Which elements in SL2(Q) are conjugated to an element in SL2(Z) - MathOverflow most recent 30 from http://mathoverflow.net 2013-05-23T15:59:58Z http://mathoverflow.net/feeds/question/43650 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/43650/which-elements-in-sl2q-are-conjugated-to-an-element-in-sl2z Which elements in SL2(Q) are conjugated to an element in SL2(Z) Karl 2010-10-26T09:43:15Z 2010-10-26T10:01:36Z <p>Dear all,</p> <p>once again my question is all about $SL_2(\mathbb{Z})$ and $SL_2(\mathbb{Q})$ ! Which elements in $M \in SL_2(\mathbb{Q})$ can you write in the following form:</p> <p>$M= NBN^{-1}$</p> <p>with $N \in GL_2(\mathbb{Q})$ and $B \in SL_2(\mathbb{Z})$?</p> <p>It is surely nor all of $SL_2(\mathbb{Q})$ (look at traces), but I do not have any guess which matrices I get!</p> <p>Thank you very much again! Karl</p> http://mathoverflow.net/questions/43650/which-elements-in-sl2q-are-conjugated-to-an-element-in-sl2z/43651#43651 Answer by Robin Chapman for Which elements in SL2(Q) are conjugated to an element in SL2(Z) Robin Chapman 2010-10-26T10:01:36Z 2010-10-26T10:01:36Z <p>You can do this if and only if the trace of $M$ is an integer. By the theory of the rational canonical form if matrices $A$ and $B$ over $\mathbb{Q}$ have the same characteristic polynomial and neither has a repeated eigenvalue they are conjugate by a matrix over $\mathbb{Q}$. This almost does it, save for some fiddling about when the eignvalue of $M$ is repeated.</p>