# On periods of algebraic integers modulo rational primes

I run, somewhat indirectly, into the following problem and I have no hints where to look in the literature in search for answers or clues.

Let $K$ be a number field, which we may assume Galois if it helps, $\cal O$ its ring of integers and for each prime number $p$ let $R_p={\cal O}/p{\cal O}$ (a finite product of finite fields of characteristic $p$ for almost all $p$). Fix $\lambda\in{\cal O}$, $\lambda\neq0$ or a root of $1$. Then $\bar\lambda\in R_p^\times$ for almost all $p$ and the period $\pi_p=\pi_p(\lambda)$ of $\lambda$ is defined as the smallest positive $d$ such that $\bar\lambda^d=1$.

It is obvious that if we fix an integer $n$ the number of $p$'s such that $\pi_p\leq n$ is finite, since $\pi_p=d$ implies that $p|(\lambda^d-1)$ and there are only finitely many of those.

On the other hand, if $n\geq2$ the number of $p$'s such that $\max\{\text{Supp}(\pi_p)\}\leq n$ (the support $\text{Supp}(N)$ of an integer $N$ is the set of prime divisors of $N$) is infinite. For instance, the set of elements $\lambda^{2^k}-1$ has an infinite set of rational prime divisors because $\lambda^{2^{k+1}}-1=(\lambda^{2^k}-1)(\lambda^{2^k}+1)$ and the only common prime divisors in $\cal O$ to the 2 factors are primes of residual characteristic 2. Thus, as k grows, a new set of primes adds up at each step, so to speak.

Now the question is: fix an arithmetic progression ${\cal P}:a,a+d,a+2d,\dots$ with $(a,d)=1$, is it true that there are infinitely many primes in $\cal P$ such that $\max\{\text{Supp}(\pi_p)\}\leq n$? Conditionally on $n$?

In particular: suppose $K$ quadratic, and let $\ell>2$ a prime. Are there infinitely many primes $p\equiv 1\bmod\ell$ such that $\max\{\text{Supp}(\pi_p)\}\leq\ell-1$?

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Is another question which would imply yours: Are there infinitely many primes p so that p-1 is \ell smooth (meaning it is divisible by only primes less than \ell) –  Ben Weiss Jan 30 '10 at 14:45
This sounds like a little like a "smooth" version of Artin's Conjecture for number fields. If I'm correct in understanding that, try looking up Hans Roskam's work on the subject "Quadratic Analogue of Artin's Conjecture." –  Ben Weiss Jan 30 '10 at 20:21
@Andrea: I suspect that this may be unknown even for K=Q, in which case one cannot expect an answer for higher number fields. –  Bjorn Poonen Feb 6 '10 at 18:04
Bjorn, thanks for your comment. I got the paper of Roskam quoted above by Ben together with a nice short paper of H.W. Lenstra (Séminaire Delange-Pisot-Poitou, 1977) and the 1967 paper of Hooley on the Artin's conjecture, and I somehow got convinced that any precise answer must use the Generalized Riemann Hypothesis. For the moment, I'll be happy to find just the right heuristics in the quadratic case toggling with the fields $K(\mu_n,\sqrt[n](\lambda),\sqrt[n](\bar{\lambda}))$ along the lines of Roskam. –  Andrea Mori Feb 7 '10 at 9:48
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