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Canonical heights and Drinfeld modules. (Hauteurs canoniques et modules de Drinfeld.) (French) Zbl 0764.11027
Let $$A$$ be the polynomial ring $$\mathbb{F}_ q[t]$$ with quotient field $$k$$, and $$\bar k$$ its algebraic closure. In the paper, two principal results about Drinfeld $$A$$-modules and $$t$$-modules (= G. Anderson’s generalization of Drinfeld modules) are obtained:
Theorem 1: If $$\varphi$$ is a $$t$$-module over $$\bar k$$ of dimension $$N$$, there exists a canonical height function $$\hat h$$ on $$(\mathbb{G}_ a)^ N(\bar k)$$ with properties (1)…(8) similar to the Néron-Tate height on an abelian variety.
Theorem 2: Let $$C$$ be the Carlitz module. There exists a real constant $$\eta$$ (effectively computable, depending only on $$q$$) such that for separable points $$\alpha$$ of $$C$$ of degree $$\delta$$ the following estimate holds: $\hat h(\alpha)\geq \eta\delta^{-1}(\log \delta/\log\log \delta)^{-3}.$ The last theorem is similar to results of Dobrowolski and Laurent about the multiplicative group and elliptic curves with complex multiplication, respectively.

##### MSC:
 11G09 Drinfel’d modules; higher-dimensional motives, etc. 11J25 Diophantine inequalities 11T55 Arithmetic theory of polynomial rings over finite fields
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##### References:
 [1] Anderson, G.:t-motives. Duke Math. J.53, 457–502 (1986) · Zbl 0679.14001 [2] Anderson, G., Thakur, D.: Tensor powers of the Carlitz module and zeta values. Ann. Math.132, 159–191 (1990) · Zbl 0713.11082 [3] Denis, L.: Théorème de Baker et modules de Drinfeld. J. Number Theory (à paraître 1990) · Zbl 0722.11031 [4] Dobrowolski, E.: On a question of Lehmer and the number of irreducible factors of a polynomial. Acta Arith.34, 391–401 (1979) · Zbl 0416.12001 [5] Gekeler, E.U.: Zur Arithmetik von Drinfeld-Moduln. Math. Ann.262, 167–182 (1983) · Zbl 0536.14028 [6] Hayes, D.: Explicit class field theory for rational function fields. Trans. Am. Math. Soc.189, 77–91 (1974) · Zbl 0292.12018 [7] Hindry, M.: Géométrie et hauteurs dans les groupes algébriques. Thèse de l’université Paris 6 (1987) [8] Lang, S.: Fundamentals of Diophantine Geometry. Berlin Heidelberg New York: Springer 1983 · Zbl 0528.14013 [9] Laurent, M.: Minoration de la hauteur de Néron-Tate. In: Bertin, M.-J. (ed.) Seminar on number theory, Paris 1981–1982. (Prog. Math., vol. 38, pp. 137–151) Boston Basel Stuttgart: Birkhäuser 1983 [10] Masser, D.: Small value of the quadratic part of the Néron-Tate height. In: Bertin M.-J. (ed.) Seminar on number theory, Paris 1970–1980. (Prog. Math., vol. 12, pp. 213–222) Boston Basel Stuttgart: Birkhäuser 1981 [11] Philippon, P.: Pour des hauteurs alternatives. Math. Ann.289, 255–283 (1991) · Zbl 0726.14017 [12] Serre, J.P.: Quelques propriétés des groupes algébriques commutatifs. Appendice à M. Waldschmidt. (Astérisque, vols. 69–70, pp. 191–202) Paris: Soc. Math. Fr. 1979 [13] Silverman, J.: Rational points on K3 surfaces; a new canonical height. Invent. Math.105, 347–373 (1991) · Zbl 0754.14023 [14] Yu, J.: Transcendence and special zeta values in characteristicp. Ann. Math. (à paraître 1991)
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