From: voloch@fireant.ma.utexas.edu (Felipe Voloch)
Newsgroups: sci.math.research
Subject: Re: Questions concerning varieties
Date: 29 Jul 1998 15:40:43 GMT

Ray Easton (ray.easton@mci.com) wrote:
: Let V be a smooth projective variety over the field Q of rational
: numbers.
: 
: (1)    Is there a V' defined over the algebraic closure of Q that is
: birationally-equivalent over this field to V and that is defined over
: Q and has a Q-point?
: 
: (2)    Let V_p denote V considered as a variety over some p-adic
: field.  Suppose there is a point on V_p.  Is there a V', with V'_p
: birationally-equivalent over the p-adic field to V_p, that is defined
: over Q and has a Q-point?
: 
: (3)    Suppose V_p has a point for all p.  Is there a V' defined over
: Q that has a Q-point, with V'_p birationally-equivalent to V_p over
: all the p-adic fields?
: 
: (4)    Assuming the answers to any of the above are affirmative, if V
: is given by a single form, can we find a V' that is as well?
: 

I am afraid the answers to your questions are all "no".
Let V be a curve of genus 3 over Q with no Q-points and no automorphisms
over the algebraic closure of Q. A random plane quartic will be like that.
Because V has no automorphisms, any V' as in (1) is automatically Q-isomorphic
to V, so has no points either. This answers (1). The same V will have Q_p
points for p large, so will give couterexamples to (2) as well. To get a
counterexample for (3), you need to choose V having Q_p points for all p
and that shouldn't be too hard. The probability that a random one will work
is positive. Now (4) answers itself.

Felipe