18.783 Elliptic Curves Spring 2013 Lecture #7 02/28/2013

E[n] ' { Z/nZ⊕ Z/nZ if p = 0 or p n, Z/nZ or {0} if p > 0 and n is a power of p. Proof. Assume p n and let ` be a prime dividing n. Then the multiplication-by-` map [`] is separable of degree `2, and therefore E[`] = ker[`] has order `2. Every nonzero element of E[`] has order `, and it follows that E[`] ' Z/`Z ⊗ Z/`Z. Thus the `-rank of E[`], and hence of E[n], is 2. If `e is the largest power of ` dividing n, then the `-Sylow subgroup of E[n] is E[`e], which must be isomorphic to Z/`eZ⊕ Z/`eZ, since it has order `2e, `-rank 2, and no elements of order greater than `e. It follows that E[n] ' Z/nZ⊕ Z/nZ. If the characteristic p is not zero, then [p] is inseparable and its kernel E[p] therefore has order strictly less than deg [p] = p2. Since E[p] is a p-group of order less than p2, it must be isomorphic to either Z/pZ or {0}. If n = pk, then [n] = [p]k, and E[pk] is isomorphism to either Z/pkZ or {0} depending on whether E[p] is isomorphic to Z/pZ or {0}, respectively.