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Let $\mathcal{O}_k$ be the ring of integers in an algebraic number field $k$ and let $\mathfrak{p}$ be a prime ideal of $\mathcal{O}_k$. I'm looking for conditions on $k$ and $\mathfrak{p}$ which will ensure that the image of the group of units $(\mathcal{O}_k)^{\ast}$ in $\mathcal{O}_k/\mathfrak{p}$ is all of $(\mathcal{O}_k/\mathfrak{p})^{\ast}$. More generally, what is known about this image?

My area of research is geometry and not number theory (the above problem arose while studying some geometric problems), so I apologize if the answer to the above is standard stuff.

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I don't know of any result that specifically looks at units but there are a lot of results on looking at the image of a fixed finitely generated subgroup $G$ of $k^*$ in the units of the residue fields. The granddad of these questions is the Artin conjecture for primitive roots. There are results of Gupta and Murty (which probably have been improved) and guarantee the image is everything for infinitely many primes provided that the rank of $G$ is big enough. You'd have to look at the papers to see if "big enough" applies when $G$ is group of units. If you are willing to assume GRH then much stronger results are known.

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    $\begingroup$ Joshua Zelinsky arxiv.org/abs/1307.2319 proves some results about the average size of the image of the unit group in $(\mathcal{O}/I)^{\ast}$ (average in the sense of a suitably normalized sum over ideals $I$ up to some norm.) He told me that these were results about units and not just arbitrary finitely generated multiplicative groups, but I didn't follow why. $\endgroup$
    – David E Speyer
    Commented Sep 11, 2013 at 18:30
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    $\begingroup$ To clarify Felipe's remark at the end about GRH: if you assume GRH then for any number field $k$ other than the rationals or imaginary quadratic fields (namely, for every $k$ with unit group that is infinite), the reduction map from the unit group to $({\mathcal O}_k/{\mathfrak p})^\times$ is surjective for a positive density of prime ideals $\mathfrak p$. This also works using $S$-units as a generalization of ordinary units, and in that case you want the $S$-unit group to be infinite (could then even include the rationals and imaginary quadratic fields). $\endgroup$
    – KConrad
    Commented Sep 11, 2013 at 19:06
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    $\begingroup$ There has been work on removing the GRH assumptions in such theorems, e.g., math.fsu.edu/~aluffi/archive/paper371.pdf for a paper with such an approach that includes a geometric application. $\endgroup$
    – KConrad
    Commented Sep 11, 2013 at 19:07
  • $\begingroup$ Thanks to all! I'm probably not going to get more answers at this point, so I'll accept this one. $\endgroup$
    – Nick
    Commented Sep 12, 2013 at 15:05

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