Math 703 schedule

Monday Wednesday Friday
Jan. 7

In Lecture:
  • No Class!
Jan. 9

In Lecture:
  • No Class!
Jan. 11

In Lecture:
  • Hensel's Lemma (Naive form)
  • Hensel's Lemma (For fields complete with respect to a non-archimedean absolute value)

Associated Reading:
  • Lang, chapter 2
Jan. 14

  • Proof of the sophisticated form of Hensel's Lemma
  • Henselian rings.
Jan. 16

In Lecture:
  • Completions are Henselian
  • Henselizations

Associated Reading:
  • Lang, chapter 2
  • Milne, "Etale cohomology," chapter 1.4

Video of class (downloadable)
Jan. 18

In Lecture:
  • Henselizations and Galois theory
  • Start of Class Field Theory

Associated Reading:
Video of class (downloadable)
Jan. 21

In Lecture:
  • No Class - MLK day.
Jan. 23

In Lecture:
  • Cohen Lenstra Heuristics
  • Artin conjecture on primitive roots, and generalizations
  • Congruence subgroup problems
  • Chevalley theorem on units

Associated Reading:
Video of class (downloadable)
Jan. 25

In Lecture:
  • Chevalley's theorem on units (continued)
  • Congruence subgroup problems (continued)
  • Behavior of Frobenius elements
  • Start of Class field theory

Associated Reading:
Video of class (downloadable)
Jan. 28

In Lecture:
  • The Artin homomorphism and the Existence Theorem
  • Quadratic reciprocity
  • The Hilbert classfield
  • Frobenius elements in non-abelian extensions

Associated Reading:
Video of class (downloadable)
Jan. 30

In Lecture:
  • Frobenius elements in non-abelian extensions and splitting of polynomials mod primes
  • Functorial properties of Class field theory
  • The Norm and Transfer theorems.

Associated Reading:
Video of class (downloadable)
Feb. 1

In Lecture:
  • The norm and transfer theorems, continued
  • Constructing dihedral and quaternion extensions

Associated Reading:
Video of class (downloadable)
Feb. 4

In Lecture:
  • Constructing dihedral and quaternion extensions, continued
  • Contructing metabelian extensions
  • Canonical classes

Associated Reading:
Video of class (downloadable)
Feb. 6

In Lecture:
  • Class field theory in the language of ideles
  • Comparing idele class groups with ray class groups

Associated Reading:
Video of class (downloadable)
Feb. 8

In Lecture:
  • Statement of the fundamental theorem of global class field theory (for all global fields)
  • The connected component of the identity in the ideles.
  • End of the comparison of idele class groups with ray class groups in the case of number fields

Associated Reading:
Video of class (downloadable)
Feb. 11

In Lecture:
  • Recap of idelic statement of global class field theory
  • The Mittag Leffler criterion and comparing idele class groups with ray class groups
  • Leopoldt's conjecture
  • Statement of local class field theory
  • Quadratic extensions of Q_p

Associated Reading:
Video of class (downloadable)
Feb. 13

No Class!
Feb. 15

No Class!
Feb. 18

In Lecture:
  • Functorial properties of local class field theory
  • Counting local metabelian extensions

Associated Reading:
Video of class (downloadable)
Feb. 20

In Lecture:
  • Texts about class field theory
  • Global class field theory for global function fields, examples, relation with the Carlitz module
  • Genus theory

Associated Reading:
Video of class (downloadable)
Feb. 22

In Lecture:
  • Genus theory, conclusion

Associated Reading:
Video of class (downloadable)
Feb. 25

In Lecture:
  • Ramification theory

Associated Reading:
  • Serre, Corps Locaux

Video of class (downloadable)
Feb. 27

In Lecture:
  • Ramification theory, continued

Associated Reading:
  • Serre, Corps Locaux.

Video of class (downloadable)
March 1

In Lecture:
  • Ramification theory, continued. Definition and properties of the upper numbered ramification groups.

Associated Reading:
  • Serre, Corps Locaux.

Video of class (downloadable)
March 4

No class - spring break!
March 6

No class - spring break!
March 8

No class - spring break!
March 11

In Lecture:
  • Artin Representation
  • Brauer's induction theorem

Associated Reading:
  • Serre, Corps Locaux

Video of class (downloadable)
March 13

In Lecture:
  • Artin representation, continued

Associated Reading:
  • Serre, Corps Locaux.

Video of class (downloadable)
March 15

In Lecture:
  • Lehmer conjecture
  • Villegas' higher Lehmer conjecture

Associated Reading:
Video of class (downloadable)
March 18

In Lecture:
  • Higher Lehmer Conjectures, continued
  • Cheeger constants

Associated Reading:
Video of class (downloadable)
March 20

In Lecture:
  • Higher Lehmer conjectures, continued

Associated Reading:
Video of class (downloadable)
March 22

No class!
March 25

In Lecture:
  • Hecke L-Series
  • Artin L-series

Associated Reading:
  • Lang, Algebraic Number Theory
  • A. Ogg, Modular forms and Dirichlet series

Video of class (downloadable)
March 27

In Lecture:
  • Functorial properties of Artin L-series
  • Riemann's proof of the functional equation of the Riemann zeta function

Associated Reading:
  • A. Ogg, Modular forms and Dirichlet series

Video of class (downloadable)
March 29

In Lecture:
  • End of Riemann's proof
  • Conductors of Artin representations
  • The functional equation of Artin L-series '

Associated Reading:
  • A. Ogg, Modular forms and Dirichlet series

Video of class (downloadable)
April 1

No class!
April 3

No class!
April 5

In Lecture:
  • Start of Hecke's proof of the functional equations of partial zeta functions '

Associated Reading:
  • Lang, Algebraic number theory

Video of class (downloadable)
April 8

Hecke's proof, continued
Fourier transforms.
Associated Reading:
  • Lang, Algebraic number theory

Video of class (downloadable)
April 10

Hecke's proof, continued
Poisson summation formula
Sketch of the entire proof
  • Lang, Algebraic number theory

Video of class (downloadable)
April 12

In Lecture:
  • End of Hecke's proof '

Associated Reading:
  • Lang, Algebraic number theory

Video of class (downloadable)
April 15

No class!
April 17

No class!
April 19

No class!
April 22

The work of Friedman and Skoruppa on regulators
Applications to unit lattices
Associated Reading:
  • Friedman, Eduardo; Skoruppa, Nils-Peter Relative regulators of number fields. Invent. Math. 135 (1999), no. 1, 115144.

Video of class (downloadable)
April 24

No class!
April 26

No class!

Last updated: 9/6//13
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