Algorithmic Number Theory: Third International Symposiun, by Noam D. Elkies (auth.), Joe P. Buhler (eds.)

By Noam D. Elkies (auth.), Joe P. Buhler (eds.)

This booklet constitutes the refereed lawsuits of the 3rd foreign Symposium on Algorithmic quantity conception, ANTS-III, held in Portland, Oregon, united states, in June 1998.
The quantity provides forty six revised complete papers including invited surveys. The papers are prepared in chapters on gcd algorithms, primality, factoring, sieving, analytic quantity thought, cryptography, linear algebra and lattices, sequence and sums, algebraic quantity fields, type teams and fields, curves, and serve as fields.

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Extra info for Algorithmic Number Theory: Third International Symposiun, ANTS-III Portland, Oregon, USA, June 21–25, 1998 Proceedings

Example text

We find in Γ ∗(1) the elements of finite order s2 = [b], s2 = [7b − 2g − bg], s2 = [7b + 2g − bg], s4 = [1 + 2b + g] (65) [NB 7b ± 2g − bg ∈ 2O] of orders 2, 2, 2, 4 with s2 s2 s2 s4 = 1, and conclude that 2 2 s2 , s2 , s2 , s4 generate Γ ∗(1) with relations determined by s22 = s2 = s2 = s44 = ∗ s2 s2 s2 s4 = 1. None of these is in Γ (1): the representatives b, 1 + 2b + g of s2 , s4 have norm 2, while s2 , s2 have representatives (7b ± 2g − bg)/2 of norm 14. The discriminants of s4 , s2 , s2 , s2 are −4, −8, −56, −56; note that −56 is not among the “idoneal” discriminants (discriminants of imaginary quadratic fields with class group (Z/2)r ), and thus that the elliptic fixed points P2 , P2 of s2 , s2 are quadratic conjugates on X ∗ (1).

Cambridge University Press, 1992. : Die Typen die Multiplikatorenringe elliptische Funktionk¨ orper, Abh. Math. Sem. Hansischen Univ. 14, 197–272 (1941). : ABC implies Mordell, International Math. Research Notices 1991 #7, 99–109.

Using the condition that the Jacobian of X (1), and any elliptic curve occurring in the Jacobian of X0 (2), have conductor at most 15 and 30 respectively, we find A = B = −3. Then X (1) has equation (76) y2 = −(3s2 + 1)(s2 + 27) (with t = −3s2 ) and Jacobian isomorphic with elliptic curve 15C (15-A1); the curve intermediate between X ∗ (2) and X0 (2) whose function field is obtained from Q(X ∗(2)) by adjoining −3(t − 1)(t − 81) has equation y2 = −3(x4 − 10x3 + 33x2 − 360x + 1296) (77) 36 Noam D.

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