Let $N\geq2$ be a positive integer. Is the canonical homomorphism $\pi$ from $SL_2(\mathbb{Z})$ to $SL_2(\mathbb{Z}/N\mathbb{Z})$ surjective?
What if we ask the same question for $SL_n$?
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3 Answers
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Yes, it is surjective. It is called the Strong Approximation Property (because of a more general way to formulate it). In your case, there is a simple proof: $SL_n(\mathbb{Z})$ contains the elementary matrices, and their projections generate $SL_n(\mathbb{Z}/N\mathbb{Z})$ because $\mathbb{Z}/N\mathbb{Z}$ is Euclidean.
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$\begingroup$ Could you please explain a bit on why $\mathbb{Z}/N\mathbb{Z}$ is Euclidean and why $SL_n(\mathbb{Z}/N\mathbb{Z})$ is then generated by the projections of elementary matrix in $SL_n(\mathbb{Z})$? $\endgroup$ Apr 24, 2014 at 16:39
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1$\begingroup$ $\mathbb{Z}/N\mathbb{Z}$ is Euclidean because $\mathbb{Z}$ is: lift the elements to be divided in $[0,N-1]$, divide, and project back. In a Euclidean ring $R$, $SL_n(R)$ is generated by the elementary matrices (those with diagonal $1$ and exactly one off-diagonal nonzero entry): start with a matrix in $SL_n(R)$, use Gaussian elimination with the Euclidean division to reduce to a diagonal matrix. To reduce to the identity, shows that it works for $2\times 2$ diagonal matrices with a few elementary operations. $\endgroup$– AurelApr 24, 2014 at 17:02
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Yes, and yes. See Yoshida's notes, Lemma 5.7 (a better reference is Morris Newman's "Integral matrices").
answered Apr 24, 2014 at 16:10
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Yes, this is included in the proof of Lemma 1.38 in Shimura: Introduction to the arithmetic theory of automorphic functions (the lemma is restricted to $n=2$, but the proof proceeds for general $n$).
answered Apr 24, 2014 at 16:17