Analytic Geometry of Space by Virgil Snyder, Charles Herschel Sisam

By Virgil Snyder, Charles Herschel Sisam

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Proof. Omitted. 8 shows that (X, Y ) → (℘(z), ℘ (z)) defines a homomorphism C[x, y] =df C[X, Y ]/(Y 2 − 4X 3 + g2 X + g3 ) → C[℘, ℘ ], where C[℘, ℘ ] is the C-algebra of meromorphic functions on C generated by ℘ and ℘ . I claim13 that the map is an isomorphism. For this, we have to show that a polynomial 13 Those who know some commutative algebra will be able to give a simpler proof. ELLIPTIC CURVES 47 g(X, Y ) ∈ C[X, Y ] for which g(℘, ℘ ) = 0 is divisible by f (X, Y ) =df Y 2 − X 3 + g2 X + g3 .

Finally, one can verify that it induces isomorphisms on the tangent spaces. S. MILNE The addition formula. Consider ℘(z + z ). It is a doubly periodic function of z, and therefore it is a rational function of ℘ and ℘ . The next result exhibits the rational function. 12. The following formula holds: 1 ℘(z + z ) = 4 ℘ (z) − ℘ (z ) ℘(z) − ℘(z ) 2 − ℘(z) − ℘(z ). Proof. Let f (z) denote the difference between the left and the right sides. Its only possible poles (in D) are at 0, or ±z , and by examining the Laurent expansion of f (z) near these points one sees that it has no pole at 0 or z, and at worst a simple pole at z .

The fact that E(Q)tors is so much smaller than E(Qal )tors shows that the image of the Galois group in the automorphism group of E(Qal )tors is large. ELLIPTIC CURVES 51 Endomorphisms. A field K of finite degree over Q is called an algebraic number field. Each α ∈ K satisfies an equation, αm + a1 αm−1 + · · · + am = 0, ai ∈ Q. If it satisfies such an equation with the ai ∈ Z, then α is said to be an (algebraic) integer of K. The algebraic integers form a subring OK of K, which is a free Z-module of rank [K : Q].

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