The prediction of Manin-Batyrev-Peyre on the number of rational points of algebraic varieties

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

It is believed that certain geometrical properties of $X$ are making the set $X(\mathbb{Q})$ “large”. We count rational points in this case, and to do so, we introduce “heights”. A height on $X$ is a function on $X(\mathbb{Q})$, 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 $B \to \infty$. 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.