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Equidistribution and integral points for Drinfeld modules. (English) Zbl 1178.11046
Consider a smooth affine curve \(C\) over a finite field \(k\). Denote the generic point of \(C\) by \(\eta \) and the function field of \(C\) by \(K\). Let \( \phi : k[t]\to\text{End}(\mathbb G_a/C)\) be an action of the polynomial ring \(k[t]\) on the additive group over \(C\) such that the fibre over \(\eta\) becomes a Drinfeld module of generic characteristic over \(K\). (In other words: the induced action of \(k[t]\) on the tangent space at zero to \(\mathbb G_{a,K}\) is faithful and does not coincide with the action on \(\mathbb G_{a,K}\) itself, when identified with its tangent space at zero.) Let \(\beta\) in \(\mathbb G_a(K)\) be a section over \(\eta\) that is non-torsion, i.e. for all \(f \in k[t] - \{0\}\) we have that \(\phi(f)(\beta) \neq 0\). The authors show the following finiteness result:
For all but finitely many \(f \in k[t]\) the zero section of \(\mathbb G_{a,C}\) meets the Zariski closure in \(\mathbb G_{a,C}\) of \(\phi(f)(\beta)\).

MSC:
11G09 Drinfel’d modules; higher-dimensional motives, etc.
11G50 Heights
11J68 Approximation to algebraic numbers
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