Canonical heights and Drinfeld modules
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)\dots(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.
scientific article; zbMATH DE number 86198