Download e-book for kindle: Algorithmic Number Theory: Third International Symposiun, by Noam D. Elkies (auth.), Joe P. Buhler (eds.)

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

ISBN-10: 3540646574

ISBN-13: 9783540646570

This booklet constitutes the refereed lawsuits of the 3rd overseas 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 concept, cryptography, linear algebra and lattices, sequence and sums, algebraic quantity fields, category teams and fields, curves, and serve as fields.

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

Example text

2 The Case Σ = {3, 5} Here the area of X ∗ (1) is 1/3. This again is small enough to show that there are only four elliptic points, but leaves two possibilities for their indices: 2,2,2,6 or 2,2,3,3. It turns out that the first of these is correct. IV:2]; it can also be checked as we did in the cases Σ = {2, p} (p = 3, 5, 7) by exhibiting appropriate elliptic elements of Γ ∗ (1) — which we need to do anyway to compute the CM points. We chose to write write O = Shimura Curve Computations Z[ 21 1 + c, e] with c2 + 3 = e2 − 5 = ce + ec = 0, 35 (72) and found the elliptic elements s2 = [4c − 3e], s2 = [5c − 3e − ce], s2 = [20c − 9e − 7ce], s6 = [3 + c] (73) [NB 20c − 9e − 7ce, 3 + c ∈ 2O] of orders 2, 2, 2, 6 with s2 s2 s2 s6 = 1.

9 Indeed, let K be the totally real cubic field Q(cos 2π/7) of minimal discriminant 49, and let A be a quaternion algebra over K ramified at two of the three real places and at no finite primes of K. Now for any totally real number field of degree n > 1 over Q, and any quaternion algebra over that field ramified at n − 1 of its real places, the group Γ (1) of norm-1 elements of a maximal order embeds as a discrete subgroup of PSL2 (R) = Aut(H), with H/Γ of finite area 9 Actually this fact is due to Fricke [F1,F2], over a century ago; but Fricke could not relate G2,3,7 to a quaternion algebra because the arithmetic of quaternion algebras had yet to be developed.

Thus X0∗(3) is a rational curve with six elliptic points all of index 2, and we may choose coordinates t, x on X ∗ (1), X0∗(3) such that t(P4 ) = ∞, t(P2 ) = 0, and x = ∞, x = 0 at the quadruple pole and double zero respectively of t. 32 Noam D. Elkies Table 4 |D| 3 8 20 40 52 120 35 27 72 43 180 88 115 280 67 148 340 520 232 760 163 D0 1 8 20 40 4 40 5 1 8 1 20 8 5 40 1 4 20 40 8 40 1 D1 1 1 1 1 13 3 7 33 32 43 32 11 23 7 67 37 17 13 29 19 163 |A| B |A − 2B| |A − 27B| 0 1 2 33 1 0 1 1 2 1 0 52 33 1 52 0 3 2 2·3 5 23 13 36 33 72 53 2·33 52 26 7 2·52 53 6 2 2 2 3 5 2·11 172 3 3 2 2 5 7 3 13 22 312 6 3 2 2 2 2 3 5 7 2·19 36 43 2·113 132 23 53 3 52 172 33 53 2·72 172 11 36 9 3 2 2 2 2 3 13 23 2·5 11 36 53 3 3 2 3 2 3 11 2·23 7 5 13 38 52 26 33 53 72 132 2·112 312 38 67 2·33 113 52 72 132 25 172 37 38 292 3 3 2 2 3 2 2 2·3 23 7 29 2 5 13 7 36 54 3 3 3 2 2 4 2 2 3 29 2 7 47 13 5 11 17 38 52 43 33 113 173 22 52 72 232 132 192 532 36 712 29 33 173 473 72 312 712 52 112 132 372 19 2·38 53 672 29 33 53 113 72 132 292 312 2·192 592 792 36 172 732 163 We next determine the action of w3 on the elliptic points of X0∗ (3).

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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.)


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