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A proof of the Mahler-Manin conjecture. (Une preuve de la conjecture de Mahler-Manin.) (French) Zbl 0853.11059
This paper answers the question of the transcendence values for the Fourier expansion at infinity of the modular invariant $$j$$ at algebraic points. More precisely, $$j(q)$$ is proved to be transcendental over the field of rationals (respectively over the field of $$p$$-adic rationals) for any non-zero algebraic point $$q$$ of the unit disk of the complex plane (respectively of the completion of the field of $$p$$-adic rationals). This result conjectured by Mahler in the complex case and by Manin in the $$p$$-adic case has numerous applications in the theory of elliptic curves and $$p$$-adic $$L$$ functions. The proof inspired by Mahler’s method is based on sharp estimates of the coefficients of the modular polynomials. Let us note that a characteristic $$p$$ analogue of this result has been proved by J. F. Voloch [J. Number Theory 58, 55-59 (1996; Zbl 0853.11061)], see also D. S. Thakur’s proof based on the theory of automata [J. Number Theory 58, 60-63 (1996; Zbl 0853.11060)].

##### MSC:
 11J91 Transcendence theory of other special functions 11G40 $$L$$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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