Big fields that are not large
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- by Barry Mazur and Karl Rubin HTML | PDF
- Proc. Amer. Math. Soc. Ser. B 7 (2020), 159-169
Abstract:
A subfield $K$ of $\bar {\mathbb {Q}}$ is large if every smooth curve $C$ over $K$ with a $K$-rational point has infinitely many $K$-rational points. A subfield $K$ of $\bar {\mathbb {Q}}$ is big if for every positive integer $n$, $K$ contains a number field $F$ with $[F:\mathbb {Q}]$ divisible by $n$. The question of whether all big fields are large seems to have circulated for some time, although we have been unable to find its origin. In this paper we show that there are big fields that are not large.References
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Additional Information
- Barry Mazur
- Affiliation: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
- MR Author ID: 121915
- ORCID: 0000-0002-1748-2953
- Email: mazur@g.harvard.edu
- Karl Rubin
- Affiliation: Department of Mathematics, UC Irvine, Irvine, California 92697
- MR Author ID: 151435
- Email: krubin@uci.edu
- Received by editor(s): May 1, 2020
- Received by editor(s) in revised form: May 18, 2020, June 11, 2020, and July 29, 2020
- Published electronically: November 13, 2020
- Communicated by: Romyar T. Sharifi
- © Copyright 2020 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- Journal: Proc. Amer. Math. Soc. Ser. B 7 (2020), 159-169
- MSC (2020): Primary 11R04, 11U05, 14G05
- DOI: https://doi.org/10.1090/bproc/57
- MathSciNet review: 4173816