We consider the expansion of the real field by a subgroup of a one-dimensional definable group satisfying a certain diophantine condition. The main example is the group of rational points of an elliptic curve over a number field. We prove a completeness result, followed by a quantifier elimination result. Moreover we show that open sets definable in that structure are semialgebraic.

Structures having ominimal open core . To appear in Trans . Amer . Math . Soc . [ 4 ] Lou van den Dries and Ayhan Günaydın . The fields of real and complex numbers with a small multiplicative group

Alfred Dolich, Chris Miller, Charles Steinhorn

Proc . London Math . Soc .

2006

The fields of real and complex numbers with a small multiplicative group

Lou van den Dries, Ayhan Günaydın

Proc. London Math. Soc

2006

2 Excerpts

Complex roots of unity on the real plane. Available at the webpage www.maths.ox.ac.uk/∼zilber

Boris Zilber

Fields Institute,

2003

1 Excerpt

The arithmetic of elliptic curves, volume 106 of Graduate Texts in Mathematics

Joseph H. Silverman

Corrected reprint of the 1986 original

1992

Number theory. III, volume 60 of Encyclopaedia of Mathematical Sciences

Serge Lang

1991

1 Excerpt

The general case of S. Lang’s conjecture

Gerd Faltings

In Barsotti Symposium in Algebraic Geometry…

1991

1 Excerpt

On groups and fields definable in o-minimal structures