Homework Assignment 4

(Math 252: Modular Abelian Varieties)

William A. Stein

Oct. 8 (Due: Oct. 15)

There are six problems and parts of problems are of equal weight. (Reminder: It is fine to try to find solutions to these problems in books, etc., and recopy them, as long as you cite your sources.)

1. Let T = V/L be a complex torus. Then we know that the map that sends an element φ ∈ End(T) = Hom(T, T) to the corresponding homomorphism L → L is injective, so there is an injective map ρ_{Z} : Hom(L, L) ≈ Z^{4d}, where d = dim T. Is it possible that ρ_{Z} is surjective?

2. Suppose L_{1} = Z + Zαi ⊂ V_{1} = C, with α^{3} = 2. Then with respect to the basis 1, αi, the matrix of complex conjugation is J_{1} = ( \begin{matrix}0 & −α \\ 1/α & 0 \end{matrix} ). Use this to prove that End(V_{1}/L_{1}) = Z.

3. Let L be a lattice in a vector space V. A subgroup M of L is saturated in L if L/M is torsion free.

(a) Suppose W is a vector space and f : L → W is a homomorphism. Prove that ker(L) is saturated in L.

(b) Suppose W ⊂ V is a subspace of V. Prove that L ∩ W is saturated in L.

(c) Suppose M is a subgroup of L. The saturation M′ of M in L is the intersection of QM with L. Prove that M has finite index in M′.

4. Let T = V/L be a complex torus and let ρ_{Z} : End(T) → End_{Z}(L) be the integral representation. Prove that the image of ρ_{Z} is saturated in End_{Z}(L), in the sense that the cokernel of ρ_{Z} is torsion free.

5. (a) Let T = V/L be a complex torus. Prove that as a real manifold T is isomorphic to a product of copies of S^{1} = R/Z. Thus topologically complex tori are boring since they are classified by their dimension; it is their complex structure that makes them interesting.

(b) (*) Construct a 2-dimensional complex torus T = V/L (so V is a 2dimensional complex vector space), such that T is not isomorphic as a complex torus to a product C/L_{1} × C/L_{2} of 1-dimensional complex tori.

6. Let L be the subgroup of V = C × C generated by (1, 0), (0, i), (1, \sqrt{2}), (i, \sqrt{2}).

(a) Prove that L is a lattice in V.

(b) With respect to the given basis for L, compute the matrix J that represents multiplication by i from V to V.

(c) (*) Let T = V/L. Compute End(T), which is the subgroup of End_{Z}(L, L) of elements that commute with J.

(d) (*) Find a complex torus T′ that is isogenous to T but not isomorphic to T.