Polynomial sums over automorphs of a positive definite binary quadratic form

Abstract

Let P( X) be a homogeneous polynomial in X = ( x, y), Q( X) a positive definite integral binary quadratic form, and G the group of integral automorphs of Q( X). Let A(m) = { N ∈ Z × Z : Q(N) = m}. It is shown that if Σ N∈ A( m) P( N) = 0 for each m = 1, 2, 3,… then Σ U∈ G P( UX) ≡ 0.