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

This e-book constitutes the refereed lawsuits of the seventh foreign Algorithmic quantity thought Symposium, ANTS 2006, held in Berlin, Germany in July 2006.

The 37 revised complete papers awarded including four invited papers have been conscientiously reviewed and chosen for inclusion within the booklet. The papers are prepared in topical sections on algebraic quantity concept, analytic and effortless quantity thought, 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

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1 − x)(1 − x2 )(1 − x3 ) · · · 48 ´ Fouvry and J. Kl¨ E. uners This formula has been extended into Lemma 3. [6, Thm 351] For any |x| < 1, we have 1 (1 − x)(1 − x2 )(1 − x3 ) · · · x x4 x9 = 1+ + + + ··· 2 2 2 2 2 (1 − x) (1 − x) (1 − x ) (1 − x) (1 − x2 )2 (1 − x3 )2 In other words, we have the formula η∞ (1/x)−1 = 2 ∞ xk 2 (1/x) . k=0 ηk By choosing x = 1/p, we proved that the sequence (xr ) satisfies the first equation of (S − ). We must continue this checking to the other equations of (S − ).

Schoof’s algorithm is no exception to this, as the two-dimensional IFl -vector space EIFp (IFp )[l] is the dual of H1 (EIFp ,et , IFl ), the first ´etale cohomology group with IFl -coefficients of EIFp . A natural question that arises is then the following: are there other interesting cases where cohomology groups can be used to construct polynomial time algorithms for counting rational points of varieties over finite fields? The answer to this question is a clear yes. Since about 6 years ago, methods based on p-adic cohomology theories have been developed mainly by Satoh, Kedlaya, Lauder and Wan.

4 (2002), 487–502. nl/~edix Abstract. We give an overview of the recent result by Jean-Marc Couveignes, Bas Edixhoven and Robin de Jong that says that for l prime the mod l Galois representation associated to the discriminant modular form Δ can be computed in time polynomial in l. As a consequence, Ramanujan’s τ (p) for prime numbers p can be computed in time polynomial in log p. The mod l Galois representation occurs in the Jacobian of the modular curve X1 (l), whose genus grows quadratically with l.

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