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Algebraic Number Theory 20112012
Book = 201112 Notes
[updated
]
... 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.
201112 Overheads in reverse chrono order:
again, these are subsumed by the
201112 Notes

05042012
Sketch of main points of homological formulation of classfield theory,
with some explanations: Tate cohomology of groups, etc,
[updated
]

05022012
Classfield theory, (co)homological formulation, cont'd:
Herbrand quotient, example: local cyclic norm index equality.
[updated
]

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

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

04252012
Classfield theory, (co)homological formulation, cont'd:
homology long exact sequences, Euler characteristics, snake lemma, examples
[updated
]

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
]

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
]

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
]

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
]

04132012
Classfield theory, cont'd: idelic statement of global classfield
theory, review of Kummer theory and independence of roots
[updated
]

04112012
Classfield theory, cont'd: review of extensions of
unramified extensions of local fields, quadratic extensions of local
fields, Kummer theory
[updated
]

04092012
Classfield theory, cont'd: review of extensions of finite fields,
unramified extensions of local fields
[updated
]

04062012
Classfield theory: basic overview
[updated
]

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
]

04022012
Completion of proof of Hecke's identity... via some representation
theory of orthogonal groups
[updated
]

03302012
Interlude: Hecke's identity for harmonic polynomial multiples of
Gaussians, cont'd... some representation theory of orthogonal groups
[updated
]

03282012
Interlude: Hecke's identity for harmonic polynomial multiples of
Gaussians, cont'd
[updated
]

03262012
Interlude: Hecke's identity for harmonic polynomial multiples of Gaussians
[updated
]

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
]

03212012
Details supporting IwasawaTate:
Convergence of global halfzeta integrals, Hecke's identity, harmonic
polynomials, harmonic analysis on spheres.
[updated
]

03192012
Details supporting IwasawaTate: good finiteprime local zeta integrals,
unramified and ramified archimedean local zeta integrals
[updated
]
 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
]

03072012
IwasawaTate executed for Dedekind zetas, and then the general
case. Comments on needed supporting material.
[updated
]

