Download e-book for iPad: 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 e-book 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 provides 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, classification 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

The correct choice is determined by the condition that the Shimura curves must be fixed by the involution of the Galois closure of K/Q Shimura Curve Computations 41 that switches the two real embeddings of K that ramify A: the image of τ under the the third (split) embedding must be√used in (86). √ We find that the simple and of double preimages of P2 , P2 are x = ±2 a2 − 3, ∓ a2 − 3, and the preimages √ P2 are 12 − 2τ − 3τ 2 (fixed by w(τ) ) and (−12 + 2τ + 3τ 2 ± (3a2 − 12) a2 − 3)/2. From this we recover as usual the tower of curves X0 ((τ )r ), whose reductions at primes of K other than τ are asymptotically optimal over the quadratic extensions of the primes’ residue fields, and which in this case is a tower of double (whence cyclic) covers unramified above the genus-3 curve X0 ((τ )4 ) and thus involved in that curve’s class-field tower.

1 The Case Σ = {2, 5} The Quaternion Algebra and the Curves X (1), X ∗ (1) For this section we let A be the quaternion algebra ramified at {2, 5}. This time A is generated over Q by elements b, e satisfying b2 + 2 = e2 − 5 = be + eb = 0, (50) and the conjugate and norm of an element α = α1 + α2 b + α3 e + α4 be ∈ A are α ¯ = α1 − α2 b − α3 e − α4 be, αα ¯=α ¯ α = α21 + 2α22 − 5α23 − 10α24 . (51) The elements b and (1 + e)/2 generate a maximal order, which we use for O. 26 Noam D. Elkies Table 1 |D| D0 D1 A B proved?

The reader will note that so far we have said nothing about computing with modular forms on Shimura curves. Not only is this an intriguing question in its own right, but solving it may also allow more efficient computation of Shimura curves and the natural maps between them, as happens in the classical modular setting. 42 Noam D. Elkies In another direction, we ask: is there a prescription, analogous to (27), for towers of Shimura curves whose levels are powers of a ramified prime of the algebra?

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