An asymptotic formula for representations of integers by indefinite hermitian forms

LAURET, EMILIO A.

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

Unión Matemática de América Latina y el Caribe

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 $kinmathbb{N}$. By applying a lattice point theorem on $n$-dimensional $mathbb{F}$-hyperbolic space, we gave an asymptotic formula with an error term, as $to+infty$, for the number $N_t(Q,-k)$ of integral solutions $xinmathcal O^{n+1}$ of the equation $Q[x]=-k$ satisfying $|x_{n+1}|leq t$. The error term depends on the first nonzero eigenvalue of the Laplace-Beltrami operator in certain hyperbolic manifolds. We also study the behavior of the error term with experimental computations, obtaining evidences on the nonexistence of exceptional eigenvalues in certain hyperbolic manifolds.