SOME METHODS FOR EVALUATING THE

REGULATOR

OF A REAL QUADRATIC FUNCTION FIELD

ANDREAS STEIN AND HUGH C. WILLIAMS

Abstract. We develop methods for the rapid computation of the regulator of a real quadratic congruence function field K = k(x)(\sqrt{D}). By extending Shanks’ infrastructure ideas in real quadratic number fields to real quadratic congruence function fields we obtain a baby step – giant step method for evaluating the regulator of K in O(|D|^{1/4}) polynomial operations. We also show the existence of an effective algorithm which computes the regulator unconditionally in O(|D|^{1/5}) polynomial operations. By implementing both methods on a computer, we found that the O(|D|^{1/5})-algorithm tends to be far better than the baby step – giant step algorithm in those cases where the regulator exceeds 10^{8}.