Download Algorithmic Number Theory: 9th International Symposium, by Henri Darmon (auth.), Guillaume Hanrot, François Morain, PDF

By Henri Darmon (auth.), Guillaume Hanrot, François Morain, Emmanuel Thomé (eds.)

ISBN-10: 3642145183

ISBN-13: 9783642145186

This e-book constitutes the refereed lawsuits of the ninth foreign Algorithmic quantity idea Symposium, ANTS 2010, held in Nancy, France, in July 2010. The 25 revised complete papers offered including five invited papers have been conscientiously reviewed and chosen for inclusion within the ebook. The papers are dedicated to algorithmic points of quantity thought, together with straight forward quantity concept, algebraic quantity idea, analytic quantity thought, geometry of numbers, algebraic geometry, finite fields, and cryptography.

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Additional info for Algorithmic Number Theory: 9th International Symposium, ANTS-IX, Nancy, France, July 19-23, 2010. Proceedings

Example text

T ÔhÔf ÕÕ and M MT ÔhÔf ÕÕ 4: f 5: end while 6: return f and M 1 é Exchange step é Normalization step The original Gauss algorithm of 1801 used actually the largest normalization at each step. The number of reduction steps is exponential on some entries. Lagarias introduced the classical normalization to obtain a quadratic complexity 38 A. Bernard and N. Gama Theorem 2 (Real bound). Let f Ôa, b, cÕ be a primary-normalized form of discriminant Δ 0. Given f as input, RedGL2 terminates after at most ¡ log Ô a ß ΔÕ © 1 5 4 iterations where ω is the gold number.

Let X be the curve y 2 = x(x − 1)(x − 2)(x − 5)(x − 6), whose Jacobian has Mordell-Weil rank 1. The curve X has good reduction at 7, and X(F7 ) = {(0, 0), (1, 0), (2, 0), (5, 0), (6, 0), (3, 6), (3, −6), ∞}. 3(2)], we know |X(Q)| ≤ 10. However, we can find 10 rational points on X: the six rational Weierstrass points, and the points (3, ±6), (10, ±120). Hence |X(Q)| = 10. Since the Chabauty condition holds, there must exist a holomorphic differQ ential ω for which ∞ ω = 0 for all Q ∈ X(Q). We can find such a differential by taking Q to be one of the rational non-Weierstrass points, then computing Q Q a := ∞ ω0 , b := ∞ ω1 and setting ω = bω0 − aω1 .

Among the two integers hf , hf the one of smallest absolute value is noted hÔf Õ: that is hÔf Õ h h¡ h¡ f if hf f , and hÔf Õ f otherwise. In other words, hÔf Õ is the shortest normalization of f . T Ôνf Õ is classically-normalized, νf being one of the integers h¡ f , hf but not necessarily the one with the smallest absolute value. Our reduction algorithm, is a variant of the Gauss reduction which operates in GL2 ÔZÕ. It alternates exchange E and the shortest normalization T ÔhÔf ÕÕ at each loop, and terminates on a largely-reduced form.

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