Transcendence of elliptic modular functions in characteristic \(p\). (English) Zbl 0853.11061

The paper under review is devoted to the study of transcendence properties of the Tate elliptic curve \(y^2+ xy= x^3+ a_4 x+a_6\), defined in positive characteristic, say \(p\), where \[ a_4= -5 \sum_{n\geq 1} n^3 q^n/ (1- q^n), \qquad a_6= (-1/12) \sum_{n\geq 1} (7n^5+ 5n^3) q^n/ (1- q^n) \] for \(q\) transcendental over the algebraic closure \(k\) of \(\mathbb{F}_p\). By using Igusa theory, the author proves the transcendence of \(q\) over \(k (a_4, a_6)\) and the transcendence over \(k(a_4, a_6)\) of parameters of algebraic points of the Tate elliptic curve. The first of these two results can be seen as the analogue of the theorem of Siegel and Schneider on the transcendence of periods of elliptic curves defined over algebraic number fields, or as the analogue of the Mahler-Manin conjecture, proved by L. Barré-Sirieix, G. Diaz, F. Gramain and G. Philibert [Invent. Math. 124, 1-9 (1996; Zbl 0853.11059)], whereas the second of these results is the analogue of the transcendence of the elliptic logarithm of algebraic points on elliptic curves. Let us note that D. S. Thakur [J. Number Theory 58, 60-63 (1996; Zbl 0853.11060)] has given another proof of the transcendence of the period based on the theory of automata.


11J89 Transcendence theory of elliptic and abelian functions
14H52 Elliptic curves
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