An asymptotic formula for representations of integers by indefinite hermitian forms

TL;DR

This paper derives an asymptotic formula with an error term for counting integral solutions to indefinite hermitian forms over maximal orders in real, complex, or quaternionic fields, as the solutions grow large.

Contribution

It introduces a new asymptotic counting formula for solutions of indefinite hermitian forms over maximal orders, extending previous results to a broader algebraic setting.

Findings

Asymptotic formula with explicit error term for solution count

Applicable to hermitian forms over real, complex, and quaternionic fields

Generalizes lattice point counting in hyperbolic spaces

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 $k\in\N$. 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 $|x_{n+1}|\leq t$.