The distribution of the multiplicative index of algebraic numbers over residue classes
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914747902853120 |
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| author | Moree, Pieter Perucca, Antonella Sgobba, Pietro |
| author_facet | Moree, Pieter Perucca, Antonella Sgobba, Pietro |
| contents | Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative group of the residue field at $\mathfrak p$. Under the Generalized Riemann Hypothesis we determine the natural density of primes of $K$ for which this index is in a prescribed set $S$ and has prescribed Frobenius in a finite Galois extension $F$ of $K$. We study in detail the natural density in case $S$ is an arithmetic progression, in particular its positivity. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2302_05165 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | The distribution of the multiplicative index of algebraic numbers over residue classes Moree, Pieter Perucca, Antonella Sgobba, Pietro Number Theory Primary: 11R45, Secondary: 11A07, 11R44 Let $K$ be a number field and $G$ a finitely generated torsion-free subgroup of $K^\times$. Given a prime $\mathfrak p$ of $K$ we denote by ${\rm ind}_{\mathfrak p}(G)$ the index of the subgroup $(G\bmod\mathfrak p)$ of the multiplicative group of the residue field at $\mathfrak p$. Under the Generalized Riemann Hypothesis we determine the natural density of primes of $K$ for which this index is in a prescribed set $S$ and has prescribed Frobenius in a finite Galois extension $F$ of $K$. We study in detail the natural density in case $S$ is an arithmetic progression, in particular its positivity. |
| title | The distribution of the multiplicative index of algebraic numbers over residue classes |
| topic | Number Theory Primary: 11R45, Secondary: 11A07, 11R44 |
| url | https://arxiv.org/abs/2302.05165 |