8-RANKS OF CLASS GROUPS OF IMAGINARY QUADRATIC NUMBER FIELDS AND THEIR DENSITIES
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For imaginary quadratic number fields F = Q(<수식>),where ε ∈ {-1, -2} and distinct primes pi ≡ 1 mod 4, we give conditions 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 = Q(<수식>), we com-pute 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 relations of pi modulo 2n, the quartic residue symbol (<수식>)4 and binary quadratic forms such as <수식>, where h+(2_(p1)) is the narrow class number of Q(<수식>). The results are also very useful for numerical computations.
Abstract
For imaginary quadratic number fields F = Q(<수식>),where ε ∈ {-1, -2} and distinct primes pi ≡ 1 mod 4, we give conditions 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 = Q(<수식>), we com-pute 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 relations of pi modulo 2n, the quartic residue symbol (<수식>)4 and binary quadratic forms such as <수식>, where h+(2_(p1)) is the narrow class number of Q(<수식>). The results are also very useful for numerical computations.
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