An asymptotic formula for representations of integers by indefinite hermitian forms

Abstract

We fix a maximal order $\mathcal{O}$ in $\F=\R,\C$ or $\mathbb{H}$, and an $\F$-hermitian form $Q$ of signature $(n,1)$ with coefficients in $\mathcal{O}$. Let $k\in \mathbb{N}$. By applying a lattice point theorem on the $\F$-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 $|x_{n+1}|\le t$.

arXiv commentTo appear in Proceedings of the American Mathematical Society

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