A singular K3 surface related to sums of consecutive cubes

Abstract

We study the surface arising from the diophantine equation $m^3+(m+1{)}^3+...+(m+k-1{)}^3=l^2$. It turns out that this is a $K3$ surface with Picard number 20. We stduy its aritmetic properties in detail. We construct elliptic fibrations on it, and we find a parametric solution to the original equation. Also, we determine the Hasse-Weil zeta function of the surface over $Q$.