Get Algorithmic Number Theory: Third International Symposiun, PDF

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

ISBN-10: 3540646574

ISBN-13: 9783540646570

This booklet constitutes the refereed court cases of the 3rd foreign 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 prepared in chapters on gcd algorithms, primality, factoring, sieving, analytic quantity concept, cryptography, linear algebra and lattices, sequence and sums, algebraic quantity fields, classification teams and fields, curves, and serve as fields.

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

Sample text

For instance, if I is the prime ideal ℘7 above the totally ramified prime 7 of Q then X (℘7 ) is the Klein curve of genus 3 with automorphism group PSL2 (F7 ) of order 168. The next-smallest example is the ideal ℘8 above the inert prime 2, which yields a curve of genus 7 with automorphism group [P]SL2 (F8 ) of order 504. 1990 letter to Abhyankar. At any rate, we obtain towers {X (℘r7 )}r>0 , {X (℘r8 )}r>0 of unramified abelian extensions which are asymptotically optimal over the quadratic extensions of residue fields10 of K other than F49 and F64 respectively, which are involved in the class field towers of exponents 7, 2 of the Klein and Macbeath curves over those fields.

In Table 4 we give the factorizations of |A|, B, |A−2B|, |A−27B|; as expected, |A| is always “almost” a perfect cube, and B, |A − 2B|, |A − 27B| “almost” a perfect square, any exceptional primes other than 2 or 5 being the unique prime in D1 , which if it occurs is listed at the end of its respective factorization. 5 Further Examples and Problems Our treatment here is briefer because most of the ideas and methods of the previous sections apply here with little change. Thus we only describe new features that did not arise for the algebras ramified at {2, 3} and {2, 5}, and exhibit the final results of our computations of modular curves and CM points.

While the curve has genus 0, no degree-1 rational function on it takes Q-rational values at all four of its elliptic points, and the towers of modular curves over this X (1) are defined over K but not over Q. Let K be the cubic field Q[τ ]/(τ 3 − 4τ + 2) and discriminant 148 = 22 37, which is minimal for a totally real non-Galois field. Let A/K be a quaternion algebra ramified at two of the three real places and at no finite primes of K. 16666 . ; thus the area is 1/6 and, since A is not in Takeuchi’s list, the curve X (1) has genus 0 and four elliptic points, one of order 3 and three of order 2.

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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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