The local-global principle for integral points on stacky curves
Authors:
Manjul Bhargava and Bjorn Poonen
Journal:
J. Algebraic Geom. 31 (2022), 773-782
DOI:
https://doi.org/10.1090/jag/796
Published electronically:
May 31, 2022
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Abstract |
References |
Additional Information
Abstract: We construct a stacky curve of genus $1/2$ (i.e., Euler characteristic $1$) over $\mathbb {Z}$ that has an $\mathbb {R}$-point and a $\mathbb {Z}_p$-point for every prime $p$ but no $\mathbb {Z}$-point. This is best possible: we also prove that any stacky curve of genus less than $1/2$ over a ring of $S$-integers of a global field satisfies the local-global principle for integral points.
References
- Manjul Bhargava, Most hyperelliptic curves over $\mathbb {Q}$ have no rational points, arXiv:1308.0395v1, 2013.
- Atticus Christensen, A Topology on Points on Stacks, ProQuest LLC, Ann Arbor, MI, 2020. Thesis (Ph.D.)–Massachusetts Institute of Technology. MR 4272330
- Brian Conrad, The Keel–Mori theorem via stacks (2005-11-27, 2005, unpublished manuscript, available at http://math.stanford.edu/~conrad/papers/coarsespace.pdf).
- Brian Conrad and Michael Temkin, Non-Archimedean analytification of algebraic spaces, J. Algebraic Geom. 18 (2009), no. 4, 731–788. MR 2524597, DOI 10.1090/S1056-3911-09-00497-4
- Henri Darmon and Andrew Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, Bull. London Math. Soc. 27 (1995), no. 6, 513–543. MR 1348707, DOI 10.1112/blms/27.6.513
- Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041, DOI 10.2307/2951828
- Andrew Kobin, Artin-Schreier root stacks, J. Algebra 586 (2021), 1014–1052. MR 4296227, DOI 10.1016/j.jalgebra.2021.07.023
- Carl-Erik Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, University of Uppsala, Uppsala, 1940 (German). Thesis. MR 0022563
- Martin Olsson, Algebraic spaces and stacks, American Mathematical Society Colloquium Publications, vol. 62, American Mathematical Society, Providence, RI, 2016. MR 3495343, DOI 10.1090/coll/062
- B. Poonen, Curves over every global field violating the local-global principle, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. Issledovaniya po Teorii Chisel. 10, 141–147, 243–244 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 6, 782–785. MR 2753654, DOI 10.1007/s10958-010-0182-9
- Hans Reichardt, Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen, J. Reine Angew. Math. 184 (1942), 12–18 (German). MR 9381, DOI 10.1515/crll.1942.184.12
- The Stacks Project authors, Stacks project, 2020-05-18. Available at http://stacks.math.columbia.edu\phantom{i}.
- John Voight and David Zureick-Brown, The canonical ring of a stacky curve, Mem. Amer. Math. Soc. 277 (2022), no. 1362, v+144. MR 4403928, DOI 10.1090/memo/1362
References
- Manjul Bhargava, Most hyperelliptic curves over $\mathbb {Q}$ have no rational points, arXiv:1308.0395v1, 2013.
- Atticus Christensen, A topology on points on stacks, ProQuest LLC, Ann Arbor, MI, 2020. Thesis (Ph.D.)—Massachusetts Institute of Technology. MR 4272330
- Brian Conrad, The Keel–Mori theorem via stacks (2005-11-27, 2005, unpublished manuscript, available at http://math.stanford.edu/~conrad/papers/coarsespace.pdf).
- Brian Conrad and Michael Temkin, Non-Archimedean analytification of algebraic spaces, J. Algebraic Geom. 18 (2009), no. 4, 731–788. MR 2524597, DOI 10.1090/S1056-3911-09-00497-4
- Henri Darmon and Andrew Granville, On the equations $z^m=F(x,y)$ and $Ax^p+By^q=Cz^r$, Bull. London Math. Soc. 27 (1995), no. 6, 513–543. MR 1348707, DOI 10.1112/blms/27.6.513
- Seán Keel and Shigefumi Mori, Quotients by groupoids, Ann. of Math. (2) 145 (1997), no. 1, 193–213. MR 1432041, DOI 10.2307/2951828
- Andrew Kobin, Artin–Schreier root stacks, J. Algebra 586 (2021), 1014–1052. MR 4296227, DOI 10.1016/j.jalgebra.2021.07.023
- Carl-Erik Lind, Untersuchungen über die rationalen Punkte der ebenen kubischen Kurven vom Geschlecht Eins, Thesis, University of Uppsala, 1940 (German). MR 0022563
- Martin Olsson, Algebraic spaces and stacks, American Mathematical Society Colloquium Publications, vol. 62, American Mathematical Society, Providence, RI, 2016. MR 3495343, DOI 10.1090/coll/062
- B. Poonen, Curves over every global field violating the local-global principle, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI) 377 (2010), no. Issledovaniya po Teorii Chisel. 10, 141–147, 243–244 (English, with English and Russian summaries); English transl., J. Math. Sci. (N.Y.) 171 (2010), no. 6, 782–785. MR 2753654, DOI 10.1007/s10958-010-0182-9
- Hans Reichardt, Einige im Kleinen überall lösbare, im Grossen unlösbare diophantische Gleichungen, J. Reine Angew. Math. 184 (1942), 12–18 (German). MR 9381, DOI 10.1515/crll.1942.184.12
- The Stacks Project authors, Stacks project, 2020-05-18. Available at http://stacks.math.columbia.edu\phantom{i}.
- John Voight and David Zureick-Brown, The canonical ring of a stacky curve, Mem. Amer. Math. Soc. 277 (2022), no. 1362. MR 4403928, DOI 10.1090/memo/1362
Additional Information
Manjul Bhargava
Affiliation:
Department of Mathematics, Princeton University, Princeton, New Jersey 08544
MR Author ID:
623882
Email:
bhargava@math.princeton.edu
Bjorn Poonen
Affiliation:
Department of Mathematics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139-4307
MR Author ID:
250625
ORCID:
0000-0002-8593-2792
Email:
poonen@math.mit.edu
Received by editor(s):
June 22, 2020
Published electronically:
May 31, 2022
Additional Notes:
The first author was supported in part by National Science Foundation grant DMS-1001828 and Simons Foundation grant #256108. The second author was supported in part by National Science Foundation grants DMS-1601946 and DMS-2101040 and Simons Foundation grants #402472 and #550033.
Article copyright:
© Copyright 2022
University Press, Inc.
