## 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

### Keywords:

order; unimodular matrix; primitive unit

### Citations:

Zbl 0931.11034; Zbl 1042.81026
Full Text: