Seminar on Topics in Arithmetic, Geometry, Etc.

STAGE is a seminar in algebraic geometry and number theory, featuring speakers talking about work that is not their own. Talks will be at a level suitable for graduate students. Everyone is welcome.
Meetings are held on Fridays, 5-7pm in MIT room 2-131, unless indicated otherwise below. Talks will generally run only 90 minutes, however.
To receive announcements by email, add yourself to the STAGE mailing list.
If you are interested in giving one of the talks, please contact the organizers.

Fall 2017 topic: Iwasawa theory.

Prerequisites:

Statements of class field theory (e.g., this summary)
Galois cohomology (e.g., I.1-2 and II.1 in Serre, Galois cohomology)
Arithmetic of elliptic curves (e.g., Chapters 1-8 and 10 of Silverman, The arithmetic of elliptic curves).

References to be followed:

Serre, Classes des corps cyclotomiques, Séminaire Bourbaki, 1958-1960, exp. no. 174, pp. 83-93. Some comments. A much more thorough account of this theory and more can be found in Greenberg, Topics in Iwasawa theory.
Coates and Sujatha, Cyclotomic fields and zeta values, Springer, 2006. Some comments.
Greenberg, Introduction to Iwasawa theory for elliptic curves, pp. 407-464 in Arithmetic algebraic geometry, eds. Conrad and Rubin, AMS, 2001.

Surveys (these contain a broad overview of Iwasawa theory, but without details):

Kato, Iwasawa theory and generalizations, Proceedings of the International Congress of Mathematicians, Madrid, Spain, 2006.
Greenberg, Iwasawa theory - past and present, pp. 335-385 in Class field theory - its centenary and prospect (Tokyo, 1998), Adv. Stud. Pure Math., 30, Math. Soc. Japan, Tokyo, 2001.
Coates and Sujatha, The main conjecture, Guwahati Workshop on Iwasawa Theory of Totally Real Fields, 113-140, Ramanujan Math. Soc. Lect. Notes Ser., 12, Ramanujan Math. Soc., Mysore, 2010 (sketches the proof of the cyclotomic main conjecture via modular forms).

Books (each of these contains the Euler system proof of the cyclotomic main conjecture, among other things):

Washington, Introduction to cyclotomic fields, Springer, 1997.
Lang, Cyclotomic fields I and II, Springer, 1990, with an appendix by Rubin.
Coates and Sujatha, Cyclotomic fields and zeta values, Springer, 2006.
Greenberg, Topics in Iwasawa theory (first two chapters of a unfinished book).

Other references:

Rubin, Global units and ideal class groups, Invent. Math. 89 (1987), 511-526.
Rubin, Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication, Invent. Math. 89 (1987), 527-559.
Rubin, The "main conjectures" of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25-68.

More about Iwasawa theory can be found on Ralph Greenberg's website. An even longer list of references is available here.

September 8: Atticus Christensen. Class groups of cyclotomic fields (Serre, pages 83-86, then start Section 6)

September 15: David Corwin. Class groups of cyclotomic fields, continued (rest of Serre)

September 22: Alex Smith. Local units (Coates and Sujatha, Cyclotomic fields and zeta values, Chapter 2, but we will omit some details of the proofs)

September 29: Borys Kadets. Iwasawa algebras and p-adic measures (Coates and Sujatha, Cyclotomic fields and zeta values, Chapter 3)

October 6: NO SEMINAR (MIT retreat)

October 13: NO SEMINAR (AGNES at Northeastern)

October 20: Nicholas Triantafillou. Cyclotomic Units and Iwasawa's Theorem (Coates and Sujatha, Cyclotomic fields and zeta values, Chapter 4, Sections 1-5)

October 27: Vishal Arul. Cyclotomic Units and Iwasawa's Theorem, continued (Coates and Sujatha, Cyclotomic fields and zeta values, Chapter 4, Sections 6-8)

November 3: Max Menzies. Euler systems (Coates and Sujatha, Cyclotomic fields and zeta values, Chapter 5, Sections 1-3)

November 10: NO SEMINAR (Veterans Day)

November 17: Lynnelle Ye. Euler systems, continued (Coates and Sujatha, Cyclotomic fields and zeta values, Chapter 5, Sections 4-5)

November 24: NO SEMINAR (Thanksgiving Vacation)

December 1: Vishal Arul. Main conjecture (Coates and Sujatha, Cyclotomic fields and zeta values, Chapter 6)

December 8: Atticus Christensen. Mordell-Weil groups (Greenberg, Introduction to Iwasawa theory for elliptic curves, Chapter 1)

December 15: Alex Smith. Selmer groups (Greenberg, Introduction to Iwasawa theory for elliptic curves, Chapter 2)

Past semesters: Fall 2009, Spring 2010, Fall 2010, Spring 2011, Fall 2011, Spring 2012, Fall 2012, Spring 2013, Fall 2013, Spring 2014, Fall 2014, Spring 2015, Fall 2015, Spring 2016, Fall 2016, Spring 2017. For semesters up to and including Spring 2009, see the old STAGE webpage.

Organizers: Vishal Arul, Bjorn Poonen