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A transcendence theorem in finite characteristic. (Un théorème de transcendance en caractéristique finie.) (French) Zbl 1049.11080
Let \(E_k(q)\) be the Fourier-expansion (\(q=e^{2\pi iz}\)) of the normalized Eisenstein series of weight \(k\). Yu. Nesterenko [Sb. Math. 187, 1319–1348 (1996); translation from Mat. Sb. 187, No. 9, 65–96 (1996; Zbl 0898.11031)] proved that if \(0<| q| <1\), then at least \(3\) of the numbers \(q\), \(E_2(q)\), \(E_4(q)\) and \(E_6(q)\) are algebraically independent over \({\mathbb Q}\). A different proof was given by P. Philippon, who also established the same result in the \(p\)-adic setting [J. Reine Angew. Math. 497, 1–15 (1998; Zbl 0887.11032)].
Now let \(K={\mathbb F}_q(T)\) and \({\mathcal C}\) be the completion of an algebraic closure of \({\mathbb F}_q((1/T))\). The analogy between elliptic curves over \({\mathbb C}\) and Drinfeld modules of rank \(2\) over \({\mathcal C}\) and between their associated modular forms leads to the following
Conjecture: If \(t\in{\mathcal C}\) with \(0<| t| <q^{-1/(q-1)}\), then at least \(3\) of the numbers \(t\), \(E(t)\), \(\overline{g}(t)\) and \(\overline{\Delta}(t)\) are algebraically independent over \(K\).
Here \(\overline{\Delta}(t)\) is the \(t\)-expansion of the normalized Drinfeld discriminant function. The author proves the following result, which because of \(j(t)=\overline{g}(t)^{q+1}/\overline{\Delta}(t)\) would be a corollary of the conjecture.
Theorem: If \(0<| t| <q^{-1/(q-1)}\), at least one of the numbers \(\overline{\Delta}(t)\) and \(j(t)\) is transcendental over \(K\).
An important ingredient of the proof are height estimates for certain modular polynomials. We also mention that another corollary of the conjecture was proved by M. Ably, L. Denis and F. Recher [Math. Z. 231, 75–89 (1999; Zbl 0935.11025)]: If \(0<| t| <q^{-1/(q-1)}\), at least one of the numbers \(t\) and \(j(t)\) is transcendental over \(K\). (Function field analog of the Mahler-Manin conjecture).
MSC:
11J93 Transcendence theory of Drinfel’d and \(t\)-modules
11F52 Modular forms associated to Drinfel’d modules
11J81 Transcendence (general theory)
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References:
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