Submission date: 1 November 2010.

Abstract:

We proceed with an analysis of 1-basedness for bounded hyperdefinable groups of the form G/G^{00} where G is the semialgebraic connected component of the K-points of an elliptic curve over a saturated real closed field K, or G a truncation of it. We then characterize one-basedness of G/G^{00} in terms of algebraic geometric reduction and the notion of internality to the value group or to the residue field of a real closed valued field.

Mathematics Subject Classification: 03C64 (primary), 03C98, 14H52, 12J10 (secondary)

Keywords and phrases: G/G00, elliptic curves, valued fields, reduction, internality, one-basedness.

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