Math 620 schedule
Monday |
Wednesday |
Friday |
Sept. 7
In Lecture:
| Sept. 9
In Lecture:
- Algebraic numbers, algebraic integers and transcendental numbers
Associated Reading:
- Lang, Chapter I, sections 1.1 and 1.2
- Hardy and Wright, section 11.7
- Samuel, chapter 2, sections 2.1 to 2.5
Video of class
Video of class (downloadable)
| Sept. 11
In Lecture:
|
Sept. 14
In Lecture:
- Norms and Traces
- Iintegral closures
- Prime ideals and maximal ideals
Associated Reading:
- Lang, Chapter I, sections 1.2, 1.3, 1.4, 1.5.
- Samuel, chapter 2.
Video of class
Video of class (downloadable)
| Sept. 16
In Lecture:
- Integral closures, continued
- Traces and norms, finiteness of integral closures, continued
- Dual bases
Associated Reading:
- Lang, chapter I, sections 1.2, 1.3, 1.4, 1.5
- Lang, chapter 3, section 3.1.
- Samuel, chapter 2.
Video of class
Video of class (downloadable)
| Sept. 18
In Lecture:
|
Sept. 21
In Lecture:
Associated Reading:
- Samuel, chapter 2, sections 2.7, 2.8.
- Lang, chapter 3, section 3.3.
Video of class
Video of class (downloadable)
| Sept. 23
In Lecture:
- Rings of cyclotomic integers
- Statement of Kronecker Weber Theorem, Hilbert's 12th problem
- Drinfeld modules
- Beginning of Dedekind rings
Associated Reading:
Video of class
Video of class (downloadable)
| Sept. 25
In Lecture:
|
Sept. 28
In Lecture:
- Fractional ideals of Dedekind rings form a group
- Integral closures of Dedekind rings
- Ideal class groups, elliptic curves, Jacobians of curves
Associated Reading:
- Samuel, chapter 3.
- Samuel, chapter 5, sections 5.1 and 5.2.
- Lang, chapter 1, section 1.6.
- Hartshorne, chapters 1.6 and 2.6.
Video of class
Video of class (downloadable)
| Sept. 30
In Lecture:
- Applications of Dedekind rings: Fermat, Curves over fields and Dedekind rings
- Decomposition of primes
Associated Reading:
- Washington, chapters 1 and 2.
- Hartshorne, chapters 1.6 and 2.6
- Samuel, chapter 5, sections 5.1 and 5.2.
- Lang, chapter 1, section 1.6.
Video of class
Video of class (downloadable)
| Oct. 2
In Lecture:
|
Oct. 5
In Lecture:
- End of the proof of main result about decomposition of primes
- Discrete valuation rings
- Decomposition of primes in Galois extensions
- Explicit decompositions of primes
Associated Reading:
- Samuel, chapter 5, sections 5.1 and 5.2.
- Lang, chapter 1, section 1.6, 1.7, 1.8.
Video of class
Video of class (downloadable)
| Oct. 7
In Lecture:
- Examples of discrete valuation rings
- Absolute values and completions
Associated Reading:
Video of class
Video of class (downloadable)
| Oct. 9
In Lecture:
- The time and location of the class will be sent by e-mail the night of Oct. 8
Associated Reading:
- Absolute values and completions, continued
Video of class
Video of class (downloadable)
|
Oct. 12
In Lecture:
Associated Reading:
- Borevich and Shafarevitch, see index for "Ostrowski's Theorem".
- Lang, chapter 2.
Video of class
Video of class (downloadable)
| Oct. 14
In Lecture:
- Ostrowski's Theorem, concluded
- The topology of local fields
- Approximation Theorems
Associated Reading:
- Borevich and Shafarevitch, see index for "Ostrowski's Theorem".
- Lang, chapter 2.
Video of class (streaming)
Video of class (downloadable)
| Oct. 16
In Lecture:
- Weak and Strong approximation theorems for linear algebraic groups
- Extension fields and absolute values
Associated Reading:
-
- Platonov and Rapinchuk, sections 1.2, Chapter 2.1, Chapter 7
- Lang, Chapter 2
Video of class (streaming)
Video of class (downloadable)
|
Oct. 19
In Lecture:
| Oct. 21
In Lecture:
- Approximation Theorems
- Extensions of absolute values
- The Manin Batryev Conjeture
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Oct. 23
In Lecture:
- Extensions of absolute values
- Hensel's Lemma
- Selberg's analogy, fields with the same zeta function
Associated Reading:
- Lang, chapter 2
- Sunada's paper on the length spectrum of manifolds
Video of class (streaming)
Video of class (downloadable)
|
Oct. 26
In Lecture:
| Oct. 30
In Lecture:
| Oct. 28
In Lecture:
|
Nov. 2
In Lecture:
- Extensions of absolute values (end)
- Hensel's Lemma
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Nov. 4
In Lecture:
- Hensel's Lemma and applications
- Dynamical systems and morphisms of varieties
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Nov. 6
In Lecture:
- End of proof of Hensel's Lemma
- Dynamical systems (continued)
Associated Reading:
- Lang, chapters 2 and 5
- Julia Sets
- Galois stochastic processes
- Notes by Larry Shepp on Martingales
Video of class (streaming)
Video of class (downloadable)
|
Nov. 9
In Lecture:
| Nov. 11
In Lecture:
- Product formula
- Geometry of numbers
- Finiteness of class numbers, finite generation of unit groups
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Nov. 13
In Lecture:
- Geometry of numbers (continued)
- Finiteness of class numbers, finite generation of unit groups (continued)
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
|
Nov. 16
In Lecture:
- Algorithms for computing idea class groups
- Quadratic field examples
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Nov. 18
In Lecture:
- Proof of the Minkowski bound
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Nov. 20
In Lecture:
- End of the proof of the Minkowski bound, and the function field case
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
|
Nov. 23
In Lecture:
- The Dirichet Unit theorem, in number fields and function fields
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Nov. 25
| Nov. 27
|
Nov. 30
In Lecture:
- Constructing small generators for discrete groups using fundamental domains
- Applications to generating S-units
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Dec. 2
In Lecture:
- Curves, codes and algebraic integers
- Beginning of class field theory via ray class groups
Associated Reading:
- Here is one Survey article
on codes constructed via algebraic geometry.
- Lang, chapter 10
Video of class (streaming)
Video of class (downloadable)
| Dec. 4
In Lecture:
- Class field theory via ray class groups
- Congruence subgroups of unit groups and the congruence subgroup problem
Associated Reading:
- Lang, chapter 10
- Here is one Survey article
on the congruence subgroup problem.
Video of class (streaming)
Video of class (downloadable)
|
Dec. 7
In Lecture:
- Frobenius elements
- Class field theory via ray class groups
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
| Dec. 9
No class due to the flu
| Dec. 11
In Lecture:
- Class field theory, continued
Associated Reading:
Video of class (streaming)
Video of class (downloadable)
|