By Milne J.S.

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Before proving the lemma, we explain why it implies the proposition. For P ∈ E n (Qp ), let x¯(P ) = x (P ) mod p5n Zp . The lemma shows that the map P → x¯(P ) : E n (Qp ) → pn Zp /p5n Zp has the property: P1 + P2 + P3 = 0 =⇒ x¯(P1 ) + x¯(P2 ) + x¯(P3 ) = 0. Since x¯(−P ) = −¯ x(P ), it is therefore a homomorphism of abelian groups. Suppose that P ∈ E 1 (Qp ) has order m divisible by p. Then Q =df mp P will also lie in E 1 (Qp ) and will have order p. Since Q = 0, for some n, Q ∈ E n (Qp ) \ E n+1 (Qp ).

Let P = (x, y) be the sum of P1 = (x1 , y1 ) and P2 = (x2 , y2 ). If P2 = −P1 , then P = O, and if P1 = P2 , we can apply the duplication formula. Otherwise, x1 = x2 , and (x, y) is determined by the following formulas: x(x1 − x2 )2 = x1 x22 + x21 x2 − 2y1 y2 + a(x1 + x2 ) + 2b and y(x1 − x2 )3 = W2 y2 − W1 y1 where W1 = 3x1 x22 + x32 + a(x1 + 3x2 ) + 4b W2 = 3x21 x2 + x31 + a(3x1 + x2 ) + 4b. Duplication formula. Let P = (x, y) and 2P = (x2 , y2 ). If y = 0, then 2P = 0. Otherwise y = 0, and (x2 , y2 ) is determined by the following formulas: x4 − 2ax2 − 8bx + a2 (3x2 + a)2 − 8xy 2 = 4y 2 4(x3 + ax + b) x6 + 5ax4 + 20bx3 − 5a2 x2 − 4abx − a3 − 8b2 = .

10 N´eron himself didn’t use schemes, but rather invented his own private version of algebraic geometry over discrete valuation rings, which makes his papers almost unreadable. S. MILNE By (c) we mean the following: for each t0 ∈ C we have a curve E(t0 ) : Y 2 Z = X 3 + a(t0 )XZ 2 + b(t0 )Z 3 , a(t0 ), b(t0 ) ∈ C, with discriminate ∆(t0 ). This is nonsingular, and hence an elliptic curve, if and only if ∆(t0 ) = 0. Otherwise, it will have a singularity, and we view it as a degenerate elliptic curve.

### Elliptic curves and algebraic geometry. Math679 U Michigan notes by Milne J.S.

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