Counting invertible matrices and uniform distribution

Christian Roettger[1]

  • [1] Iowa State University 396 Carver Hall 50011 Ames, IA

Journal de Théorie des Nombres de Bordeaux (2005)

  • Volume: 17, Issue: 1, page 301-322
  • ISSN: 1246-7405

Abstract

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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, SL 2 ( O K , t ) = C t 2 n + O ( t 2 n - 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.

How to cite

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Roettger, Christian. "Counting invertible matrices and uniform distribution." Journal de Théorie des Nombres de Bordeaux 17.1 (2005): 301-322. <http://eudml.org/doc/249421>.

@article{Roettger2005,
abstract = {Consider the group $\operatorname\{SL\}_2(\mathbf\{O\}_K)$ 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 $\operatorname\{SL\}_2(\mathbf\{O\}_K,t)$ be the number of matrices in $\operatorname\{SL\}_2(\mathbf\{O\}_K)$ with height bounded by $t$. We determine the asymptotic behaviour of $\operatorname\{SL\}_2(\mathbf\{O\}_K,t)$ as $t$ goes to infinity including an error term,\[ \operatorname\{SL\}\_2(\mathbf\{O\}\_K,t)= C t^\{2n\} + O(t^\{2n-\eta \}) \]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.},
affiliation = {Iowa State University 396 Carver Hall 50011 Ames, IA},
author = {Roettger, Christian},
journal = {Journal de Théorie des Nombres de Bordeaux},
keywords = {matrices; uniform distribution; counting problems},
language = {eng},
number = {1},
pages = {301-322},
publisher = {Université Bordeaux 1},
title = {Counting invertible matrices and uniform distribution},
url = {http://eudml.org/doc/249421},
volume = {17},
year = {2005},
}

TY - JOUR
AU - Roettger, Christian
TI - Counting invertible matrices and uniform distribution
JO - Journal de Théorie des Nombres de Bordeaux
PY - 2005
PB - Université Bordeaux 1
VL - 17
IS - 1
SP - 301
EP - 322
AB - Consider the group $\operatorname{SL}_2(\mathbf{O}_K)$ 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 $\operatorname{SL}_2(\mathbf{O}_K,t)$ be the number of matrices in $\operatorname{SL}_2(\mathbf{O}_K)$ with height bounded by $t$. We determine the asymptotic behaviour of $\operatorname{SL}_2(\mathbf{O}_K,t)$ as $t$ goes to infinity including an error term,\[ \operatorname{SL}_2(\mathbf{O}_K,t)= C t^{2n} + O(t^{2n-\eta }) \]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.
LA - eng
KW - matrices; uniform distribution; counting problems
UR - http://eudml.org/doc/249421
ER -

References

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