A conjecture of Erdos, supersingular primes and short character sums

If $k$ is a sufficiently large positive integer, we show that the Diophantine equation $$n(n+d)\cdots (n+(k-1)d)=y^{\mathrm{\ell }}$$ has at most finitely many solutions in positive integers $n,d,y$ and $\mathrm{\ell }$, with $\gcd (n,d)=1$ and $\mathrm{\ell }\ge 2$. 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.