Interesting Esoterica

Big fields that are not large

Article by Mazur, Barry and Rubin, Karl
  • Published in 2020
  • Added on
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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.

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Other information

key
Bigfieldsthatarenotlarge
type
article
date_added
2020-11-17
date_published
2020-06-10
journal
Proceedings of the American Mathematical Society, Series B
issn
2330-1511
volume
7
issue
14
doi
10.1090/bproc/57
pages
159-169

BibTeX entry

@article{Bigfieldsthatarenotlarge,
	key = {Bigfieldsthatarenotlarge},
	type = {article},
	title = {Big fields that are not large},
	author = {Mazur, Barry and Rubin, Karl},
	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. },
	comment = {},
	date_added = {2020-11-17},
	date_published = {2020-06-10},
	urls = {https://www.ams.org/journals/bproc/2020-07-14/S2330-1511-2020-00057-8/,https://www.ams.org/bproc/2020-07-14/S2330-1511-2020-00057-8/S2330-1511-2020-00057-8.pdf},
	collections = {attention-grabbing-titles},
	url = {https://www.ams.org/journals/bproc/2020-07-14/S2330-1511-2020-00057-8/ https://www.ams.org/bproc/2020-07-14/S2330-1511-2020-00057-8/S2330-1511-2020-00057-8.pdf},
	year = 2020,
	urldate = {2020-11-17},
	journal = {Proceedings of the American Mathematical Society, Series B},
	issn = {2330-1511},
	volume = 7,
	issue = 14,
	doi = {10.1090/bproc/57},
	pages = {159-169}
}