1,406 research outputs found
Computing Hilbert Class Polynomials
We present and analyze two algorithms for computing the Hilbert class
polynomial . The first is a p-adic lifting algorithm for inert primes p
in the order of discriminant D < 0. The second is an improved Chinese remainder
algorithm which uses the class group action on CM-curves over finite fields.
Our run time analysis gives tighter bounds for the complexity of all known
algorithms for computing , and we show that all methods have comparable
run times
Numerical verification of the Cohen-Lenstra-Martinet heuristics and of Greenberg's -rationality conjecture
In this paper we make a series of numerical experiments to support
Greenberg's -rationality conjecture, we present a family of -rational
biquadratic fields and we find new examples of -rational multiquadratic
fields. In the case of multiquadratic and multicubic fields we show that the
conjecture is a consequence of the Cohen-Lenstra-Martinet heuristic and of the
conjecture of Hofmann and Zhang on the -adic regulator, and we bring new
numerical data to support the extensions of these conjectures. We compare the
known algorithmic tools and propose some improvements
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