The Manin-Peyre's conjectures for Châtelet surfaces

For a projective variety containing infinitely many rational points, a natural question is to count the number
of such points of height less than some bound $B$. The Manin-Peyre’s conjectures predict, for Fano varieties,
the distribution of rational points of bounded height in terms of geometric invariants of the variety. We will
discuss in this talk the Manin-Peyre’s conjectures in the case of certain Châtelet surfaces, namely minimal
proper smooth models of affine varieties given by $$Y^2 -aZ^2 =F(X,1)$$ where $F \in \mathbf{Z}[x_1,x_2]$
is a degree 4 polynomial without repeated roots and $a$ is a squarefree non zero integer.