Counting invertible matrices and uniform distribution

Roettger, Christian

Journal de théorie des nombres de Bordeaux, Tome 17 (2005) no. 1, pp. 301-322.

On considère le groupe ${SL}_2\left({\mathbf{O}}_K\right)$ sur l’anneau des entiers d’un corps de nombres $K$. La hauteur d’une matrice est définie comme le maximum de tous les conjugués de ses éléments en valeur absolue. Soit ${SL}_2({\mathbf{O}}_K,t)$ le nombre de matrices de ${SL}_2\left({\mathbf{O}}_K\right)$ dont la hauteur est inférieure à $t$. Nous déterminons le comportement asymptotique de ${SL}_2({\mathbf{O}}_K,t)$, ainsi qu’un terme d’erreur. Plus précisemment,

$${SL}_2({\mathbf{O}}_K,t)=C{t}^{2n}+O\left(t^{2n-\eta }\right)$$

où $n$ est le degré de $K$. La constante $C$ dépend du discriminant de $K$, d’une intégrale ne dépendant que de la signature de $K$, et de la valeur de la fonction zêta de Dedekind relative à $K$ pour $s=2$. Nous faisons appel à la théorie de distribution uniforme et de la discrépance pour obtenir le terme d’erreur. Enfin, nous discuterons trois applications concernant le nombre asymptotique de matrices de ${GL}_2\left({\mathbf{O}}_K\right)$, d’unités dans certains anneaux de groupe entiers, et de bases normales intégrales.

Consider the group ${SL}_2\left({\mathbf{O}}_K\right)$ over the ring of algebraic integers of a number field $K$. Define the height of a matrix to be the maximum over all the conjugates of its entries in absolute value. Let ${SL}_2({\mathbf{O}}_K,t)$ be the number of matrices in ${SL}_2\left({\mathbf{O}}_K\right)$ with height bounded by $t$. We determine the asymptotic behaviour of ${SL}_2({\mathbf{O}}_K,t)$ as $t$ goes to infinity including an error term,

$${SL}_2({\mathbf{O}}_K,t)=C{t}^{2n}+O\left(t^{2n-\eta }\right)$$

with $n$ being the degree of $K$. The constant $C$ involves the discriminant of $K$, an integral depending only on the signature of $K$, and the value of the Dedekind zeta function of $K$ at $s=2$. We use the theory of uniform distribution and discrepancy to obtain the error term. Then we discuss applications to counting problems concerning matrices in the general linear group, units in certain integral group rings and integral normal bases.

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