Supplement: Addition on Elliptic Curves

18.786: Topics in Algebraic Number Theory Supplement: addition on elliptic curves Let K be a field and let K be an algebraic closure of K. Then any homogeneous poly­ nomial P ∈ K[x, y, z] defines a closed subvariety V (P ) of the projective space P3K . Actually it’s a closed subscheme, but I’ll often assume that P has no repeated factors, so that I can neglect this. I say P is nonsingular at a point [a : b : c] ∈ V (P ) (the a, b, c being homogeneous coordinates) if the partial derivatives of P do not all vanish at (x, y, z) = (a, b, c). Then P has a unique tangent line at that point. For P, Q ∈ K[x, y, z] homogeneous polynomials with no factors in common, I define the intersection multiplicity of P, Q at a point [a : b : c] ∈ V (P ) ∩ V (Q) to be the K­dimension of the local ring of the scheme V (P ) ∩ V (Q) at [a, b, c]. Concretely, take K[x, y, z]/(P, Q), invert any homogeneous polynomial not vanishing at [a, b, c], then pull out the bit of degree zero. If P and Q are both nonsingular, then the intersection multiplicity is 1. If only P is nonsingular, then the intersection multiplicity is the order of vanishing of Q along the tangent line of P . Theorem 1 (Bézout) Let P, Q ∈ K[x, y, z] be homogeneous polynomials with no repeated factors and no factors in common. Then the intersection multiplicities of all points of V (P )∩ V (Q) add up to deg(P ) deg(Q). Let P ∈ K[x, y, z] be a polynomial with no repeated factors. Let Div(P ) be the free abelian group generated by V (P ); we refer to elements of Div(P ) as divisors on P and define the degree of a divisor as the sum of its coefficients. For any Q ∈ K[x, y, z] with no factor in common with P , write (Q) for the divisor consisting of each point in V (P ) ∩ V (Q) with multiplicity equal to the intersection multiplicity. By Bézout, this divisor has degree deg(P ) deg(Q). Let Div0 (P ) be the subgroup of Div(P ) consisting of divisors of degree 0. Define the Picard group Pic(P ) of P (or better, of the algebraic curve V (P ) over K) to be the quotient of Div0 (P ) by the subgroup generated by (Q1 )−(Q2 ) for all homogeneous polynomials Q1 , Q2 of the same degree. Now suppose P has degree 3 and is nonsingular everywhere, and that O ∈ V (P ) is a point with coefficients in K. The pair (V (P ), O) is an example of an elliptic curve. In this case, for any points T, U ∈ V (P ), you can draw a line through T and U which hits V (P ) in a third point S, and thus get a relation (S) + (T ) + (U ) = �, where � is the divisor of any fixed line. Consequence: every element of Pic(P ) can be represented by a pair (T ) − (O) for some T ∈ V (P ). Moreover, this T is unique: that amounts to saying that (T ) − (U ) can never occur as the divisor of a polynomial. That’s a little exercise in the theory of algebraic curves: such a divisor would give rise to an isomorphism between V (P ) and P1K , but the former has genus 1 and the latter has genus 0. (Concretely: there is a rational differential 1 form on V (P ) with no poles anywhere, but any rational differential on PK has two more poles than zeroes, when counting with multiplicity.) 1 In other words, there is an addition law for points on V (P )! Moreover, you can compute this law as follows: given two points T, U , take the third intersection S of the line through them with V (P ), then take the third intersection of the line through S and O with V (P ). In particular, K­rational points form a subgroup under addition. Aside: the uniqueness argument doesn’t work if P is degree 3 but singular, but you can still use the geometric addition law on nonsingular points, as long as O itself is nonsingular: you can prove the associativity by degeneration from the nonsingular case. (The hangup with a singular point is that a line through it always has intersection multiplicity greater than 1 with V (P ).) But in this case you can sometimes identify the result more simply; see exercises. For more, see Silverman, The Arithmetic of Elliptic Curves. I may have more to say on this topic later. Exercise (not to be turned in): 1. Let P = x3 + y 2 z and let O = [0 : 1 : 0]. Give an isomorphism of the group of nonsingular points of V (P ) with the additive group of K. 2. Let P = x3 + x2 z + y 2 z and let O = [0 : 1 : 0]. Give an isomorphism of the group of nonsingular points of V (P ) with the multiplicative group of K. 2