Geodesic
Modular functions and transcendence questions
Sbornik. Mathematics, Tome 187 (1996) no. 9, pp. 1319-1348

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We prove results on the transcendence degree of a field generated by numbers connected with the modular function $j(\tau )$. In particular, we show that $\pi$ and $e^\pi$ are algebraically independent and we prove Bertrand's conjecture on algebraic independence over $\mathbb Q$ of the values at algebraic points of a modular function and its derivatives. 
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     author = {Yu. V. Nesterenko},
     title = {Modular functions and transcendence questions},
     journal = {Sbornik. Mathematics},
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     number = {9},
     year = {1996},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/SM_1996_187_9_a3/}
}
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Yu. V. Nesterenko. Modular functions and transcendence questions. Sbornik. Mathematics, Tome 187 (1996) no. 9, pp. 1319-1348. http://geodesic.mathdoc.fr/item/SM_1996_187_9_a3/