By Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian Pauli, Michael Pohst (eds.)

ISBN-10: 3540360751

ISBN-13: 9783540360759

This e-book constitutes the refereed complaints of the seventh overseas Algorithmic quantity concept Symposium, ANTS 2006, held in Berlin, Germany in July 2006.

The 37 revised complete papers offered including four invited papers have been conscientiously reviewed and chosen for inclusion within the publication. The papers are equipped in topical sections on algebraic quantity idea, analytic and common quantity concept, lattices, curves and types over fields of attribute 0, curves over finite fields and functions, and discrete logarithms.

**Read or Download Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings PDF**

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**Additional resources for Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings**

**Sample text**

Cremona, Computing the degree of the modular parametrization of a modular elliptic curve, Math. Comp. 64 no. 211 (1995), 1235–1250. 9. J. E. uk/personal/jec/ftp/data/. 10. J. E. uk/personal/jec/mwrank/. 11. J. E. Cremona and S. Siksek, Computing a Lower Bound for the Canonical Height on Elliptic Curves over Q, ANTS VII Proceedings 2006, Springer 2006. 12. W. Duke, Elliptic curves with no exceptional primes, C. R. Acad. Sci. Paris S´er. I Math. 325 (1997), no. 8, 813–818. 13. B. Edixhoven , On the Manin constants of modular elliptic curves, in Arithmetic algebraic geometry (Texel, 1989), Progr.

When D ranges over the set of positive fundamental discrimi- Conjecture 4. For every non-negative integer r, for every prime p, the density 2 ) = r is equal to of positive fundamental discriminant D such that rkp (CD p−r(r+1) η∞ (p)ηr (p)−1 ηr+1 (p)−1 . 9) of [1, p. 56 & 57], respectively. Note that for p ≥ 3, we have the equality 2 2 ) = rkp (CD ), and, by deﬁnition we have rk2 (CD ) = rk4 (CD ), the 4–rank rkp (CD of CD . Very little is known about these conjectures : Conjectures 1 and 3 are trivially true for any p and α = 0.

In order to simplify we will consider 4–ranks of the narrow class group CD . In order to present our results we need the following deﬁnition. Deﬁnition 1. Let (a|b) : Q∗ × Q∗ → {0, 1}, where (a|b) = 1 if and only if the equation x2 − ay 2 − bz 2 = 0 has a solution (x, y, z) ∈ Q3 \ {(0, 0, 0)}. The 4–rank of the narrow class group can be described by the following theorem which is already implicitly contained in [8, p. 56]. Theorem 3 2rk4 (CD ) = 1 #{a | a > 0 squarefree, a | D, (a| − b) = 1}, 2 where b ∈ Z is squarefree such that aD = bc2 for a suitable c ∈ Z.

### Algorithmic Number Theory: 7th International Symposium, ANTS-VII, Berlin, Germany, July 23-28, 2006. Proceedings by Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian Pauli, Michael Pohst (eds.)

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