18.787 - Topics in Number Theory (Fall 2012)



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Let's thank Holden Lee for graciously sharing his typed notes.

Semester Timeline
Date Topic Reference Remark
09-06definition of group schemes[Mum] sec 11 (pp.89-91,96-98), [BLR] 4.1, etc
09-11quotients by group schemes 1[vdGM] sec 4, [MFK] sec 0.1-0.4, cf. [Mum] sec 7, 12 Geom quot in [vdGM] are a little different from those in [MFK]. I followed [vdGM].
09-13quotients by group schemes 2 [vdGM] Th 4.16, [Mum] sec 12
09-18equivariant sheaves[Mum] sec 12, [MFK] sec 1.3, [Vis] sec 3.8 [Vis] is for generalists as it discusses equivariant objects in a category.
09-20definition of abelian schemes [MFK] pp.115-117, [Lan], [Mum] sec 4
09-25rigidity lemma, isogenies [MFK] Prop 6.1, Lem 6.12, [Lan]
09-27more on isogenies, theorem of the cube [Mum] sec 6, [vdGM] sec 2 narrow focus to ab. var. over fields for a while
10-02Th of square, Seesaw and K(L) [Mum] sec 6, 10; start of sec 13 discusses K(L) [vdGM] sec 2 Seesaw => Th of cube is in [Mum] sec 10. It's a nice proof to read.
10-04Projectivity, rank of A[n] [Mum] sec 6, [vdGM] sec 2, 5 see [Mil] Th 7.1 for reduction to alg closed and another proof of projectivity.
10-09MIT holiday (no class)
10-11Picard schemes, dual abelian schemes [BLR] Ch 8, [MFK] p.120, [Kle], [FC] pp.2-7
10-16No class, hooray! SWS is out of town (to Fields Inst.)
10-18Duality continued [Mum] sec 13, [vdGM] sec 7.2 Black boxes are [Mum] sec 12, thm 2; sec 13, cor 1,2.
10-23Poincare reducibility, Cartier duality [Mum] thm 19.1, sec 14, [vdGM] thm 12.2, sec 3.22
10-25 ker of dual isogeny, Tate modules [Mum] thm 15.1, sec 18 [vdGM] thm 7.5, sec 10
10-30 Hom, End and deg function [Mum] sec 19
11-01 characteristic poly [Mum] sec 19-20
11-06 Riemann forms [Mum] sec 20
11-08 Rosati involution [Mum] sec 20-21
11-13 Complex abelian varieties [Mum] sec 1-3, 9, 24
11-15 CM abelian varieties [Mum] sec 22, [Mil3] sec 2,3,6,7 [Mil4] sec 1, [ST] See "Remarks" page for Zariski density of l-power torsions.
11-20 End algebra over finite fields, I [Tat4], [Mum] Appendix A
11-22 Happy Thanksgiving!
11-27 End algebra over finite fields, II [Tat4], [Mum] Appendix A
11-29 Finite group schemes over a field of char p [Gro] I.5-II.3, [Tat3], [Mum] sec 14
12-04 Dieudonne theory, p-divisible groups [Gro] I.1-4,II.4,III.5, [Tat1], [Dem] Ch 3
12-06 class holiday SWS is away to Berkeley
12-11 Honda-Tate theory [Tat2], [MW], [Dem] Ch 4