Modular functions and transcendence questions

Yu. V. Nesterenko

Sbornik. Mathematics, Tome 187 (1996) no. 9, pp. 1319-1348

Résumé

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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