Elliptic Curves : A Computational Approach

Elliptic Curves : A Computational Approach

Susanne Schmitt

Language: English

Pages: 367

ISBN: 9380250010

Format: PDF / Kindle (mobi) / ePub


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over Q b) Y 2 + XY = X3 + 2X + 1 over Q c) Y 2 = X3 − 15 over F2 , over F3 , and over F7 5) Prove Theorem 1.7. 6) Why are the given birational transformations the only variable transformations which leave the Weierstraß normal form fixed? 7) Find a birational transformation to short Weierstraß normal form for E : Y 2 Z + 4XY Z = X3 + 2X2 Z + XZ 2 + 3Z 3 . 8) Prove Proposition 1.11 9) Check the different addition formulas for the different representations of points on elliptic curves. 10) a)

u)2 = (λ2 − a2 )2 + 2(λ2 − a2 )u + u2 = 2uλ2 + 4(a4 − 2νλ) − 2a2 u + u2 = 2uλ2 − 8νλ + (u2 − 2a2 u + 4a4 ) √ 4ν √ = ( 2uλ)2 − 2 √ ( 2uλ) + (u2 − 2a2 u + 4a4 ). 2u (5.2) The left hand side of this equation is a square. The right hand side is a square if and only if its discriminant (divided by 4) 4ν √ 2u 2 − (u2 − 2a2 u + 4a4 ) = 8ν 2 − (u2 − 2a2 u + 4a4 ) = 0. u Multiplying by u, and using ν 2 = a6 (= 0, if P = (0, 0)), we get the equation 0 = 8a6 − u3 + 2a2 u2 − 4a4 u = −u3 + 2a2 u2 − 4a4

determine explicitly the corresponding curves E and fields K in the finiteness cases because the orders of possible torsion groups were again too high. Proof. a) See the paper of Müller, Ströher, and Zimmer [149]. The finitely many elliptic curves and the finitely many quadratic fields found by means of the SIMATH package are listed there. Schmitt [189] has computed the exact torsion group for all curves and fields listed in this article. b) This part is proved in the paper of Peth˝o, Weis, and

that there is at least one such solution. An example of this situation is Case 14 of Theorems 6.18, 6.19, where the torsion group E(K)tors ∼ = Z/5Z is treated. For cubic fields K, the norm equations (6.3) read N (α 2 N (α) = ±1, − 11α − 1) = ±5m , 0 ≤ m ≤ 9. (6.7) This case requires some extra endeavor as the subsequent proposition (see Peth˝o, Weis, Zimmer [162]) shows. Proposition 6.22. An integer α ∈ OK \Z with minimal polynomial H (Z) ∈ Z[Z] and polynomial discriminant Dα is a solution of

1632 132 132 This way all the norm equations can be solved if the degree of K, n = [K : Q], is sufficiently small. Then, there exist only finitely many elliptic curves E (up to isomorphisms) with integral j -invariant over a finite set of number fields K such that the Mordell–Weil group E(K) contains a certain given torsion group E(K)tors (aside from groups of very small order) (see Abel-Hollinger and Zimmer [1], [100], Müller, Ströher, Zimmer [149], Peth˝o, Weis, Zimmer [162], and the theses of

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