MAT 1210 (Arithmetic of Elliptic Curves), Winter 2015
Main reference: Silverman, The Arithmetic of Elliptic Curves
Homework 1 (due Feb 25):
Problems 1.10, 2.3, 2.4, 3.3(d) + show this is an elliptic curve + find a Weierstrass equation (hint: recall how we found Weierstrass coordinates for a general elliptic curves), 3.5, 3.8
Consider E: y^2 = x^3 - 43x + 166 over the rationals and P = (3,8). Show by hand that P has finite order. (Hint: it may be more
efficient to compute some 2^i P first.)
MAT 1200/415 (Algebraic Number Theory), Winter 2014
Main reference: Milne's notes
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.