×

Transcendence and Drinfeld modules. (English) Zbl 0586.12010

Let \(k\) be a function field in one variable over a finite field, \(\infty\) a fixed prime of \(k\) and \(A\) the ring of functions regular away from \(\infty\). Let \(K=k_{\infty}\) be the completion of \(k\) at \(\infty\). This paper is the culmination of a long study by the author of the transcendence properties of Drinfeld \(A\)-modules. In it the author establishes an elegant function field version of the classical theorem of Schneider-Lang.
He is then able to deduce, among other things, the following very important corollaries:
1) Let \(\phi\) be a Drinfeld module defined over the algebraic closure, \(k^{\text{ac}}\), of \(k\) and let \(M\) be the associated lattice. Let \(0\neq \omega \in M\). Then \(\omega\) is transcendental over \(k\).
2) Let \(\phi_ 1\) and \(\phi_ 2\) be two Drinfeld modules over \(k^{\text{ac}}\) with different ranks, and associated lattices \(M_ 1\) and \(M_ 2\). Let \(0\neq \omega_ i\in M_ i\), \(i=1,2\). Then the quotient of \(\omega_ 1\) by \(\omega_ 2\) is transcendental over \(k\).
3) \((A=\mathbb F_ q[T])\) Let \(\tau \in k^{\text{ac}}\cap (K^{\text{ac}}-K)\). Then the value of the modular function \(j\) (defined in analogy with classical elliptic theory) at \(\tau\) is transcendental over \(k\).

MSC:

11J93 Transcendence theory of Drinfel’d and \(t\)-modules
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11R58 Arithmetic theory of algebraic function fields
14L05 Formal groups, \(p\)-divisible groups
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Carlitz, L.: On certain functions connected with polynomials in a Galois field. Duke Math. J.1, 137-168 (1935) · Zbl 0012.04904 · doi:10.1215/S0012-7094-35-00114-4
[2] Deligne, P., Husemoller, D.: Survey of Drinfeld modules. (Preprint, IHES, 1977) · Zbl 0627.14026
[3] Drinfeld, V.G.: Elliptic modules (Russian). Math. Sbornik94, 594-627 (1974) (English translation, Math. USSR Sbornik23, No.4 (1974)
[4] Geijsel, J.M.: Transcendence in fields of positive characteristic, Mathematical centre tracts 91. Amsterdam: Mathematisch Centrum 1979 · Zbl 0422.10025
[5] Gekeler, E.-U.: Zur Arithmetik von Drinfeld-Moduln. Math. Ann.262, 167-182 (1983) · Zbl 0536.14028 · doi:10.1007/BF01455309
[6] Goss, D.: The algebraist’s upper half-plane. Bull. Am. Math. Soc.2, #3, 391-415 (1980) · Zbl 0433.14017 · doi:10.1090/S0273-0979-1980-14751-5
[7] Goss, D.: The ?-ideal and special zeta-values. Duke Math. J.47, 345-364 (1980) · Zbl 0441.12002 · doi:10.1215/S0012-7094-80-04721-3
[8] Goss, D.: On a new type ofL-functions for algebraic curves over finite fields. Pac. J. Math.105, 143-181 (1983) · Zbl 0571.14010
[9] Hayes, D.: Explicit class field theory in global function fields.In: G.-C Rota (ed.), Studies in Algebra and Number Theory. New York: Academic Press 1979 · Zbl 0476.12010
[10] Lang, S.: Introduction to transcendental numbers. Reading, M.A.: Addison-Wesley 1966 · Zbl 0144.04101
[11] Wade, L.I.: Certain quantities transcendental overGF(p n ,x). Duke Math. J.8, 701-729 (1941) · Zbl 0063.08101 · doi:10.1215/S0012-7094-41-00860-8
[12] Waldschmidt, M.: Transcendence methods. Queens papers in pure and applied mathematics, No. 52, 1979
[13] Yu, J.: Transcendental numbers arising from Drinfeld modules. Mathematika30, 61-66 (1983) · Zbl 0523.12013 · doi:10.1112/S0025579300010408
[14] Yu, J.: Transcendence theory over function fields. Duke Math. J.52, 517-527 (1985) · Zbl 0574.12015 · doi:10.1215/S0012-7094-85-05226-3
[15] Yu, J.: A six exponentials theorem in finite characteristic. Math. Ann.272, 91-98 (1985) · Zbl 0574.12014 · doi:10.1007/BF01455930
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.