By Noam D. Elkies (auth.), Joe P. Buhler (eds.)
This ebook constitutes the refereed complaints of the 3rd overseas Symposium on Algorithmic quantity idea, ANTS-III, held in Portland, Oregon, united states, in June 1998.
The quantity offers forty six revised complete papers including invited surveys. The papers are equipped in chapters on gcd algorithms, primality, factoring, sieving, analytic quantity idea, cryptography, linear algebra and lattices, sequence and sums, algebraic quantity fields, type teams and fields, curves, and serve as fields.
Read Online or Download Algorithmic Number Theory: Third International Symposiun, ANTS-III Portland, Oregon, USA, June 21–25, 1998 Proceedings PDF
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Additional info for Algorithmic Number Theory: Third International Symposiun, ANTS-III Portland, Oregon, USA, June 21–25, 1998 Proceedings
Of the remaining elliptic points, P6 is complete ramified, and each of P2 , P2 has one simple and one double preimage. So we may choose coordinates x, t on X0∗ (2) and X ∗(1) such that t = x(x − 3)2 /4, with t(P6 ) = ∞, t(P2 ) = 0, t(P2 ) = 1. To determine t(P2 ) we use the involution w2 , which switches x = ∞ (the triple pole) with x = 0 (the simple zero), x = 4 (the simple preimage of P2 ) with one of the preimages x1 of P2 (the one parametrizing the isogeny from P2 to P2 ), and the other two preimages of P2 with each other.
Thus K is the compositum of K with an imaginary quadratic field, which must have unique factorization. We check that of the nine such fields √ only five retain unique factorization when composed with K. One of these, Q( −7 ), yields the cyclotomic field Q(e2πi/7 ), whose ring of integers is the CM ring for the elliptic point P7 : t = ∞; two subrings still have unique factorization and yield CM points ℘7 - and ℘8 -isogenous to that elliptic point, which again are not only K- but even Q-rational thanks to the Galois invariance of ℘7 , ℘8 .
These CM points are 5-isogenous with the elliptic points Shimura Curve Computations 23 t = ∞, t = 1 respectively, and thus have discriminants −3 · 52 and −4 · 52 . Similarly on X0∗(7) we have w7 (∞) = −9/20 at which t = −1073152081/3024000000, a CM point 7-isogenous with t = ∞ and thus of discriminant −3 · 72 . For each of l = 5, 7, 13 the two fixed points of wl on X0∗ (l) are rational and yields two new CM points of discriminants −cl for some factors c of 24. For X0∗(5) these fixed points are x5 = −3/5 and x5 = 7/30, at which t = 2312/125 and t = 5776/3375 respectively; these CM points have discriminants −40, −120 by the supersingular test.
Algorithmic Number Theory: Third International Symposiun, ANTS-III Portland, Oregon, USA, June 21–25, 1998 Proceedings by Noam D. Elkies (auth.), Joe P. Buhler (eds.)