Download Algorithmic Number Theory: 7th International Symposium, by Nigel Boston, Harris Nover (auth.), Florian Hess, Sebastian PDF

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

This ebook constitutes the refereed court cases of the seventh overseas Algorithmic quantity concept Symposium, ANTS 2006, held in Berlin, Germany in July 2006.

The 37 revised complete papers provided including four invited papers have been rigorously reviewed and chosen for inclusion within the ebook. The papers are equipped in topical sections on algebraic quantity conception, analytic and trouble-free quantity conception, lattices, curves and forms over fields of attribute 0, curves over finite fields and functions, and discrete logarithms.

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

Example text

Numbers of isogeny and isomorphism classes of curves range of N # isogeny classes # isomorphism classes 0-9999 10000-19999 20000-29999 30000-39999 40000-49999 50000-59999 60000-69999 70000-79999 80000-89999 90000-99999 100000-109999 110000-119999 120000-129999 38042 43175 44141 44324 44519 44301 44361 44449 44861 43651 44274 44071 44655 64687 67848 66995 66561 66275 65393 65209 64687 64864 63287 63410 63277 63467 0-129999 568824 845960 23 24 J. Cremona Table 3. 01 between curves in the class. For each possible value of D, there is a uniquely determined shape of the graph of curves and isogenies of prime degree between them.

Theorem 2. There exists a probabilistic algorithm that computes the mod l Galois representation associated to Δ in time polynomial in l. More precisely, on input a prime number l it gives: 1. , the ai,j,k in Q such that ei ej = k ai,j,k ek are given); 2. a list of the elements σ of Gal(Kl /Q), where each σ is given as its matrix with respect to e; 3. an injective morphism ρl from Gal(Kl /Q) into GL2 (IFl ), such that Kl is unramified outside {l}, and for all prime numbers p different from l we have trace(ρl Frobp ) = τ (p) mod l and det(ρl Frobp ) = p11 mod l.

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. Let us further simplify and concentrate on the case of negative discriminants which are congruent to 1 modulo 4.

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