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 Ktheory, 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, 44Ð80.
 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 Ktheory, 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, 44Ð80.
 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 twodimensional local field.
Associated Reading:
 Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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

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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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
 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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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 semilocal form of the KatoSaito classfield theory on curves:
 Spec(K) when K = Frac(A) and A is a 2dimensional 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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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:

Feb. 21
In Lecture:
 The first semilocal form of the KatoSaito 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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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:
 Feb. 25
In Lecture:

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 semilocal form of the KatoSaito 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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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 semilocal form of the KatoSaito 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, 44Ð80.
 Invitation to Higher Local Fields, I. Fesenko (see gigapedia).
 K. Kato, "A generalizatin of local class field theory by using Kgroups," 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:
 Mar. 9
In Lecture:
 Mar. 11
In Lecture:

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, 44Ð80.
 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 nonnormal 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:

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 Kgroups 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, 44Ð80.
 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:
 April 1
In Lecture:

May 9
In Lecture:
 The Brauer Manin obstruction and higher reciprocity laws
Associated Reading:
 Heuristics for the BrauerManin obstruction for curves. B. Poonen, Experiment. Math. 15 (2006), no. 4, 415Ð420.
 A generalizatin of local class field theory by using Kgroups 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, 44Ð80.
Video of class (streaming)
Video of class (downloadable)
 May 11
In Lecture:
 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 Kgroups 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, 44Ð80.
Video of class (streaming)
Video of class (downloadable)
