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

We proceed with an analysis of 1-basedness for bounded hyperdefinable groups of the form G/G where G = E(K) 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 E(K); we follow the method developed in [1]. We then relate the map G→ G/G with the algebraic geometric notion of reduction, and we characterize 1-basedness of G/G 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.