9.1 Elliptic Curves with a given Endomorphism Ring

and we know that this ring is isomorphic to Z or an order O in an imaginary quadratic field K; in fact, the ring on the right is equal to Z or O (viewed as a subring of C).1 To simplify the discussion, we shall treat the isomorphism in (1) as an equality and view elements of End(EL) as elements of Z or O. How might we construct an elliptic curve with endomorphism ring O? An obvious way is to use the lattice L = O. If α ∈ End(EO), then αO ⊆ O, by (1), and therefore α ∈ O, since the ring O contains 1. Conversely, if α ∈ O, then αO ⊆ O, since O is closed under multiplication, and therefore α ∈ End(EO), by (1); thus End(EO) = O. But are there any other (non-isomorphic) examples of elliptic curves with End(E) = O? To answer this question, we would like to classify, up to homethety, the lattices L for which {α : αL ⊆ L} = O. Without loss of generality, we may assume L = [1, τ ], and O = [1, ω]. If End(EL) = O, then we must have ω · 1 = ω ∈ L, so ω = m + nτ , for some m,n ∈ Z. Thus nL = [n, ω −m] = [n, ω] (and O = [1, nτ + m] = [1, nτ ]). So L is homothetic to a sublattice of O, and this sublattice must be closed under multiplication by O; equivalently, L is homothetic to an O-ideal (a subring of O closed under multiplication by O). For any O-ideal L, the set {α ∈ C : αL ⊆ L} is an order that contains O, which we denote O(L). The same is true for any lattice homothetic to an O-ideal, since O(L) depends only on the homethety class of L. We are interested in the cases where O(L) = O, since these are precisely the (homethety classes of) lattices that give rise to elliptic curves EL/C with End(EL) = O. When the condition O(L) = O holds, we say that L is a proper O-ideal. Note that O(L) is always contained in the maximal order OK , so when O = OK every O-ideal is proper, but otherwise this is not true (Problem Set 9 asks for a counter example). Given that O(L) depends only on the homethety class of L, we shall regard two Oideals as equivalent if they are homothetic as lattices; it follows that the ideals a and b are equivalent if and only if (α)a = (β)b for some α, β ∈ O. Since the elliptic curves EL and EL′ are isomorphic if and only if the lattices L and L ′ are homothetic, two proper O-ideals a and b are equivalent if and only if Ea ' Eb. As shown in Problem Set 9, the set cl(O) of equivalence classes of proper O-ideals form a finite abelian group that is isomorphic to the group cl(D) formed by the SL2(Z)-equivalence classes of binary quadratic forms