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, 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 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, 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 two-dimensional 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 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, 44Ð80.
- 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, 44Ð80.
- 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
|
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 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
| 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 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, 44Ð80.
- 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:
|
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, 44Ð80.
- 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:
| 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 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, 44Ð80.
- 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, 44Ð80.
- 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:
| 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 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:
|
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, 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 Brauer-Manin obstruction for curves. B. Poonen, Experiment. Math. 15 (2006), no. 4, 415Ð420.
- 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, 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 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, 44Ð80.
Video of class (streaming)
Video of class (downloadable)
|