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A generic example is ${}_2 F_1(\frac{1}{3},\frac{2}{3},\frac{5}{6};\frac{27}{32})=\frac{8}{5}$. So my question: Is there any description of the set of rational points at which the hypergeometric function ${}_m F_n$ takes rational values?

The references to the literature where a lot of such examples listed are apreciated very much too.

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    $\begingroup$ A classical problem is to determine the cases when ${}_mF_n$ is an algebraic/rational function (this automatically implies the algebraicity of the values at algebraic points). This question was addressed in several papers by F. Beukers and his coauthors (Paula Cohen-Tretkoff is one of them). A general statement about algebraicity is in the Beukers-Heckman 1989 paper but you may find it as Theorem 1.3 in arxiv.org/abs/0812.1134 . $\endgroup$
    – Wadim Zudilin
    Commented Jun 22, 2010 at 14:14
  • $\begingroup$ Just found this question -- the "generic [sic] example" is (18) in mathworld.wolfram.com/HypergeometricFunction.html , and attributed there only to personal communication from M. Trott, without proof. Do you have another source for that one? It's actually quite special, in that most examples one sees come from CM points on elliptic modular curves, but this one is connected with a Shimura curve. $\endgroup$
    – Noam D. Elkies
    Commented Jan 3, 2017 at 2:47

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There are theorems that give conditions for the set to be finite even when "rational" is replaced by "algebraic". See the work of Paula Cohen Tretkoff. E.g the following paper is a survey and Theorem 2 is about this:

http://www.math.tamu.edu/~ptretkoff/martinpub_final.pdf

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