2,626 research outputs found
The Discrete Logarithm Problem
No full textFor large prime numbers p, computing discrete logarithms of elements of the multiplicative group (Z/pZ) ∗ is at present a very difficult problem. The security of certain cryptosystems is based on the difficulty of this computation. In this expository paper we discuss several generalizations of the discrete logarithm problem and we describe various algorithms to compute discrete logarithms
Catalan's conjecture
No full textThis is a selfcontained presentation of Mihailescu's proof of Catalan's conjectur
Abelian varieties over Q with bad reduction in one prime only
No full textWe show that for the primes l = 2, 3, 5, 7 or 13, there do not exist any non-zero abelian varieties over Q that have good reduction at every prime different from 1 and are semi-stable at l. We show that any semi-stable abelian variety over Q with good reduction outside l = 11 is isogenous to a power of the Jacobian variety of the modular curve X-0(11). In addition, we show that for l = 2,3 and 5, there do not exist any non-zero abelian varieties over Q with good reduction outside l that acquire semi-stable reduction at l over a tamely ramified extension
Semistable abelian varieties with good reduction outside 15
No full textWe show that there are no non-zero semi-stable abelian varieties over with good reduction outside 3 and we show that the only semi-stable abelian varieties over Q with good reduction outside 15 are, up to isogeny over Q, powers of the Jacobian of the modular curve X (0)(15)
Visibility of ideal classes
No full textAbstractCremona, Mazur, and others have studied what they call visibility of elements of Shafarevich–Tate groups of elliptic curves. The analogue for an abelian number field K is capitulation of ideal classes of K in the minimal cyclotomic field containing K. We develop a new method to study capitulation and use it and classical methods to compute data with the hope of gaining insight into the elliptic curve case. For example, the numerical data for number fields suggests that visibility of non-trivial Shafarevich–Tate elements might be much more common for elliptic curves of positive rank than for curves of rank 0
A refined counter-example to the support conjecture for abelian varieties
No full textIf A/K is an abelian variety over a number field and P and Q are rational points, the original support conjecture asserted that if the order of Q (mod p) divides the order of P (mod P) for almost all primes p of K, then Q is obtained from P by applying an endomorphism of A. This is now known to be untrue. In this note we prove that it is not even true modulo the torsion of A. (c) 2005 Elsevier Inc. All rights reserved
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