Algorithmic Number Theory

Counting rational points on curves and abelian varieties over finite fields.- Computing cubic fields in quasi-linear time.- Fast ideal arithmetic via lazy localization.- A comparative study of algorithms for computing continued fractions of algebraic numbers.- Computing ray class groups, conductors and discriminants.- Computing l-isogenies using the p-torsion.- On computing Hilbert class fields of prime degree.- On the reduction of composed relations from the number field sieve.- Checking the $$\mathfrak{p}$$ -adic stark conjecture when $$\mathfrak{p}$$ is archimedean.- A multiple polynomial general number field sieve.- Construction of high-rank elliptic curves over ? and ?(t) with non-trivial 2-torsion.- The height on an abelian variety.- On lattices over number fields.- Minimum discriminants of primitive sextic fields.- A new algorithm and refined bounds for extended gcd computation.- Application of thue equations to computing power integral bases in algebraic number fields.- Computing S-integral points on elliptic curves.- Probabilistic computation of the Smith normal form of a sparse integer matrix.- Ray class field constructions of curves over finite fields with many rational points.- Computing isogenies in $$\mathbb{F}_{2^n } $$ .- A computational technique for determining relative class numbers of CM-fields.- Old and new deterministic factoring algorithms.- Efficient algorithms for computing the Jacobi symbol.- The number field database on the World Wide web server http://hasse.mathematik.tu-muenchen.de/.- An algorithm of subexponential type computing the class group of quadratic orders over principal ideal domains.- Computational aspects of Kummer theory.- On integral basis reduction in global function fields.- Computational aspects of curves of genus at least 2.- The complexity of approximate optima for greatest common divisor computations.- Compact representation in real quadratic congruence function fields.- Discrete logarithms: The effectiveness of the index calculus method.- How difficult is it to solve a thue equation?.- Elliptic congruence function fields.- Algebraic geometry lattices and codes.- Computing discrete logarithms with the general number field sieve.