Some new infinite families of non-$p$-rational real quadratic fields
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916296272117760 |
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| author | McConnell, Gary |
| author_facet | McConnell, Gary |
| contents | Fix a finite collection of primes $\{ p_j \}$, not containing $2$ or $3$. Using some observations which arose from attempts to solve the SIC-POVMs problem in quantum information, we give a simple methodology for constructing an infinite family of simultaneously non-$p_j$-rational real quadratic fields, unramified above any of the $p_j$. Alternatively these may be described as infinite sequences of instances of $\mathbb{Q}(\sqrt{D})$, for varying $D$, where every $p_j$ is a $k$-Wall-Sun-Sun prime, or equivalently a generalised Fibonacci-Wieferich prime.
One feature of these techniques is that they may be used to yield fields $K=\mathbb{Q}(\sqrt{D})$ for which a $p$-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-$p$-extension of $K$ unramified outside primes above $p$, is of size $p^a$ for $a\geq1$ arbitrarily large. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14632 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Some new infinite families of non-$p$-rational real quadratic fields McConnell, Gary Number Theory Quantum Physics 11B39 (Primary), 11R11, 11R37, 81P45 (Secondary) Fix a finite collection of primes $\{ p_j \}$, not containing $2$ or $3$. Using some observations which arose from attempts to solve the SIC-POVMs problem in quantum information, we give a simple methodology for constructing an infinite family of simultaneously non-$p_j$-rational real quadratic fields, unramified above any of the $p_j$. Alternatively these may be described as infinite sequences of instances of $\mathbb{Q}(\sqrt{D})$, for varying $D$, where every $p_j$ is a $k$-Wall-Sun-Sun prime, or equivalently a generalised Fibonacci-Wieferich prime. One feature of these techniques is that they may be used to yield fields $K=\mathbb{Q}(\sqrt{D})$ for which a $p$-power cyclic component of the torsion group of the Galois groups of the maximal abelian pro-$p$-extension of $K$ unramified outside primes above $p$, is of size $p^a$ for $a\geq1$ arbitrarily large. |
| title | Some new infinite families of non-$p$-rational real quadratic fields |
| topic | Number Theory Quantum Physics 11B39 (Primary), 11R11, 11R37, 81P45 (Secondary) |
| url | https://arxiv.org/abs/2406.14632 |