• A
  • A
  • A
  • ABC
  • ABC
  • ABC
  • А
  • А
  • А
  • А
  • А
Regular version of the site

Events

The prediction of Manin-Batyrev-Peyre on the number of rational points of algebraic varieties (Ratko Darda)

Event ended

Talk of Ratko Darda (University of Paris) at the Seminar of the Laboratory on Algebraic Transformation Groups

Let X be an algebraic variety over Q. The set of rational points of X, denoted by X(Q), is the set of solutions of the equations defining X with all coordinates lying in Q. 

It is believed that certain geometrical properties of X are making the set X(Q) “large”. We count rational points in this case, and to do so, we introduce “heights”. A height on X is a function on X(Q), which in a certain way measures “arithmetic complexity” of a rational point. It satisfies the following property: if B > 0, the number of rational points of X of the height less than B is finite, and we ask: what is the number of such rational points? A theory initiated by Manin, and later developed by Batyrev, Peyre, Tschinkel, Chambert-Loir and others, gives a prediction of the asymptotic behaviour of the number when B → ∞. The prediction is valid in many important cases.

We will state a version of the conjecture due to Peyre. We will try to see why is it true in some simple cases.