Distribution of units of real quadratic number fields

Abstract

Let $k$ be a real quadratic field and $\mathfrak{o}_{k}$, $E$ the ring of integers and the group of units in $k$. Denoting by $E(\mathfrak{p})$ the subgroup represented by $E$ of $(\mathfrak{o}_{k}/\mathfrak{p})^{\times}$ for a prime ideal $\mathfrak{p}$, we show that prime ideals $\mathfrak{p}$ for which the order of $E(\mathfrak{p})$ is theoretically maximal have a positive density under the Generalized Riemann Hypothesis.