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

Abstract This paper studies the 4 $4$ 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 Subscript lamda Baseline left parenthesis n right parenthesis $s_\lambda (n)$ s λ ( n ) depending on n and the sign lamda $\lambda $ λ of the fundamental discriminant D , such that for any choice of s Subscript lamda Baseline left parenthesis n right parenthesis $s_\lambda (n)$ s λ ( n ) integers t 1 comma midline horizontal ellipsis comma t Subscript s Sub Subscript lamda Subscript left parenthesis n right parenthesis Baseline $t_1, \ldots , t_{s_\lambda (n)}$ t 1 , ⋯ , t s λ ( n ) , there are infinitely many D for which the narrow class group of double struck upper Q left parenthesis StartRoot upper D plus t Subscript i Baseline EndRoot right parenthesis $\mathbb {Q}(\sqrt {D + t_i})$ Q ( D + t i ) has