Hilbert’s tenth problem in anticyclotomic towers of number fields
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- by Anwesh Ray and Tom Weston;
- Trans. Amer. Math. Soc. 377 (2024), 3577-3597
- DOI: https://doi.org/10.1090/tran/9147
- Published electronically: March 20, 2024
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Abstract:
Let $K$ be an imaginary quadratic field and $p$ be an odd prime which splits in $K$. Let $E_1$ and $E_2$ be elliptic curves over $K$ such that the $\operatorname {Gal}(\bar {K}/K)$-modules $E_1[p]$ and $E_2[p]$ are isomorphic. We show that under certain explicit additional conditions on $E_1$ and $E_2$, the anticyclotomic $\mathbb {Z}_p$-extension $K_{\operatorname {anti}}$ of $K$ is integrally diophantine over $K$. When such conditions are satisfied, we deduce new cases of Hilbert’s tenth problem. In greater detail, the conditions imply that Hilbert’s tenth problem is unsolvable for all number fields that are contained in $K_{\operatorname {anti}}$. We illustrate our results by constructing an explicit example for $p=3$ and $K=\mathbb {Q}(\sqrt {-5})$.References
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Bibliographic Information
- Anwesh Ray
- Affiliation: Chennai Mathematical Institute, H1, SIPCOT IT Park, Kelambakkam, Siruseri, Tamil Nadu 603103, India
- MR Author ID: 1376642
- ORCID: 0000-0001-6946-1559
- Email: ar2222@cornell.edu
- Tom Weston
- Affiliation: Department of Mathematics, University of Massachusetts, Amherst, Massachusetts
- MR Author ID: 691364
- Email: weston@math.umass.edu
- Received by editor(s): July 16, 2023
- Received by editor(s) in revised form: December 18, 2023
- Published electronically: March 20, 2024
- Additional Notes: When the project was started, the author was a Simons postdoctoral fellow at the Centre de recherches mathematiques in Montreal, Canada. At this time, the author’s research was supported by the CRM Simons postdoctoral fellowship.
- © Copyright 2024 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 377 (2024), 3577-3597
- MSC (2020): Primary 11R23, 11U05
- DOI: https://doi.org/10.1090/tran/9147
- MathSciNet review: 4744788