By Milne J.S.

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P → P¯ : E(Qp ) → E(F We shall define a filtration E(Qp ) ⊃ E 0 (Qp ) ⊃ E 1 (Qp ) ⊃ · · · ⊃ E n (Qp ) ⊃ · · · and identify the quotients. First, define E 0 (Qp ) = {P | P¯ is nonsingular}. It is a subgroup because, as we observed on p26, a line through two nonsingular points on a cubic (or tangent to a nonsingular point), will meet the cubic again at a nonsingular point. Write E¯ ns for E¯ \ {any singular point}. The reduction map P → P¯ : E 0 (Qp ) → E¯ ns (Fp ) is a homomorphism, and we define E 1 (Qp ) be its kernel.

For any elliptic curve E over an algebraically closed field k of characteristic zero, E(k)n is a free Z/nZ-module of rank 2. Proof. There will exist an algebraically closed subfield k0 of finite transcendance degree over Q such that E arises from a curve E0 over k0 . Now k0 can be embedded into C, and so we can apply the next lemma (twice). 19. Let E be an elliptic curve over an algebraically closed field k, and let Ω be an algebraically closed field containing k. Then the map E(k) → E(Ω) induces an isomorphism on the torsion subgroups.

3 27 3 27 a = 2cd + ELLIPTIC CURVES 37 ´ron Models 9. Ne Consider an elliptic curve over Qp E : Y 2 Z = X 3 + aXZ 2 + bZ 3 , a, b ∈ Qp , ∆ = 4a3 + 27b2 = 0. After making a change of variables X → X/c2 , Y → Y /c3 , Z → Z, we can suppose that a, b ∈ Zp and ordp (∆) is minimal. We can think of E as defining a curve over Zp , which will be the best “model” of E over Zp among plane projective curves when p = 2, 3. However, when p = 2 or 3 we may be able to get a better model of E over Zp by allowing a more complicated equation.