SIMULTANEOUS BOUNDING OF $4$-RANKS IN NARROW CLASS GROUPS OF QUADRATIC FIELDS

Abstract This paper studies the $4$ -ranks of narrow class groups in certain families of quadratic fields. We prove that for any positive integer n , there exists an integer $s_\lambda (n)$ depending on n and the sign $\lambda $ of the fundamental discriminant D , such that for any choice of $s_\lambda (n)$ integers $t_1, \ldots , t_{s_\lambda (n)}$ , there are infinitely many D for which the narrow class group of $\mathbb {Q}(\sqrt {D + t_i})$ has $4$ -rank bounded by n for all i . This result extends previous work on $3$ -ranks to the case of $4$ -ranks.