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On the order of unimodular matrices modulo integers. (English) Zbl 1030.11048

Given coprime integers \(b\) and \(n\), let ord\(_n(b)\) be the multiplicative order of \(b\) modulo \(n\). In other words, ord\(_n(b)\) is the smallest positive integer \(k\) such that \(b^k\equiv 1\pmod n\). The idea is that for a ‘typical’ \(b\) and \(n\) the multiplicative order should be large. P. Erdős and M. Ram Murty [CRM Proc. Lect. Notes 19, 87-97 (1999; Zbl 0931.11034)] proved that if \(b\neq 0,\pm 1\), then there exists a \(\delta>0\) so that ord\(_p(b)\) is at least \(p^{1/2}\exp((\log p)^{\delta})\) for a full density subset of primes \(p\). Under the Generalized Riemann Hypothesis (GRH) they proved (in the same paper) the much stronger result that the order of \(b\) modulo \(p\) is greater than \(p/f(p)\) for a full density subset of primes \(p\).
In the present paper the latter result is extended to integers and to hyperbolic unimodular matrices \(A\in \text{SL}_2(\mathbb Z)\). It is shown, assuming GRH, that the set of integers \(N\) such that ord\(_N(b)\gg N^{1-\varepsilon}\) has density one. Given an hyperbolic matrix \(A\in \text{SL}_2(\mathbb Z)\) it is shown, assuming GRH, that the set of integers \(N\) such that ord\(_N(A)\gg N^{1-\varepsilon}\) has density one. In an earlier paper P. Kurlberg and Z. Rudnick [Commun. Math. Phys. 222, 201-227 (2001; Zbl 1042.81026)] showed that ord\(_N(A)\) is slightly larger than \(N^{1/2}\) for a full density subset of the integers, which they needed to establish a strong form of quantum ergodicity for toral automorphisms.
The proofs involve a nice mix of techniques from both algebraic and analytic number theory.

MSC:

11N37 Asymptotic results on arithmetic functions
11A07 Congruences; primitive roots; residue systems
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