 10:01:36 CST]
- 02-13-2012
Toward Iwasawa-Tate's modernization of Hecke's treatment of
L-functions: unitary duals
of Qp, A,
and A/Q.
...
[updated
Monday, 13-Feb-2012 15:40:48 CST]
- 02-10-2012
General argument for uniqueness of invariant functionals. Introduction
to Iwasawa-Tate's modernization of Hecke's extensions of Riemann's
argument for meromorphic continuation and functional equation.
...
[updated
Friday, 10-Feb-2012 15:37:11 CST]
- 02-08-2012
measure and integration on Qp and A. General
uniqueness proof.
...
[updated
Thursday, 09-Feb-2012 18:06:24 CST]
- 02-06-2012
volumes of arithmetic quotients, special values of L-functions
...
[updated
Tuesday, 07-Feb-2012 08:48:01 CST]
- [hiatus]
- 01-23-2012
measures on quotients, volume of SL(n,Z)\SL(n,R)
...
[updated Sunday, 22-Jan-2012 20:49:30 CST]
- 01-20-2012
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, 20-Jan-2012 13:13:54 CST]
- 01-18-2012
(Recap of Fujisaki's lemma, ideal class groups as idele class groups,
units theorem.) Invariant measures on quotients, iterated integrals.
...
[updated Thursday, 19-Jan-2012 14:31:52 CST]
- 12-14-2011
Interlude/preview... toward Iwasawa-Tate theory: self-duality
of Qp and A, mutual duality
of A/k and k, compact-open
topology, no-small-subgroups, ...
[updated Tuesday, 13-Dec-2011 19:35:15 CST]
- 12-12-2011
Generalized ideal class groups are images of the idele class group,
classification of closed subgroups of Rn,
other supporting stuff...
[updated Monday, 12-Dec-2011 18:16:25 CST]
- 12-09-2011
Proof of Fujisaki's compactness lemma, measure-theory pigeon-hole
principle, finiteness of class number, Dirichlet
Units Theorem, extensions
[updated Saturday, 10-Dec-2011 12:03:20 CST]
- 12-07-2011
Proof of Fujisaki's compactness lemma, finiteness of class number, Dirichlet
Units Theorem, extensions
[updated Wednesday, 07-Dec-2011 14:15:45 CST]
- 12-05-2011
Fujisaki's compactness lemma, finiteness of class number, Dirichlet
Units Theorem
[updated Monday, 05-Dec-2011 12:25:09 CST]
- 12-02-2011
Ostrowski's theorem, Approximation Theorem, adelic solenoid A/k.
[updated Friday, 02-Dec-2011 12:19:57 CST]
- 11-30-2011
Product formula, Ostrowski's theorem, Approximation Theorem
[updated Wednesday, 30-Nov-2011 18:11:20 CST]
- 11-28-2011
... completions, finite-dimensional topological vector
spaces, product formula
[updated Tuesday, 29-Nov-2011 11:34:52 CST]
- 11-23-2011
... completions, finite-dimensional topological vector
spaces, product formula
[updated Saturday, 26-Nov-2011 09:31:26 CST]
- 11-21-2011
Absolute values, completions, finite-dimensional topological vector spaces
[updated Sunday, 20-Nov-2011 11:41:54 CST]
- some class number data
class numbers of the first few hundred complex quadratic fields
[updated Monday, 21-Nov-2011 07:32:40 CST]
- 11-18-2011
class number formula for complex quadratic fields
[updated Friday, 18-Nov-2011 12:39:10 CST]
- 11-16-2011
... Dedekind zeta functions, class numbers, residues of Epstein zetas,
class number formula for complex quadratic fields
[updated Thursday, 17-Nov-2011 10:18:40 CST]
- 11-14-2011
... ramification degrees, residue class field degrees, ideal norms,
Dedekind zeta functions
[updated Tuesday, 15-Nov-2011 10:22:59 CST]
- 11-11-2011
... basic big theorem characterizing Dedekind domains, cont'd + sequel
[updated Friday, 11-Nov-2011 14:03:09 CST]
- 11-09-2011
... basic big theorem on factorization in Dedekind domains
[updated Thursday, 10-Nov-2011 10:13:56 CST]
- 11-07-2011
... more: Galois theory of primes lying over, intro to Dedekind rings.
