Pries: 605C Elliptic Curves 2011. Homework 2.
Assignment: Do 4-5 of the problems below by Friday 3/4.
Complex multiplication - less theory
√
1. Let RK be the ring of integers of K = Q( −7). Let E = C/RK .
(i) Find an element of norm 2 in RK .
(ii) Find two endomorphisms γ, γ 0 ∈ End(E) of degree 2.
(iii) What is the kernel of γ ◦ γ 0 ?
2. We proved that C/Z[i] is isomorphic to E : y 2 = x3 + x. Let σ : E → E be given by
σ((x, y)) = (−x, iy).
(i) Prove that σ corresponds to multiplication by −i map on C/Z[i] by showing
p(−iz) = −p(z) and p0 (−iz) = ip0 (z).
(ii) Prove that i 7→ σ gives the right embedding Z[i] ⊂ End(E): show d(σ(x))
= i dx
.
d(σ(y))
y
P
3. For a lattice L ⊂ C, let G2k (L) = 06=ω∈L ω −2k , let g2 (L) = 60G4 and g3 (L) = 140G6 .
Define
∆(L) = g2 (L)3 − 27g3 (L)2 and j(L) = 1728g2 (L)3 /∆(L).
(i) If γ ∈ C∗ , prove that g2 (γL) = γ −4 g2 (L) and g3 (γL) = γ −6 g3 (L) and so ∆(γL) =
γ −12 ∆(L) and j(γL) = j(L).
(ii) Prove that j(L1 ) = j(L2 ) if and only if there exists γ ∈ C∗ such that γL1 = L2 .
(iii) If L is invariant under complex conjugation cc, show that cc(j(L)) = j(cc(L)).
(iv) Prove that E is defined over R (i.e., g2 , g3 ∈ R) if and only if there exists γ ∈ C∗
such that cc(γL) = γL.
4. Consider the set SK of elliptic curves E = C/L with complex multiplication by the ring
of integers RK of a quadratic imaginary field K. Recall that the class group C(RK )
acts on SK : if A is a fractional ideal of RK , then A · E = C/A−1 L. Prove that this
action is transitive.
Complex multiplication - more theory
√
1. Let√ K = Q( −6). How many elliptic curves E have complex multiplication by
Z[ −6]? Find a lattice L representing each isomorphism class.
2. Let E and E 0 be complex elliptic curves. Suppose End(E) ⊗ Q = K for some quadratic
imaginary field K. Prove that E and E 0 are isogenous if and only if End(E 0 ) ⊗ Q = K.
3. How many complex elliptic curves have an endomorphism of degree 2? If n ∈ N is not
a square, prove there are only finitely many isomorphism classes of complex elliptic
curves with an endomorphism of degree n.
4. Consider the differential form ω =
dx
y
on E : y 2 = x3 + ax + b.
(i) Prove that ω is a holomorphic differential (no zeros or poles).
(ii) Prove that ω is translation invariant: given Q ∈ E, let τQ : E → E be the
translation map τQ (P ) = P +E Q. Show that τ ∗ ω = ω.