Formes homogènes de degré 3 et puissances p-ièmes

In this paper, we are interested in diophantine equations of type F(x,y)=dzp where F is a separable homogeneous form of degree ⩾3 with integer coefficients, d a fixed integer ⩾1 and p a prime number ⩾7. As a consequence of the abc conjecture, if p is sufficiently large and (a,b,c) is a nontrivial proper solution of the above equation, we have c=±1. In the case where F has degree 3, we associate to (a,b,c) an elliptic curve defined over Q called the Frey curve or Hellegouarch–Frey curve. This allows us to deduce our conjecture from another one about elliptic curves attributed to G. Frey and B. Mazur (which is itself a consequence of the abc conjecture). We then applied our construction to the study of an explicit form. We give some results about the set of nontrivial proper solutions of the equation considered for several values of d.