On the p-ranks of the ideal class groups of imaginary quadratic fields

Autor: Jaitra Chattopadhyay, Anupam Saikia
Rok vydání: 2022
Předmět:
Zdroj: The Ramanujan Journal.
ISSN: 1572-9303
1382-4090
DOI: 10.1007/s11139-022-00667-0
Popis: For a prime number $p \geq 5$, we explicitly construct a family of imaginary quadratic fields $K$ with ideal class groups $Cl_{K}$ having $p$-rank ${\rm{rk}_{p}(Cl_{K})}$ at least $2$. We also quantitatively prove, under the assumption of the $abc$-conjecture, that for sufficiently large positive real numbers $X$ and any real number $\varepsilon$ with $0 < \varepsilon < \frac{1}{p - 1}$, the number of imaginary quadratic fields $K$ with the absolute value of the discriminant $d_{K}$ $\leq X$ and ${\rm{rk}_{p}(Cl_{K})} \geq 2$ is $\gg X^{\frac{1}{p - 1} - \varepsilon}$. This improves the previously known lower bound of $X^{\frac{1}{p} - \varepsilon}$ due to Byeon and the recent bound $X^{\frac{1}{p}}/(\log X)^{2}$ due to Kulkarni and Levin.
10 pages. Comments are welcome
Databáze: OpenAIRE
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