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arXiv · 1312.1237

8-rank of the class group and isotropy index

Qing Lu
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Abstract

Suppose $F=\mathbb{Q}(\sqrt{-p_1\dotsm p_t})$ is an imaginary quadratic number field with distinct primes $p_1,\dots,p_{t}$, where $p_i\equiv 1\pmod{4}$ ($i=1,\dots,t-1$) and $p_t\equiv 3\pmod{4}$. 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 $Q$.

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Qing Lu. 2013-12-04. 8-rank of the class group and isotropy index. https://doi.org/10.1007/s11425-014-4898-8

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