An asymptotic formula for representations of integers by indefinite hermitian forms

Abstract:

We fix a maximal order $\mathcal O$ in $\mathbb {F}=\mathbb {R},\mathbb {C}$ or $\mathbb {H}$, and an $\mathbb {F}$-hermitian form $Q$ of signature $(n,1)$ with coefficients in $\mathcal O$. Let $k\in \mathbb {N}$. By applying a lattice point theorem on an $n$-dimensional $\mathbb {F}$-hyperbolic space, we give an asymptotic formula with an error term, as $t\to +\infty$, for the number $N_t(Q,-k)$ of integral solutions $x\in \mathcal O^{n+1}$ of the equation $Q[x]=-k$ satisfying $|x_{n+1}|\leq t$.