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

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.