Department of Mathematics
University of Toronto
40 St. George Street, #6290
Toronto, ON M5S 2E4
Office: Bahen Centre 6186
E-mail: my last name at math dot toronto dot edu
MAT 1200/415 (Algebraic Number Theory); main reference: Milne's notes
Office hours: Thursdays 4-6pm (no office hours on January 30)
- Homework 1 (due Feb 27): Marcus chapter 2, exercises 11, 28, 30, 40, 41, 42(ab).
- Homework 2 (due Apr 4):
Determine the class groups of Q(root(d)) for d equal to 6, 10, -19, -23, -14, -21.
Show that the class number of Q(a) is 1, where a^3-a+1 = 0.
Marcus chapter 5, exercises 33, 47. Determine the unit group of O_K where K = Q(root(d)) for d equal to 5, 14.
Marcus chapter 3, exercises 16, 17, 30. Marcus chapter 4, exercises 5, 6, 7.
- Homework 3 (due Apr 18):
Let K be the 527-th cyclotomic field. Determine all quadratic and all cubic subextensions of K.
Give a description of the primes that split completely in the cubic subextensions.
Cassels chapter 4, exercises 3, 5, 6. Also, answer the following question: for the cubic
field K = Q(a), where a has minimal polynomial F(X) given in Cassels exercise 6, find all e_i and f_i for p = 2, 3, 5, 7.
Cassels chapter 7, exercises 5 (only (i)-(iii)), 10, 13, 18.
Milne, appendix B, problem 6.
PDF files may differ slightly from published versions.
- Weight cycling and Serre-type conjectures for unitary groups (with Matthew Emerton and Toby Gee).
Duke Mathematical Journal 162 (2013), no. 9, 1649-1722.
- Adequate subgroups
(with Robert Guralnick,
Richard Taylor, and
Appendix to On the automorphy of l-adic Galois representations with small residual image by Jack Thorne.
Journal de l'Institut de Mathématiques de Jussieu 11 (2012), no. 4, 907-920.
- The classification of irreducible admissible mod p representations of a p-adic GLn.
Inventiones Mathematicae 186 (2011), no. 2, 373-434.
- A Satake isomorphism in characteristic p.
Compositio Mathematica 147 (2011), no. 1, 263-283.
- Conjecture de type de Serre et formes compagnons pour GSp4 (with Jacques Tilouine), 36 pages.
Journal für die reine und angewandte Mathematik (Crelle) 676 (2013), 1-32.
- The weight in a Serre-type conjecture for tame n-dimensional Galois representations.
Duke Mathematical Journal 149 (2009), no. 1, 37-116. (Typos.)
- Linear Algebraic Groups
Notes for my graduate course, Spring 2013 (typed by Joshua Seaton).
- p-Modular Representations of p-adic Groups
Notes for my minicourse at IMS Singapore, April 2013 (typed by Karol Koziol). These notes focus on GL2(Qp).
- The Mod p Representation Theory of p-adic Groups
Notes for my graduate course at Fields Institute, spring 2012 (typed by Christian Johansson).
- Classification of irreducible mod p representations of a p-adic GLn (scanned).
Notes for mini-course at the Galois Trimester, Paris (Jan 4-Mar 27, 2010)
- Serre-type conjecture for tame n-dimensional mod p Galois representations (scanned).
Serre-type conjectures, mod p cohomology, and representation theory of GLn(Qp) (scanned).
Notes for half of a mini-course with Matthew Emerton at the summer school
Représentations p-adiques des groupes p-adiques, Paris (July 7-12, 2008)
- Smooth representations of p-adic reductive groups.
Notes for the Instructional Conference on Representation Theory and Arithmetic, Northwestern (May 5-9, 2008)