Cusp Volumes and Rational Points on Spheres

In recent work on the Fourier coefficients of light-cone Eisenstein series arising in the problem of counting rational points on spheres, Kelmer and Yu conjectured that the leading coefficient in the asymptotic formula for the counting function is rational up to a volume factor. We prove this conjecture by showing that every cusp volume appearing in the corresponding light-cone Eisenstein series is rational.