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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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