Math 721 schedule

Monday Wednesday Friday
Jan. 17

No Lecture (Martin Luther King Day)
Jan. 19

No Lecture
Jan. 21

In Lecture
  • Classfield theory for two dimensional complete local rings:
  • Computing the prime to p part of K_2 of a complete discretely valued field.

Associated Reading:
  • "The Milnor ring of a glogal field", H. Bass and J. Tate, in "Classical" Algebraic K-theory, and connections with arithmetic, Springer Lecture Notes in Math 342.
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).

Video of class (downloadable)
Jan. 24

In Lecture
  • Computing K_2 of a complete discretely valued field, continued.

Associated Reading:
  • "The Milnor ring of a glogal field", H. Bass and J. Tate, in "Classical" Algebraic K-theory, and connections with arithmetic, Springer Lecture Notes in Math 342.
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).

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

No Lecture
Jan. 28

In Lecture:
  • Proof of the prime to p part of Kato's class field theory isomorphism for
  • the maximai abelian extension of a two-dimensional local field.

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.

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

In Lecture (room 4C6, 1 - 2 p.m.):
  • The p part of Kato's class field theory isomorphism

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.

Video of class (streaming)
Video of class (downloadable)
Feb. 2

In Lecture
  • The p part of Kato's class field theory isomorphism(continued)

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.

Video of class (streaming)
Video of class (downloadable)
Feb. 4

  • No lecture
Feb. 7

In Lecture (room 4C6, 1 - 2 p.m.):
  • Review of Etale cohomology and torsors
  • The Cartier operator

Associated Reading:
  • J. Milne, Etale cohomology.
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.

Video of class (streaming)
Video of class (downloadable)
Feb. 9

  • no lecture
Feb. 11

In Lecture:
  • The Cartier operator, continued
  • Classifying torsors for Z/p, \mu_p and \alpha_p

Associated Reading:
  • J. Milne, Etale cohomology.
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.

Video of class (streaming)
Video of class (downloadable)
Feb. 14

In Lecture (room 4C6, 1 - 2 p.m.):
  • The Cartier operator, continued
  • Classifying torsors for Z/p, \mu_p and \alpha_p
  • Torsion subgroup schemes of elliptic curves

Associated Reading:
  • J. Milne, Etale cohomology.
  • N. Katz and B. Mazur, Arithmetic moduli of elliptic curves.

Video of class (streaming)
Video of class (downloadable)
Feb. 16

In Lecture:
  • The first semi-local form of the Kato-Saito classfield theory on curves:
  • Spec(K) when K = Frac(A) and A is a 2-dimensional complete local ring with finite residue field.

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.

Video of class (streaming)
Video of class (downloadable)
Feb. 18

In Lecture:
  • No class
Feb. 21

In Lecture:
  • The first semi-local form of the Kato-Saito class field theory on curves, continued

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.
  • J. Milne, Etale cohomology, and R. Hartshorne, Algebraic geometry (on flatness)

Video of class (streaming)
Video of class (downloadable)
Feb. 23

In Lecture:
  • no class
Feb. 25

In Lecture:
  • No class
Feb. 28

In Lecture:
  • Finite extensions of fields in which every discrete valuation splits.

Associated Reading:
  • J. Milne, Etale cohomology, and R. Hartshorne, Algebraic geometry (on flatness)

Video of class (streaming)
Video of class (downloadable)
Mar. 2

In Lecture:
  • The first semi-local form of the Kato-Saito class field theory on curves, an example

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.
  • J. Milne, Etale cohomology, and R. Hartshorne, Algebraic geometry (on flatness)

Video of class (streaming)
Video of class (downloadable)
Mar. 4

In Lecture:
  • The first semi-local form of the Kato-Saito class field theory on curves, more examples.
  • Reciprocity laws.

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
  • K. Kato, "A generalizatin of local class field theory by using K-groups," J. Fac. Sci. of the Univ. of Tokyo, Sect. 1A (Mathematics), Vol. 26, Issue 2 (1979), p. 303 - 376.
  • J. Milne, Etale cohomology, and R. Hartshorne, Algebraic geometry (on flatness)

Video of class (streaming)
Video of class (downloadable)
Mar. 7

In Lecture:
  • No class - spring Break!
Mar. 9

In Lecture:
  • No class - spring Break!
Mar. 11

In Lecture:
  • No class - spring Break!
Mar. 14

In Lecture:
  • Unramified class field theory on curves, SK_1
  • Etale fundamental groups

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • J. Milne, Etale cohomology (see section on fundamental groups)
  • A. Grothendieck, SGA1 (for fundamental groups)

Video of class (streaming)
Video of class (downloadable)
Mar. 16

In Lecture:
  • Etale fundamental groups of non-normal curves

Associated Reading:
  • J. Milne, Etale cohomology (see section on fundamental groups)
  • A. Grothendieck, SGA1 (for fundamental groups)

Video of class (streaming)
Video of class (downloadable)
Mar. 18

In Lecture:
  • No class
Mar. 21

In Lecture:
  • No class due to conference in Luminy
Mar. 23

In Lecture:
  • No class due to conference in Luminy
Mar. 25

In Lecture:
  • No class due to conference in Luminy
Mar. 28

In Lecture:
  • Preparation for proof of Kato's reciprocity law for 2 dimensional local fields

Associated Reading:
  • A generalizatin of local class field theory by using K-groups II, K. Kato, J. Fac. Sci. Univ. Tokyo, vol. 27 (3), 1980, 603 - 683.
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • J. Milne, Etale cohomology (see section on fundamental groups)
  • A. Grothendieck, SGA1 (for fundamental groups)

Video of class (streaming)
Video of class (downloadable)
Mar. 30

In Lecture:
  • No class due to illness
April 1

In Lecture:
  • No class due to illness
May 9

In Lecture:
  • The Brauer Manin obstruction and higher reciprocity laws

Associated Reading:
  • Heuristics for the Brauer-Manin obstruction for curves. B. Poonen, Experiment. Math. 15 (2006), no. 4, 415420.
  • A generalizatin of local class field theory by using K-groups II, K. Kato, J. Fac. Sci. Univ. Tokyo, vol. 27 (3), 1980, 603 - 683.
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming)
Video of class (downloadable)
May 11

In Lecture:
  • No class due to illness
May 13

In Lecture:
  • Final sketch of the proof of the base cases of Kato's reciprocity law for 2 dimensional local fields

Associated Reading:
  • A generalization of local class field theory by using K-groups II, K. Kato, J. Fac. Sci. Univ. Tokyo, vol. 27 (3), 1980, 603 - 683.
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming)
Video of class (downloadable)

Last updated: 3/28/11
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