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| Format: | Preprint |
| Veröffentlicht: |
2024
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| Schlagworte: | |
| Online-Zugang: | https://arxiv.org/abs/2408.11916 |
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| _version_ | 1866913476532764672 |
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| author | Babu, C. G. K. Bera, R. Sivaraman, J. Sury, B. |
| author_facet | Babu, C. G. K. Bera, R. Sivaraman, J. Sury, B. |
| contents | Gerth generalised Cohen-Lenstra heuristics to the prime $p=2$. He conjectured that for any positive integer $m$, the limit
$$
\lim_{x \to \infty} \frac{\sum_{0 < D \le X, \atop{ \text{squarefree} }} |{\rm Cl}^2_{\Q(\sqrt{D})}/{\rm Cl}^4_{\Q(\sqrt{D})}|^m}{\sum_{0 < D \le X, \atop{ \text{squarefree} }} 1}
$$
exists and proposed a value for the limit. Gerth's conjecture was proved by Fouvry and Kluners in 2007. In this paper, we generalize their result by obtaining lower bounds for the average value of $|{\rm Cl}^2_Ł/{\rm Cl}^4_Ł|^m$, where $Ł$ varies over an infinite family of quadratic extensions of certain Galois number fields. As a special case of our theorem, we obtain lower bounds for the average value when the base field is any Galois number field with class number $1$ in which $2\Z$ splits. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11916 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Gerth's heuristics for a family of quadratic extensions of certain Galois number fields Babu, C. G. K. Bera, R. Sivaraman, J. Sury, B. Number Theory Gerth generalised Cohen-Lenstra heuristics to the prime $p=2$. He conjectured that for any positive integer $m$, the limit $$ \lim_{x \to \infty} \frac{\sum_{0 < D \le X, \atop{ \text{squarefree} }} |{\rm Cl}^2_{\Q(\sqrt{D})}/{\rm Cl}^4_{\Q(\sqrt{D})}|^m}{\sum_{0 < D \le X, \atop{ \text{squarefree} }} 1} $$ exists and proposed a value for the limit. Gerth's conjecture was proved by Fouvry and Kluners in 2007. In this paper, we generalize their result by obtaining lower bounds for the average value of $|{\rm Cl}^2_Ł/{\rm Cl}^4_Ł|^m$, where $Ł$ varies over an infinite family of quadratic extensions of certain Galois number fields. As a special case of our theorem, we obtain lower bounds for the average value when the base field is any Galois number field with class number $1$ in which $2\Z$ splits. |
| title | Gerth's heuristics for a family of quadratic extensions of certain Galois number fields |
| topic | Number Theory |
| url | https://arxiv.org/abs/2408.11916 |