Density computations for real quadratic 2-class groups

Abstract

We present an algorithm to compute the 2-part of the class group of quadratic orders of large, factored discriminants and its application to density questions concerning the norm of the fundamental unit in real quadratic orders. If D -0, 1 mod 4 is a large integer that is not a square, then it is computationally not feasible to calculate the strict class group C of the quadratic order O of discriminant D. However, if the factorization of D is known, it is possible to compute the 2-part of the finite abelian group g in random polynomial time [1]. This is done by an algorithm that essentially goes back to Gauss. It is based on the fact that the factorization of D provides us with a set of generators for the 2-torsion subgroup g[2] of g and an explicit description of the quadratic characters on g. Using an algorithm to extract square roots in g that uses the reduction of ternary quadratic forms, one can now find the complete 2-Sylow subgroup of g. As our algorithm finds a non-trivial relation between the generators of g[2], it computes the norm of the fundamental unit in O without computing the unit itself, which cannot even be written down in polynomial time. We have assembled numerical data for a large number of discriminants with an implementation of the algorithm using the computer algebra system MAGMA. The results enable us to check numerically the conjectural densities in [2], which predict that a fraction 1 l--Ij odd(1-2-j) ~ .58 of all real quadratic fields without discriminantal prime divisors congruent to 3 mod 4 will have a fundamental unit of norm 1. More precisely, we use a theorem of R~dei to generate a large family of discriminants for which g has a fixed 4-rank e, for several values of e, and show that, exactly as predicted in [2], approximately 1 out of 2 e+l 1 fields in such a family has a unit of norm -1 .