In 1990, the ninth Fermat number was factored into primes by means of a new algorithm, the “number field sieve”, which was proposed by John Pollard. The present paper is devoted to the description… Expand

0. In this paper we will continue to study the arithmetic of elliptic curves with complex multiplication by Q ( 1 / ~ ) , which we began in [5]. Chapter I reviews the basic facts on Q-curves, and… Expand

Let f(x) = Σaixi be a monic polynomial of degree n whosecoefficients are algebraically independent variables over a base field k of characteristic 0. We say that a polynomial g(x) isgenerating (for… Expand

1. Solving Pell's equation Hendrik Lenstra 2. Basic algorithms in number theory Joe Buhler and Stan Wagon 3. Elliptic curves Bjorn Poonen 4. The arithmetic of number rings Peter Stevenhagen 5. Fast… Expand

Using the LLL-algorithm for finding short vectors in lattices, it is shown how to compute a Jacobi sum for the prime field Fp in Q(e2πi/n) in time O(log3p), useful in the construction of hyperelliptic cryptosystems.Expand

The elliptic curve y2 = 4x3 28x + 25 has rank 3 over Q. Assuming the WeilTaniyama conjecture for this curve, we show that its L-series L(s) has a triple zero at s = 1 and compute lim, _I L(s)/(s 1)3… Expand

Abstract. Let h(`) denote the class number of the maximal totally real subfield Q(cos(2π/`n)) of the field of `n-th roots of unity. The goal of this paper is to show that (speculative extensions of)… Expand