An asymptotic formula for representations of integers by indefinite hermitian forms

EMILIO LAURET

Córdoba

Congreso; IV Congreso Latinoamericano de Matemáticos; 2012

Unión Matemática Argentina

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 $n$-dimensional $\mathbb{F}$-hyperbolic space, we gave 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 $|x_{n+1}|\leq t$. 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.