Sur les courbes hyperelliptiques cyclotomiques et les équations ${x^p-y^p=cz^2}$
Volume 444 / 2007
Abstract
Let $p$ be a prime number $\geq 11$ and $c$ be a square-free integer $\geq 3$. In this paper, we study the diophantine equation $x^p-y^p = cz^2$ in the case where $p$ belongs to $\{11,13,17\}$. More precisely, we prove that for those primes, there is no integer solution $(x,y,z)$ to this equation satisfying gcd$(x,y,z)=1$ and $xyz \neq 0$ if the integer $c$ has the following property: if $\ell$ is a prime number dividing $c$ then $\ell \not \equiv 1\bmod p$. To obtain this result, we consider the hyperelliptic curves $C_p : y^2 = {\mit\Phi} _p(x)$ and $D_p : py^2 = {\mit\Phi} _p(x)$, where ${\mit\Phi} _p$ is the $p$th cyclotomic polynomial, and we determine the sets $C_p({\sym Q})$ and $D_p({\sym Q})$. Using the elliptic Chabauty method, we prove that $C_p({\sym Q})=\lbrace (-1,-1),(-1,1),(0,-1),(0,1)\rbrace$ and $D_p({\sym Q})=\lbrace (1,-1),(1,1)\rbrace$ for $p\in \{11,13,17\}$.