[updated Monday, 07-Nov-2011 12:20:49 CST]
- 11-04-2011
...better localization... Galois theory of primes lying over, intro to
Dedekind rings.
[updated Saturday, 05-Nov-2011 10:07:11 CDT]
- 11-02-2011
A better version of localization.
[updated Thursday, 03-Nov-2011 10:26:23 CDT]
- 10-31-2011
Primes lying over, Galois action. Review of localization.
[updated Saturday, 29-Oct-2011 11:29:10 CDT]
- 10-28-2011
Brief recollection of quadratic norm-residue, Hilbert symbols, reciprocity laws. Primes
lying over, Galois action.
[updated Saturday, 29-Oct-2011 11:29:10 CDT]
- 10-26-2011
Quadratic norm-residue, Hilbert symbols, reciprocity laws. Primes
lying over, Galois action.
[updated Thursday, 27-Oct-2011 12:41:38 CDT]
- 10-24-2011
function fields over finite fields ... quadratic norm-residue, Hilbert
symbols, reciprocity laws
[updated Sunday, 23-Oct-2011 12:21:32 CDT]
- 10-21-2011
Cont'd: function fields... especially over finite
fields... Artin-Schreier extensions... quadratic norm/Hilbert
reciprocity laws
[updated Sunday, 23-Oct-2011 11:24:42 CDT]
- 10-19-2011
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, 21-Oct-2011 07:59:15 CDT]
- 10-17-2011
Cont'd: function fields, especially extensions
of C(X) and C((X)). Formal Puiseux
expansions, Newton polygons.
[updated Tuesday, 18-Oct-2011 11:44:09 CDT]
- 10-14-2011
function fields, especially extensions
of C(X) and C((X)). Formal Puiseux
expansions, Newton polygons.
[updated Saturday, 15-Oct-2011 13:52:27 CDT]
- 10-12-2011
function fields
[updated Thursday, 13-Oct-2011 09:36:36 CDT]
- 10-10-2011
commutative algebra: integral extensions, Noetherian-ness, function
field case
[updated Monday, 10-Oct-2011 12:20:01 CDT]
- 10-07-2011
commutative algebra: integral extensions, Noetherian-ness
[updated Friday, 07-Oct-2011 12:56:29 CDT]
- 10-05-2011
commutative algebra: integral extensions, algebraic integers
[updated Friday, 07-Oct-2011 13:02:16 CDT]
- 10-03-2011
some commutative algebra: integral extensions, algebraic integers
[updated Sunday, 09-Oct-2011 16:55:37 CDT]
- 09-30-2011
p-adic numbers, projective limits, colimits, adeles
[updated Friday, 30-Sep-2011 14:44:47 CDT]
- 09-28-2011
more on Hensel's Lemma, p-adic numbers, projective limits
[updated Thursday, 29-Sep-2011 09:06:27 CDT]
- 09-26-2011
more on Hensel's Lemma, p-adic numbers
[updated Sunday, 25-Sep-2011 17:04:29 CDT]
- 09-23-2011
Hensel's Lemma, p-adic numbers
[updated Sunday, 25-Sep-2011 11:07:41 CDT]
- 09-21-2011
Cont'd: more factoring Dedekind zeta functions
[updated Wednesday, 21-Sep-2011 12:21:22 CDT]
- 09-19-2011
Cont'd: factoring Dedekind zeta functions into Dirichlet L-functions
[updated Tuesday, 20-Sep-2011 07:49:20 CDT]
- 09-16-2011
factorization of some
Dedekind zeta functions into Dirichlet L-functions
[updated Saturday, 17-Sep-2011 15:06:55 CDT]
- 09-14-2011
Quadratic Reciprocity over Q, by Gauss sums
[updated Wednesday, 14-Sep-2011 14:15:36 CDT]
- 09-12-2011
meromorphic continuation and functional equation of zeta, Poisson
summation and functional equation of theta, integral representation
of zeta in terms of theta
[updated Sunday, 11-Sep-2011 18:00:38 CDT]
- 09-09-2011
Riemann's explicit formula
[updated Sunday, 11-Sep-2011 18:17:10 CDT]
Older notes
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This work
by Paul Garrett is licensed
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Unported License.
...
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[this page is http://www.math.umn.edu/~garrett/m/number_theory/]
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