The first systematic account of the theory of elliptic functions and the state of the art around the turn of the century. Preceding general class field theory and therefore incomplete. Contains a… Expand

The number field sieve is a factoring algorithm that tries to factor a ‘hard’ composite number by exploiting factorizations of smooth numbers in a well-chosen algebraic number field.Expand

We study the density of the setof real quadratic fields for which the norm of the fundamental unit equals −1 inside the set of real quadRatic fields containing elements of norm −1.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

We present an algorithm that, on input of a CM-field K, aninteger k ≥ 1, and a prime r ≡ 1 mod k, constructs a q-Weil numberπ ∈ OK corresponding to an ordinary, simple abelian variety A overthe field F of q elements that has an F-rational point of order r andembedding degree k with respect to r.Expand

En 1927, E. Artin proposait une densite conjecturale pour l'ensemble des nombres premiers p pour lesquels un entier donne g est une racine primitive modulo p. Des calculs effectues en 1957 par D. H.… Expand

The abelian extensions of an imaginary quadratic field can theoretically be generated by the values of the modular j-function, but these values are too large to be useful in practice.Expand

It follows from the work of Artin and Hooley that, under assumption of the generalized Riemann hypothesis, the density of the set of primes $q$ for which a given non-zero rational number $r$ is a… Expand

We describe the main structural results on number rings, i.e., integral domains for which the eld of fractions is a number eld. Whenever possible, the algorithmically undesirable hypothesis that the… Expand