A conjecture of Erdős, supersingular primes and short character sums

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation \[ n(n+d)\cdots(n+(k-1)d) = y^\ell \] has at most finitely many solutions in positive integers $n$, $d$, $y$ and $\ell$, with $\mathrm{gcd}(n,d)=1$ and $\ell \ge 2$. Our proof relies upon Frey-Hellegouarch curves and results on supersingular primes for elliptic curves without complex multiplication, derived from upper bounds for short character sums and sieves, analytic and combinatorial.