Exercise Set 5: Galois Cohomology

Math 582e, Winter 2010, University of Washington

Due Wednesday, February 24, 2010

1. (a) Suppose f : A → A is a homomorphism and A is a finite abelian group. Prove that # ker(f) = # coker(f).

(b) Is it necessarily the case that ker(f) ≈ coker(f) as abelian groups?

2. Give examples of central simple ℚ-algebras that correspond to elements of order 1, 2, and 3 in Br_{ℚ}. (Hint: Orders 1 and 2 are very easy. I’m not so sure how to give an explicit element of order 3, so do some digging.)

3. Give examples of genus one curves X over ℚ that correspond to elements of order 1, 2, and 3 in the H^{1}(ℚ, E), where E = Jac(X).