Counting invertible matrices and uniform distribution
Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 301-322.

Consider the group SL 2 (O K ) 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 (O K ,t) be the number of matrices in SL 2 (O K ) with height bounded by t. We determine the asymptotic behaviour of SL 2 (O K ,t) as t goes to infinity including an error term,

SL2(OK,t)=Ct2n+O(t2n-η)

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.

On considère le groupe SL 2 (O K ) 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 (O K ,t) le nombre de matrices de SL 2 (O K ) dont la hauteur est inférieure à t. Nous déterminons le comportement asymptotique de SL 2 (O K ,t), ainsi qu’un terme d’erreur. Plus précisemment,

SL2(OK,t)=Ct2n+O(t2n-η)

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 (O K ), d’unités dans certains anneaux de groupe entiers, et de bases normales intégrales.

Published online:
DOI: 10.5802/jtnb.492
Christian Roettger 1

1 Iowa State University 396 Carver Hall 50011 Ames, IA
@article{JTNB_2005__17_1_301_0,
     author = {Christian Roettger},
     title = {Counting invertible matrices and uniform distribution},
     journal = {Journal de Th\'eorie des Nombres de Bordeaux},
     pages = {301--322},
     publisher = {Universit\'e Bordeaux 1},
     volume = {17},
     number = {1},
     year = {2005},
     doi = {10.5802/jtnb.492},
     zbl = {1101.11011},
     mrnumber = {2152226},
     language = {en},
     url = {https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.492/}
}
TY  - JOUR
TI  - Counting invertible matrices and uniform distribution
JO  - Journal de Théorie des Nombres de Bordeaux
PY  - 2005
DA  - 2005///
SP  - 301
EP  - 322
VL  - 17
IS  - 1
PB  - Université Bordeaux 1
UR  - https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.492/
UR  - https://zbmath.org/?q=an%3A1101.11011
UR  - https://www.ams.org/mathscinet-getitem?mr=2152226
UR  - https://doi.org/10.5802/jtnb.492
DO  - 10.5802/jtnb.492
LA  - en
ID  - JTNB_2005__17_1_301_0
ER  - 
%0 Journal Article
%T Counting invertible matrices and uniform distribution
%J Journal de Théorie des Nombres de Bordeaux
%D 2005
%P 301-322
%V 17
%N 1
%I Université Bordeaux 1
%U https://doi.org/10.5802/jtnb.492
%R 10.5802/jtnb.492
%G en
%F JTNB_2005__17_1_301_0
Christian Roettger. Counting invertible matrices and uniform distribution. Journal de Théorie des Nombres de Bordeaux, Volume 17 (2005) no. 1, pp. 301-322. doi : 10.5802/jtnb.492. https://jtnb.centre-mersenne.org/articles/10.5802/jtnb.492/

[1] A. F. Beardon, The geometry of discrete groups. Springer, 1983. | MR | Zbl

[2] R. W. Bruggeman, R. J. Miatello, Estimates of Kloosterman sums for groups of real rank one. Duke Math. J. 80 (1995), 105–137. | MR | Zbl

[3] C. J. Bushnell, Norm distribution in Galois orbits. J. reine angew. Math. 310 (1979), 81–99. | MR | Zbl

[4] W. Duke, Z. Rudnick, P. Sarnak, Density of integer points on affine homogeneous varieties. Duke Math. J. 71 (1993), 143–179. | MR | Zbl

[5] G. Everest, Diophantine approximation and the distribution of normal integral generators. J. London Math. Soc. 28 (1983), 227–237. | MR | Zbl

[6] G. Everest, Counting generators of normal integral bases. Amer. J. Math. 120 (1998), 1007–1018. | MR | Zbl

[7] G. Everest, K. Györy, Counting solutions of decomposable form equations. Acta Arith. 79 (1997), 173–191. | MR | Zbl

[8] E. Hlawka, Funktionen von beschränkter Variation in der Theorie der Gleichverteilung (German). Ann. Mat. Pura Appl., IV. Ser. (1961), 325–333. | MR | Zbl

[9] E. Hlawka, Theorie der Gleichverteilung. Bibliographisches Institut, Mannheim 1979. | MR | Zbl

[10] L. Kuipers and H. Niederreiter, Uniform distribution of sequences. Wiley, New York 1974. | MR | Zbl

[11] P. Lax and R. Phillips, The asymptotic distribution of lattice points in Euclidean and non-Euclidean spaces. J. Funct. Anal. 46 (1982), 280–350. | MR | Zbl

[12] R. W. K. Odoni, P. G. Spain, Equidistribution of values of rational functions (modp). Proc. R. Soc. Edinb. Sect. A 125 (1995), 911–929. | MR | Zbl

[13] I. Pacharoni, Kloosterman sums on number fields of class number one. Comm. Algebra 26 (1998), 2653–2667. | MR | Zbl

[14] S. J. Patterson, The asymptotic distribution of Kloosterman sums. Acta Arith. 79 (1997), 205–219. | MR | Zbl

[15] C. Roettger, Counting normal integral bases in complex S 3 -extensions of the rationals. Tech. Rep. 416, University of Augsburg, 1999.

[16] C. Roettger, Counting problems in algebraic number theory. PhD thesis, University of East Anglia, Norwich, 2000.

[17] P. Samuel, Algebraic Number Theory. Hermann, Paris 1970. | Zbl

[18] C. L. Siegel, Lectures on the geometry of numbers. Springer, 1989. | MR | Zbl

Cited by Sources: