8-rank of the class group and isotropy index

Abstract

Suppose $F=\mathbb{Q}(\sqrt{-p_1\cdots p_t})$ is an imaginary quadratic number field with distinct primes $p_1,\dots ,p_t$, where $p_i\equiv 1(\mathrm{mod}4)$ ($i=1,\dots ,t-1$) and $p_t\equiv 3(\mathrm{mod}4)$. We express the possible values of the 8-rank $r_8$ of the class group of $F$ in terms of a quadratic form $Q$ over ${\mathbb{F}}_2$ which is defined by quartic symbols. In particular, we show that $r_8$ is bounded by the isotropy index of $Q$.

15 pages. Typos corrected

Cited by in corpus (2)

• A necessary condition for a congruent number of the form $8k+3$
• On Hilbert $2$-class fields and $2$-towers of imaginary quadratic number fields