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This is a re-post from MSE as I did not get even a single comment there.

Wikipedia mentions that the transcendence of $\Gamma(1/3), \Gamma(1/4)$ was proved by G. V. Chudnovsky. Does anyone have a reference to that proof? Or maybe some details on the essential ideas involved in the proof would be greatly helpful.

Also is there any simpler proof available when we restrict the conclusion to just "irrationality" instead of "transcendence"?

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Algebraic Independence of Values of Exponential and Elliptic Functions, G. V. Chudnovsky (1978)

(a.i. = algebraically independent)

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