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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)].

11J91 Transcendence theory of other special functions
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
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