Conditional algorithmic Mordell
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866910571982487552 |
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| author | Alpöge, Levent Lawrence, Brian |
| author_facet | Alpöge, Levent Lawrence, Brian |
| contents | We specify a Turing machine $T_{\text{Mordell}}$ with the following properties.
1. On input $(K,C/K)$, with $K/\mathbb{Q}$ a number field and $C/K$ a smooth projective hyperbolic curve, if $T_{\text{Mordell}}$ terminates, then it outputs $C(K)$.
2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\text{Mordell}}$ always terminates.
Similarly we specify a Turing machine $T_{\text{Shafarevich}}$ with the following properties.
1. On input $(g, K,S, d)$, with $g, d \in \mathbb{Z}^+$, $K/\mathbb{Q}$ a number field, and $S$ a finite set of places of $K$, if $T_{\text{Shafarevich}}$ terminates, then it outputs the finitely many polarized $g$-dimensional abelian varieties $A/K$, with polarization of degree $d$, having good reduction outside $S$.
2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\text{Shafarevich}}$ always terminates. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2408_11653 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Conditional algorithmic Mordell Alpöge, Levent Lawrence, Brian Number Theory Algebraic Geometry We specify a Turing machine $T_{\text{Mordell}}$ with the following properties. 1. On input $(K,C/K)$, with $K/\mathbb{Q}$ a number field and $C/K$ a smooth projective hyperbolic curve, if $T_{\text{Mordell}}$ terminates, then it outputs $C(K)$. 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\text{Mordell}}$ always terminates. Similarly we specify a Turing machine $T_{\text{Shafarevich}}$ with the following properties. 1. On input $(g, K,S, d)$, with $g, d \in \mathbb{Z}^+$, $K/\mathbb{Q}$ a number field, and $S$ a finite set of places of $K$, if $T_{\text{Shafarevich}}$ terminates, then it outputs the finitely many polarized $g$-dimensional abelian varieties $A/K$, with polarization of degree $d$, having good reduction outside $S$. 2. The Hodge, Tate, and Fontaine-Mazur conjectures imply that $T_{\text{Shafarevich}}$ always terminates. |
| title | Conditional algorithmic Mordell |
| topic | Number Theory Algebraic Geometry |
| url | https://arxiv.org/abs/2408.11653 |