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

TL;DR

This paper introduces a new intrinsic reduction concept for elliptic curves over real closed fields, compares it with traditional reduction, and classifies the group of points based on reduction type and geometric complexity.

Contribution

It presents a novel intrinsic reduction notion for elliptic curves over real closed fields and compares it with classical algebro-geometric reduction methods.

Findings

Classification of elliptic curve reductions over real closed fields

Comparison between intrinsic and algebro-geometric reductions

Insights into the structure of elliptic curve point groups

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 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.