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Computing the cardinality of CM elliptic curves using torsion points

François Morain (2007)

Journal de Théorie des Nombres de Bordeaux

Let ℰ / ℚ ¯ be an elliptic curve having complex multiplication by a given quadratic order of an imaginary quadratic field 𝕂 . The field of definition of ℰ is the ring class field Ω of the order. If the prime p splits completely in Ω , then we can reduce ℰ modulo one the factors of p and get a curve E defined over 𝔽 p . The trace of the Frobenius of E is known up to sign and we need a fast way to find this sign, in the context of the Elliptic Curve Primality Proving algorithm (ECPP). For this purpose, we propose...

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 .

Concentration function of additive functions on shifted twin primes

Simon Wong (1998)

Acta Arithmetica

0. Introduction. The content of this paper is part of the author's Ph.D. thesis. The two new theorems in this paper provide upper bounds on the concentration function of additive functions evaluated on shifted γ-twin prime, where γ is any positive even integers. Both results are generalizations of theorems due to I. Z. Ruzsa, N. M. Timofeev, and P. D. T. A. Elliott.

Concordant sequences and integral-valued entire functions

Jonathan Pila, Fernando Rodriguez Villegas (1999)

Acta Arithmetica

A classic theorem of Pólya shows that the function 2 z is the “smallest” integral-valued entire transcendental function. A variant due to Gel’fond applies to entire functions taking integral values on a geometric progression of integers, and Bézivin has given a generalization of both results. We give a sharp formulation of Bézivin’s result together with a further generalization.

Condition nécessaire et suffisante pour que certain groupe de Galois soit métacyclique

Abdelmalek Azizi, Mohammed Taous (2009)

Annales mathématiques Blaise Pascal

Soient d est un entier sans facteurs carrés, K = Q ( d , i ) , i = - 1 , K 2 ( 1 ) le 2 -corps de classes de Hilbert de K , K 2 ( 2 ) le 2 -corps de classes de Hilbert de K 2 ( 1 ) et G = Gal ( K 2 ( 2 ) / K ) le groupe de Galois de K 2 ( 2 ) / K . Notre but est de montrer qu’il existe une forme de d tel que le 2 -groupe G est non métacyclique et de donner une condition nécessaire et suffisante pour que le groupe G soit métacyclique dans le cas où d = 2 p avec p un nombre premier tel que p ≡ 1 ( mod 4 ) .

Currently displaying 421 – 440 of 902