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# Get A Characteristic Property of the Algebra C(Q)B PDF By Karakhanyan M.I., Khor'kova T.A.

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E LLIPTIC CURVES These are the only irreducible projective curves having a group structure defined by polynomial maps. x; y/ 7! x C y W k k ! k: We sometimes write Ga for A1 endowed with this group structure. 55 3. x; y/ 7! k : We sometimes write Gm for A1 X f0g endowed with this group structure. Note that the map x 7! x; x 1 / identifies Gm with the affine plane curve X Y D 1. T WISTED MULTIPLICATIVE GROUPS p Let a be a nonsquare in k , and let L D kŒ a. k/ D f 2 L j NmL=k D 1g: p Let ˛ D a, so that f1; ˛g is a basis for L as a k-vector space.

P ROOF. 1, only a little more complicated. k/. Without this assumption, it may only be possible to realize the curve as a (singular) plane curve of (possibly much) higher degree. 54 CHAPTER II. BASIC THEORY OF ELLIPTIC CURVES N OTES Cassels (1991, p. 4 in a letter to Cassels (Tate 1975), which has been copied (and, on occasion, miscopied) by all later authors. ” 3 Reduction of an elliptic curve modulo p Consider an elliptic curve E W Y 2 Z D X 3 C aXZ 2 C bZ 3 ; a; b 2 Q; D 4a3 C 27b 2 ¤ 0: After a change of variables X 7!

52 CHAPTER II. 2 Two elliptic curves can have the same j -invariant and yet not be isomorphic over k. a; b/, but it is not isomorphic to it. coefficient of X deg f 1 /. x2 ; y2 /. If P2 D P1 , then P D O, and if P1 D P2 , we can apply the duplication formula below. x2 ; y2 /. If y D 0, then 2P D 0. 2y/3 x2 D 53 2. 1 for a general field k. 4 Let k be an arbitrary field. 0 W 1 W 0/. 0 W 1 W 0/ an elliptic curve) if and only if ¤ 0. a10 ; : : :/ ! u2 x C rzW u3 y C su2 x C tzW z/. x W y W z/ 7! a10 ; : : :/ !