Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2024
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| _version_ | 1866913375313723392 |
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| author | Müller, Katharina Ray, Anwesh |
| author_facet | Müller, Katharina Ray, Anwesh |
| contents | Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Kriz--Li and Bhargava et al., we demonstrate that for primes $ p = 3, 11, 13, 31, 37 $, a positive proportion of imaginary quadratic fields meet our criteria. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_01443 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields Müller, Katharina Ray, Anwesh Number Theory 11R23, 11G05, 11U05 Via a novel application of Iwasawa theory, we study Hilbert's tenth problem for number fields occurring in $\mathbb{Z}_p$-towers of imaginary quadratic fields $K$. For a odd prime $p$, the lines $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ are identified with $\mathbb{Z}_p$-extensions $ K_{a,b}/K $. Under certain conditions on $ K $ that involve explicit elliptic curves, we identify a line $(a_0,b_0) \in \mathbb{P}^1(\mathbb{Z}/p\mathbb{Z})$ such that for all $(a,b) \in \mathbb{P}^1(\mathbb{Z}_p)$ with $(a, b)\not\equiv (a_0, b_0)\pmod{p}$, Hilbert's tenth problem has a negative answer in all finite layers of $ K_{a,b} $. Using results of Kriz--Li and Bhargava et al., we demonstrate that for primes $ p = 3, 11, 13, 31, 37 $, a positive proportion of imaginary quadratic fields meet our criteria. |
| title | Hilbert's tenth problem for families of $ \mathbb{Z}_p $-extensions of imaginary quadratic fields |
| topic | Number Theory 11R23, 11G05, 11U05 |
| url | https://arxiv.org/abs/2406.01443 |