Harvard Math 129: Algebraic Number Theory Homework Assignment 9

William Stein

Due: Thursday, April 28, 2005

The problems have equal point value (except 4 below), and multi-part problems are of the same value. In any problem where you use a computer, include in your solution the exact commands you type and their output. You may use any software, including (but not limited to) MAGMA and PARI.

1. Suppose G is a finite group and A is a finite G-module. Prove that for any q, the group H^{q}(G, A) is a torsion abelian group of exponent dividing the order #A of A.

2. Let K = ℚ(\sqrt{5}) and let A = U_{K} be the group of units of K, which is a module over the group G = Gal(K/ℚ). Compute the cohomology groups H^{0}(G, A) and H^{1}(G, A). (You shouldn’t use a computer, except maybe to determine U_{K}.)

3. Let K = ℚ(\sqrt{−23}) and let C be the class group of ℚ(\sqrt{−23}), which is a module over the Galois group G = Gal(K/ℚ). Determine H^{0}(G, C) and H^{1}(G, C). (Use of a computer is fine.)

4. [This problem is double credit, i.e., it counts as two problems.] Let E be the elliptic curve y^{2} = x^{3} + x + 1. Let E[2] be the group of points of order dividing 2 on E. Let

\overline{ρ}_{E,2} : Gal(\overline{ℚ}/ℚ) → Aut(E[2])

be the mod 2 Galois representation associated to E.

(a) Find the fixed field K of ker(\overline{ρ}_{E,2}).

(b) Is \overline{ρ}_{E,2} surjective?

(c) Find the group Gal(K/ℚ).

(d) Which primes are ramified in K?

(e) Let I be an inertia group above 2, which is one of the ramified primes. Determine E[2]^{I} explicitly for your choice of I. What is the characteristic polynomial of Frob_{2} acting on E[2]^{I}.

(f) What is the characteristic polynomial of Frob_{3} acting on E[2]?