Math 703 schedule
| Monday |
Wednesday |
Friday |
Jan. 7
In Lecture:
| Jan. 9
In Lecture:
| 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:
|
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:
- Milne, "Etale cohomology," chapter 1.4
- Lang, chapter 10.
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
Video of class (downloadable)
| Jan. 21
In Lecture:
| Jan. 23
In Lecture:
- Cohen Lenstra Heuristics
- Artin conjecture on primitive roots, and generalizations
- Congruence subgroup problems
- Chevalley theorem on units
Associated Reading:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
Video of class (downloadable)
| Feb. 1
In Lecture:
- The norm and transfer theorems, continued
- Constructing dihedral and quaternion extensions
Associated Reading:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
Video of class (downloadable)
| Feb. 4
In Lecture:
- Constructing dihedral and quaternion extensions, continued
- Contructing metabelian extensions
- Canonical classes
Associated Reading:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
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:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
Video of class (downloadable)
| Feb. 22
In Lecture:
Associated Reading:
- Lang, chapter 2
- Notes on class field theory by J.P. Serre, recorded by J. Weisinger
Video of class (downloadable)
| Feb. 25
In Lecture:
Associated Reading:
Video of class (downloadable)
| Feb. 27
In Lecture:
- Ramification theory, continued
Associated Reading:
Video of class (downloadable)
| March 1
In Lecture:
- Ramification theory, continued. Definition and properties of the upper numbered ramification groups.
Associated Reading:
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:
Video of class (downloadable)
| March 13
In Lecture:
- Artin representation, continued
Associated Reading:
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:
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, 115 - 144.
Video of class (downloadable)
| April 24
No class!
| April 26
No class!
|