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

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

This ebook constitutes the refereed court cases of the 3rd overseas Symposium on Algorithmic quantity thought, 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 geared up 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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Extra info for Algorithmic Number Theory: Third International Symposiun, ANTS-III Portland, Oregon, USA, June 21–25, 1998 Proceedings

Example text

But such a simple explanation probably cannot work for Shimura curves which have neither cusps nor integrality of CM points. e. given the quaternion algebra A), the height is inversely proportional to the area of the curve; does this remain true in some sense when A is varied? As a special case we might ask: how does the minimal polynomial of a CM point of discriminant −D factor modulo the primes contained in D1 ? That the minimal polynomials for CM j-invariants are almost squares modulo prime factors of the discriminant was a key component of our results on supersingular reduction of elliptic curves [E2,E3]; analogous results on Shimura curves may Shimura Curve Computations 43 likewise yield a proof that, for instance, for every t ∈ Q there are infinitely many primes p such that the point on the (2, 4, 6) curve with coordinate t reduces to a supersingular point mod p.

We write the discriminant D of each of them as −D0 D1 where D0 |24 and D1 is coprime to 6. In Table 1 we give, for each |D| = D0 D1 , the integers A, B with B ≥ 0 such that (A : B) is the t-coordinate of a CM point of discriminant D. In the last column of this table we indicate whether the point was obtained algebraically (via an isogeny of degree 5, 7, or 13) and thus proved correct, or only computed numerically. The CM points are listed in order of increasing height max(|A|, B). In Table 2 we give, for each except the first three cases, the factorizations of |A|, B, |C| where C = A − B, and also the associated “ABC ratio” [E1] defined by r = log N (ABC)/ log max(|A|, B, |C|).

On the other hand, by (10) a curve of area as small as 1/6 cannot have more than four elliptic points, and if it has exactly four then their orders must be 2, 2, 2, 3. Indeed we find in Γ ∗(1) the elements of finite order s2 = [b], s2 = [2e + 5b − be], s2 = [5b − be], s3 = [2b − e − 1] (52) [NB 2e + 5b − be, 5b − be, 2b − e − 1 ∈ 2O] of orders 2, 2, 2, 3 with s2 s2 s2 s3 = 1. As in the case of the G2,4,6 we conclude that here Γ ∗(1) has the presentation 2 2 s2 , s2 , s2 , s3 |s22 = s2 = s2 = s33 = s2 s2 s2 s3 = 1 .

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