One-basedness and reductions of elliptic curves over real closed fields
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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
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Additional Information
- Davide Penazzi
- Affiliation: School of Mathematics, University of Leeds, Leeds, LS2 9Jt, United Kingdom
- Address at time of publication: School of Computing, Engineering and Physical Sciences, University of Lancashire, Leighton Building, Preston PR1 2HE, United Kingdom
- Email: D.Penazzi@leeds.ac.uk, dpenazzi@uclan.ac.uk
- Received by editor(s): October 12, 2011
- Received by editor(s) in revised form: July 3, 2012, November 23, 2012, and February 4, 2013
- Published electronically: September 4, 2014
- Additional Notes: This research was supported by EPSRC grant EP/F009712/1
- © Copyright 2014 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 367 (2015), 1827-1845
- MSC (2010): Primary 03C98, 14H52; Secondary 03C45, 12J10
- DOI: https://doi.org/10.1090/S0002-9947-2014-06099-6
- MathSciNet review: 3286500