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Counting elements of the congruence subgroup

Part of: Special matrices Polynomials and matrices

Published online by Cambridge University Press:  22 May 2024

Kamil Bulinski  and
Igor E. Shparlinski Open the ORCID record for Igor E. Shparlinski [Opens in a new window]
Kamil Bulinski
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, UNSW 2052, Australia e-mail: k.bulinski@unsw.edu.au
Igor E. Shparlinski*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney, UNSW 2052, Australia e-mail: k.bulinski@unsw.edu.au
*
e-mail: igor.shparlinski@unsw.edu.au
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Abstract

We obtain asymptotic formulas for the number of matrices in the congruence subgroup

$$\begin{align*}\Gamma_0(Q) = \left\{ A\in\operatorname{SL}_2({\mathbb Z}):~c \equiv 0 \quad\pmod Q\right\}, \end{align*}$$
which are of naive height at most X. Our result is uniform in a very broad range of values Q and X.

Keywords

SL2(ℤ) matricescongruence subgroupmodular hyperbola

MSC classification

Primary: 11C20: Matrices, determinants 15B36: Matrices of integers 15B52: Random matrices

Information

Type
Article
Information
Canadian Mathematical Bulletin , Volume 67 , Issue 4 , December 2024 , pp. 955 - 969
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

During the preparation of this work, the authors were supported in part by the Australian Research Council Grants DP230100530 and DP230100534.

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