8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES

Title & Authors
8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES
Jung, Hwan-Yup; Yue, Qin;

Abstract
For imaginary quadratic number fields F = $\small{\mathbb{Q}(\sqrt{{\varepsilon}p_1{\ldots}p_{t-1}})}$, where $\small{{\varepsilon}{\in}}${-1,-2} and distinct primes $\small{p_i{\equiv}1}$ mod 4, we give condition of 8-ranks of class groups C(F) of F equal to 1 or 2 provided that 4-ranks of C(F) are at most equal to 2. Especially for F = $\small{\mathbb{Q}(\sqrt{{\varepsilon}p_1p_2)}$, we compute densities of 8-ranks of C(F) equal to 1 or 2 in all such imaginary quadratic fields F. The results are stated in terms of congruence relation of $\small{p_i}$ modulo $\small{2^n}$, the quartic residue symbol $\small{(\frac{p_1}{p_2})4}$ and binary quadratic forms such as $\small{p_2^{h+(2_{p_1})/4}=x^2-2p_1y^2}$, where $\small{h+(2p_1)}$ is the narrow class number of $\small{\mathbb{Q}(\sqrt{2p_1})}$. The results are also very useful for numerical computations.
Keywords
class group;unramified extension;quartic residue;density;
Language
English
Cited by
1.
Congruent elliptic curves with non-trivial Shafarevich-Tate groups, Science China Mathematics, 2016, 59, 11, 2145
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