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Calcul du nombre de classes d'un corps quadratique imaginaire ou réel, d'après Shanks, Williams, McCurley, A. K. Lenstra et Schnorr

Henri Cohen (1989)

Journal de théorie des nombres de Bordeaux

Dans cette note nous décrivons différentes méthodes utilisées en pratique pour calculer le nombre de classes d'un corps quadratique imaginaire ou réel ainsi que pour calculer le régulateur d'un corps quadratique réel. En particulier nous décrivons l'infrastructure de Shanks ainsi que la méthode sous-exponentielle de McCurley.

Computations of Galois representations associated to modular forms of level one

Peng Tian (2014)

Acta Arithmetica

We propose an improved algorithm for computing mod ℓ Galois representations associated to a cusp form f of level one. The proposed method allows us to explicitly compute the case with ℓ = 29 and f of weight k = 16, and the cases with ℓ = 31 and f of weight k = 12,20,22. All the results are rigorously proved to be correct. As an example, we will compute the values modulo 31 of Ramanujan's tau function at some huge primes up to a sign. Also we will give an improved uper bound on...

Computing the number of certain Galois representations mod p

Tommaso Giorgio Centeleghe (2011)

Journal de Théorie des Nombres de Bordeaux

Using the link between Galois representations and modular forms established by Serre’s Conjecture, we compute, for every prime p 2593 , a lower bound for the number of isomorphism classes of Galois representation of Q on a two–dimensional vector space over F ¯ p which are irreducible, odd, and unramified outside p .

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