One-basedness and reductions of elliptic curves over real closed fields

Penazzi, Davide

doi:10.1090/S0002-9947-2014-06099-6

One-basedness and reductions of elliptic curves over real closed fields
HTML articles powered by AMS MathViewer

by Davide Penazzi PDF

Trans. Amer. Math. Soc. 367 (2015), 1827-1845 Request permission

Abstract:

Building on the positive solution of Pillay’s conjecture we present a notion of “intrinsic” reduction for elliptic curves over a real closed field $K$. We compare such a notion with the traditional algebro-geometric reduction and produce a classification of the group of $K$-points of an elliptic curve $E$ with three “real” roots according to the way $E$ reduces (algebro-geometrically) and the geometric complexity of the “intrinsically” reduced curve.

References

Antonio J. Engler and Alexander Prestel, Valued fields, Springer Monographs in Mathematics, Springer-Verlag, Berlin, 2005. MR 2183496
Ehud Hrushovski, Ya’acov Peterzil, and Anand Pillay, Groups, measures, and the NIP, J. Amer. Math. Soc. 21 (2008), no. 2, 563–596. MR 2373360, DOI 10.1090/S0894-0347-07-00558-9
Assaf Hasson and Alf Onshuus, Embedded o-minimal structures, Bull. Lond. Math. Soc. 42 (2010), no. 1, 64–74. MR 2586967, DOI 10.1112/blms/bdp098
Julia F. Knight, Anand Pillay, and Charles Steinhorn, Definable sets in ordered structures. II, Trans. Amer. Math. Soc. 295 (1986), no. 2, 593–605. MR 833698, DOI 10.1090/S0002-9947-1986-0833698-1
James Loveys and Ya’acov Peterzil, Linear o-minimal structures, Israel J. Math. 81 (1993), no. 1-2, 1–30. MR 1231176, DOI 10.1007/BF02761295
Dugald Macpherson, David Marker, and Charles Steinhorn, Weakly o-minimal structures and real closed fields, Trans. Amer. Math. Soc. 352 (2000), no. 12, 5435–5483. MR 1781273, DOI 10.1090/S0002-9947-00-02633-7
T. Mellor, Imaginaries in real closed valued fields, Ann. Pure Appl. Logic 139 (2006), no. 1-3, 230–279. MR 2206257, DOI 10.1016/j.apal.2005.05.014
D. Penazzi, Hyperdefinable groups and modularity, PhD Thesis, University of Leeds, 2011.
Davide Penazzi, One-basedness and groups of the form $G/G^{00}$, Arch. Math. Logic 50 (2011), no. 7-8, 743–758. MR 2847544, DOI 10.1007/s00153-011-0246-7
Ya’acov Peterzil and Sergei Starchenko, A trichotomy theorem for o-minimal structures, Proc. London Math. Soc. (3) 77 (1998), no. 3, 481–523. MR 1643405, DOI 10.1112/S0024611598000549
Anand Pillay, Type-definability, compact Lie groups, and o-minimality, J. Math. Log. 4 (2004), no. 2, 147–162. MR 2114965, DOI 10.1142/S0219061304000346
A. Pillay, Canonical bases in o-minimal and related structures, To appear in J. American Math. Soc.
Vladimir Razenj, One-dimensional groups over an $o$-minimal structure, Ann. Pure Appl. Logic 53 (1991), no. 3, 269–277. MR 1129780, DOI 10.1016/0168-0072(91)90024-G
Joseph H. Silverman, The arithmetic of elliptic curves, Graduate Texts in Mathematics, vol. 106, Springer-Verlag, New York, 1986. MR 817210, DOI 10.1007/978-1-4757-1920-8
Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
B. I. Zil′ber, The structure of models of uncountably categorical theories, Proceedings of the International Congress of Mathematicians, Vol. 1, 2 (Warsaw, 1983) PWN, Warsaw, 1984, pp. 359–368. MR 804692