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October 13, 2025Open Access

On imaginary quadratic fields with non-cyclic class groups

Key Points

Asymptotic lower bounds for square-free integers indicate growth rates linked to their class group structure.
In the first case, results show N_H(X) grows at a rate of roughly X^(1/2 + 3/(2g_1 + 2) - ε) for fixed subgroup structures.
The second case, subject to conjectures, establishes growth at roughly N_H(X) ≈ X^(1/(g - 1) - ε) for certain abelian configurations.
These findings adapt and build upon existing results from Soundararajan, Byeon, and others, advancing our understanding of class groups.

Abstract

For a fixed abelian group H, let NH (X) be the number of square-free positive integers d X such that H is a subgroup of CL (Q (-d) ). We obtain asymptotic lower bounds for NH (X) as X in two cases: H=Z/g₁Z (Z/2Z) ˡ for l 2 and 2 g₁ 3, H= (Z/gZ) ² for 2 g 5. More precisely, for any >0, we showed NH (X) X^1{2+32g₁+2-} when H=Z/g₁Z (Z/2Z) ˡ for l 2 and 2 g₁ 3. For the second case, under a well known conjecture for square-free density of integral multivariate polynomials, for any >0, we showed NH (X) X^1{g-1-} when H= (Z/gZ) ² for g 5. The first case is an adaptation of Soundararajan's results for H=Z/gZ, and the second conditionally improves the bound X^1{g-} due to Byeon and the bound X^1{g}/ (X) ^2 due to Kulkarni and Levin.

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