An asymptotic formula for representations of integers by indefinite hermitian forms

Abstract

We fix a maximal order O in F = R, C or H, and an F-hermitian form Q of signature (n, 1) with coefficients in O. Let k ∈ N. By applying a lattice point theorem on an n-dimensional F-hyperbolic space, we give an asymptotic formula with an error term, as t → +∞, for the number Nt;(Q,-k) of integral solutions x ∈ On+1 of the equation Q[x] = -k satisfying |xn+1| ≤ t.