The rational points of a definable set

DEFINABLE SETS, MOTIVES AND p-ADIC INTEGRALS

0.1. Let X be a scheme, reduced and separated, of finite type over Z. For p a prime number, one may consider the set X(Zp) of its Zp-rational points. For every n in N, there is a natural map πn : X(Zp)→ X(Z/p) assigning to a Zp-rational point its class modulo p. The image Yn,p of X(Zp) by πn is exactly the set of Z/p-rational points which can be lifted to Zp-rational points. Denote by Nn,p the ...

Subfields of Ample Fields I. Rational Maps and Definability

Pop proved that a smooth curve C over an ample field K with C(K) 6= ∅ has |K| many rational points. We strengthen this result by showing that there are |K| many rational points that do not lie in a given proper subfield, even after applying a rational map. As a consequence we gain insight into the structure of existentially definable subsets of ample fields. In particular, we prove that a perfe...

The Real Field with the Rational Points of an Elliptic Curve

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.

Groups Definable in Separably Closed Fields

We consider the groups which are infinitely definable in separably closed fields of finite degree of imperfection. We prove in particular that no new definable groups arise in this way: we show that any group definable in such a field L is definably isomorphic to the group of L-rational points of an algebraic group defined over L.

Distribution of Rational Points: A Survey