Math 621 schedule

Monday Wednesday Friday
Jan. 11

In Lecture:
  • No Class
Jan. 13

In Lecture:
  • Functorial properties of class field theory

Associated Reading:
  • Lang, chapter 7

Video of class (streaming)
Video of class (downloadable)
Jan. 15

In Lecture:
  • No Class
Jan. 18

In Lecture:
  • No Class - Martin Luther King holiday
Jan. 20

In Lecture:
  • Applying class field theory to construct solvable extensions
  • Ideles and Adeles
  • Idele class groups
  • Class field theory in the language of Ideles

Associated Reading:
  • Lang, chapter 7

Video of class (streaming)
Video of class (downloadable)
Jan. 22

In Lecture:
  • Ideles and Adeles
  • Idele class groups
  • Class field theory in the language of Ideles

Associated Reading:
  • Lang, chapter 7

Video of class (streaming)
Video of class (downloadable)
Jan. 25

In Lecture:
  • Comparing idele class groups with ray class groups
  • The connected component of the identify of the idele classes

Associated Reading:
  • Lang, chapter 7

Video of class (streaming)
Video of class (downloadable)
Jan. 27

In Lecture:
  • The idele class group of Q and of F_q(t)
  • Leopold't conjecture

Associated Reading:
  • Lang, Chapter 7

Video of class (streaming)
Video of class (downloadable)
Jan. 29

In Lecture:
  • Global class field theory over function fields

Associated Reading:
  • Lang Chapter 7

Video of class (streaming)
Video of class (downloadable)
Feb. 1

In Lecture:
  • Local class field theory (statments)
  • Genus theory

Associated Reading:
  • Lang, chapter 7

Video of class (streaming)
Video of class (downloadable)
Feb. 3

In Lecture:
  • Genus theory, continued
  • Ramification groups

Associated Reading:
  • Serre, Corps Locaux

Video of class (streaming)
Video of class (downloadable)
Feb. 5

In Lecture:
  • Ramification groups, continued

Associated Reading:
  • Serre, Corps Locaux.

Video of class (streaming)
Video of class (downloadable)
Feb. 8

No class
Feb. 10

No class
Feb. 12

No class
Feb. 15

In Lecture:
  • Differents

Associated Reading:
  • Samuel, chapter 2, sections 2.7, 2.8.
  • Lang, chapter 3, section 3.3.

Video of class (streaming)
Video of class (downloadable)
Feb. 17

In Lecture:
  • Differents and lower ramification groups

Associated Reading:
  • Serre, Corps Locaux

Video of class (streaming)
Video of class (downloadable)
Feb. 19

In Lecture:
  • Upper numbering of ramification groups

Associated Reading:
  • Serre, Corps Locaux.

Video of class (streaming)
Video of class (downloadable)
Feb. 22

In Lecture:
  • Hasse-Arf Theorem
  • Herbrand's Theorem
  • Functorality of upper and lower numbering of the ramification groups
  • Conductors, the Artin Representation

Associated Reading:
  • Serre, Corps Locaux.

Video of class (streaming)
Video of class (downloadable)
Feb. 24

In Lecture:
  • Proof that the Artin character is the character of a representation

Associated Reading:
  • Serre, Corps Locaux
  • Lang, Algebraic number theory

Video of class (streaming)
Video of class (downloadable)
Feb. 26

In Lecture:
  • End of the proof that the Artin character is the character of a representation
  • The Artin character and actions of finite groups on Jacobians

Associated Reading:
  • Serre, Corps Locaux
  • Lang, Algebraic number theory

Video of class (streaming)
Video of class (downloadable)
Mar. 1

In Lecture:
  • The Artin character and actions of finite groups on Jacobians
  • Artin L-functions

Associated Reading:
  • Serre, Corps Locaux
  • Lang, Algebraic number theory

Video of class (streaming)
Video of class (downloadable)
Mar. 3

In Lecture:
  • Abelian L-functions and Artin L-functions

Associated Reading:
  • Lang, Algebraic number theory

Video of class (streaming)
Video of class (downloadable)
Mar. 5

In Lecture:
  • Functional equations of Artin L-functions

Associated Reading:
  • Lang, Algebraic number theory

Video of class (streaming)
Video of class (downloadable)
Mar. 8

In Lecture:
  • No class - spring break!
Mar. 10

In Lecture:
  • No class - spring break!
Mar. 12

In Lecture:
  • No class - spring break!
Mar. 15

Rademacher Lecture by John Coates:
  • Iwasawa Theory

Video of the lecture (streaming)
Video of the lecture (downloadable)
Mar. 16

Rademacher Lecture by John Coates:
  • Cyclotomic Iwasawa theory

Video of the lecture (streaming)
Video of the lecture (downloadable)
Mar. 17

Rademacher Lecture by John Coates:
  • The general Main Conjecture

Video of the lecture (streaming)
Video of the lecture(downloadable)

Rademacher Lecture by John Coates:
  • The Tate-Shafarevich group and Iwasawa theory

Video of the lecture (streaming)
Video of the lecture(downloadable)
Mar. 22

In Lecture:
  • Artin L-series
  • Relations between zeta and L-functions

Associated Reading:
  • Lang, ``Algebraic Number theory," Chapter XII.