03052012
IwasawaTate executed for Dirichlet Lfunctions, for Dedekind zetas
[updated
]

03022012
IwasawaTate executed for EulerRiemann
zeta, for Dirichlet Lfunctions
[updated
]

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
]
 02272012
Toward IwasawaTate: compact operators, HilbertSchmidt integral operators.
...
[updated
]
 02242012
Toward IwasawaTate: decomposition of L^{2} of compact
abelian groups by compact operators: HilbertSchmidt integral operators.
...
[updated
]
 02222012
Toward IwasawaTate: Schwartz spaces, Fourier inversion,
decomposition of L^{2} of compact abelian groups, by compact
operators. Toward adelic Poisson summation.
...
[updated
]
 02202012
Toward IwasawaTate: padic Fourier transform, Fourier inversion.
...
[updated
]
 02172012
Toward IwasawaTate: Fourier transform, Fourier inversion, on
archimedean and padic completions
...
[updated
]
 02152012
Toward IwasawaTate: unitary duals, characters, Fourier inversion
...
[updated
]
 02132012
Toward IwasawaTate's modernization of Hecke's treatment of
Lfunctions: unitary duals
of Q_{p}, A,
and A/Q.
...
[updated
]
 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
]
 02082012
measure and integration on Q_{p} and A. General
uniqueness proof.
...
[updated
]
 02062012
volumes of arithmetic quotients, special values of Lfunctions
...
[updated
]
 [hiatus]
 01232012
measures on quotients, volume of SL(n,Z)\SL(n,R)
...
[updated ]
 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 ]
 01182012
(Recap of Fujisaki's lemma, ideal class groups as idele class groups,
units theorem.) Invariant measures on quotients, iterated integrals.
...
[updated ]
 12142011
Interlude/preview... toward IwasawaTate theory: selfduality
of Q_{p} and A, mutual duality
of A/k and k, compactopen
topology, nosmallsubgroups, ...
[updated ]
 12122011
Generalized ideal class groups are images of the idele class group,
classification of closed subgroups of R^{n},
other supporting stuff...
[updated ]
 12092011
Proof of Fujisaki's compactness lemma, measuretheory pigeonhole
principle, finiteness of class number, Dirichlet
Units Theorem, extensions
[updated ]
 12072011
Proof of Fujisaki's compactness lemma, finiteness of class number, Dirichlet
Units Theorem, extensions
[updated ]
 12052011
Fujisaki's compactness lemma, finiteness of class number, Dirichlet
Units Theorem
[updated ]
 12022011
Ostrowski's theorem, Approximation Theorem, adelic solenoid A/k.
[updated ]
 11302011
Product formula, Ostrowski's theorem, Approximation Theorem
[updated ]
 11282011
... completions, finitedimensional topological vector
spaces, product formula
[updated ]
 11232011
... completions, finitedimensional topological vector
spaces, product formula
[updated ]
 11212011
Absolute values, completions, finitedimensional topological vector spaces
[updated ]
 11182011
class number formula for complex quadratic fields
[updated ]
 11162011
... Dedekind zeta functions, class numbers, residues of Epstein zetas,
class number formula for complex quadratic fields
[updated ]
 11142011
... ramification degrees, residue class field degrees, ideal norms,
Dedekind zeta functions
[updated ]
 11112011
... basic big theorem characterizing Dedekind domains, cont'd + sequel
[updated ]
 11092011
... basic big theorem on factorization in Dedekind domains
[updated ]
 11072011
... more: Galois theory of primes lying over, intro to Dedekind rings.
[updated ]
 11042011
...better localization... Galois theory of primes lying over, intro to
Dedekind rings.
[updated ]
 11022011
A better version of localization.
[updated ]
 10312011
Primes lying over, Galois action. Review of localization.
[updated ]
 10282011
Brief recollection of quadratic normresidue, Hilbert symbols, reciprocity laws. Primes
lying over, Galois action.
[updated ]
 10262011
Quadratic normresidue, Hilbert symbols, reciprocity laws. Primes
lying over, Galois action.
[updated ]
 10242011
function fields over finite fields ... quadratic normresidue, Hilbert
symbols, reciprocity laws
[updated ]
 10212011
Cont'd: function fields... especially over finite
fields... ArtinSchreier extensions... quadratic norm/Hilbert
reciprocity laws
[updated ]
 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 ]
 10172011
Cont'd: function fields, especially extensions
of C(X) and C((X)). Formal Puiseux
expansions, Newton polygons.
[updated ]
 10142011
function fields, especially extensions
of C(X) and C((X)). Formal Puiseux
expansions, Newton polygons.
[updated ]
 10122011
function fields
[updated ]
 10102011
commutative algebra: integral extensions, Noetherianness, function
field case
[updated ]
 10072011
commutative algebra: integral extensions, Noetherianness
[updated ]
 10052011
commutative algebra: integral extensions, algebraic integers
[updated ]
 10032011
some commutative algebra: integral extensions, algebraic integers
[updated ]
 09302011
padic numbers, projective limits, colimits, adeles
[updated ]
 09282011
more on Hensel's Lemma, padic numbers, projective limits
[updated ]
 09262011
more on Hensel's Lemma, padic numbers
[updated ]
 09232011
Hensel's Lemma, padic numbers
[updated ]
 09212011
Cont'd: more factoring Dedekind zeta functions
[updated ]
 09192011
Cont'd: factoring Dedekind zeta functions into Dirichlet Lfunctions
[updated ]
 09162011
factorization of some
Dedekind zeta functions into Dirichlet Lfunctions
[updated ]
 09142011
Quadratic Reciprocity over Q, by Gauss sums
[updated ]
 09122011
meromorphic continuation and functional equation of zeta, Poisson
summation and functional equation of theta, integral representation
of zeta in terms of theta
[updated ]
 09092011
Riemann's explicit formula
[updated ]
Older notes