Video of class (streaming)
Video of class (downloadable)
Mar. 24

In Lecture:
  • The Burnside ring
  • Zeta functions of curves over finite fields

Associated Reading:
Mar. 26

In Lecture:
  • L-functions of one-dimensional representations over an imaginary quadratic fields
  • L-functions of two-dimensional Galois representations over Q and modular forms of weight 1

Associated Reading:
  • Lang, Algebraic number theory
  • G. Frey (editor), On Artin's conjecture for odd 2-dimensional representations, Lecture notes in math. 1585 (1994), Springer-Verlag (available from gigapedia)

Video of class (streaming)
Video of class (downloadable)
Mar. 29

In Lecture:
  • Zeta and L-functions over real quadratic fields
  • Shintani formulas

Associated Reading:
  • Lang, ``Algebraic Number theory."

Video of class (streaming)
Video of class (downloadable)
Mar. 31

In Lecture:
  • The Residue of the zeta functions of a number field at s = 1

Associated Reading:
  • Lang, "Algebraic number theory," Chapters VI, VIII.

Video of class (streaming)
Video of class (downloadable)
April 2

In Lecture:
  • The Residue of the zeta functions of a number field at s = 1, continued

Associated Reading:
  • Lang, "Algebraic number theory," Chapters VI, VIII.

Video of class (streaming)
Video of class (downloadable)
April 5

In Lecture:
  • Functional equations of L-series
  • Leading terms in the Taylor expansions of L-series at non-positive integers

Associated Reading:
  • Lang, ``Algebraic Number theory," Chapters XIII, XIV
  • Washingon, L., "Introduction to cyclotomic fields," chapter 4. (Available on gigapedia)
  • Ogg, Modular forms and Dirichlet series. (Available on gigapedia)

Video of class (streaming)
Video of class (downloadable)
April 7

In Lecture:
  • Functional equation of the Riemann zeta function (the classical proof)
  • The functional equations of Artin L-functions

Associated Reading:
  • Lang, "Algebraic number theory,"
  • Ogg, "Modular forms and Dirichlet Series"

Video of class (streaming)
Video of class (downloadable)
April 9

In Lecture:
  • Leading term of the zeta function at s = 0
  • Statement of Stark's conjecture

Associated Reading:
  • Lang, Algebraic number theory
  • Tate, Les conjectures de Stark sur les fonctions L en s = 0

Video of class (streaming)
Video of class (downloadable)
April 12

In Lecture:
  • Computations of Stark regulators

Associated Reading:
  • Lang, Algebraic number theory
  • Tate, Les conjectures de Stark sur les fonctions L en s = 0

Video of class (streaming)
Video of class (downloadable)
April 14

In Lecture:
  • No class due to multiple oral exams!
April 16

In Lecture:
  • Constructing number fields using Stark's conjecture

Associated Reading:
  • Lang, Algebraic number theory
  • Tate, Les conjectures de Stark sur les fonctions L en s = 0

Video of class (streaming)
Video of class (downloadable)
April 19

In Lecture:
  • The first order zero case of Stark's Conjecture (continued)
  • Salem numbers and Stark's conjeccture

Associated Reading:
  • Chinburg, T.: "Salem numbers and $L$-functions." J. Number Theory 18 (1984), no. 2, 213--214.
  • Tate, Les conjectures de Stark sur les fonctions L en s = 0

Video of class (streaming)
Video of class (downloadable)
April 21

In Lecture:
  • Salem numbers and Stark's Conjecture (continued)

Associated Reading:
  • Chinburg, T.: "Salem numbers and $L$-functions." J. Number Theory 18 (1984), no. 2, 213--214.
  • Tate, Les conjectures de Stark sur les fonctions L en s = 0

Video of class (streaming)
Video of class (downloadable)
April 23

In Lecture:
  • Survey on Lehmer's Conjecture
  • Lehmer's conjecture and closed geodesics on hyperbolic surfaces

Associated Reading:
  • Ghate, Eknath; Hironaka, Eriko "The arithmetic and geometry of Salem numbers." Bull. Amer. Math. Soc. (N.S.) 38 (2001), no. 3, 293--314

Video of class (streaming)
Video of class (downloadable)
April 26

In Lecture:
  • Arithmetic hyperbolic surfaces and Salem numbers (continued)
  • The Selberg trace formula and Weil's explicit formulas

Associated Reading:
  • Sunada, Toshikazu, ``Riemannian coverings and isospectral manifolds." Ann. of Math. (2) 121 (1985), no. 1, 169--186.
  • Lang, Algebraic number theory

Video of class (streaming)
Video of class (downloadable)
April 28

In Lecture:
  • No class.
April 30

In Lecture:
  • No class.

Last updated: 5.5/10
Send e-mail comments to: ted@math.upenn.edu