On primitive roots for rank one Drinfeld modules
Let \(k\) be the function field of a smooth projective geometrically irreducible curve \(X\) over a finite field, \(\infty\) a closed point on \(X\) and \(A\) the ring of functions of the complement \(X-\infty\). Let \(K\) be an \(A\)-field of finite characteristic (that is, the map \(A\to K\) is not injective). Assume that \(K\) is a global function field. Let \(\phi\) be a Drinfeld module of rank one over \(K\) and \(\alpha \in K\). For all but finitely many places \(v\) of \(K\) one may reduce the pair \((\phi, \alpha)\) modulo \(v\) and ask if the reduced point \(\alpha \mod v\) generates the module of rational points of \(\phi \mod v\). If this happens, we call \(\alpha\) a primitive root modulo \(v\) (by analogy with primitive roots for the multiplicative group). In this paper, it is shown that the set of \(v\) so that \(\alpha\) is a primitive root modulo \(v\) has a density. They also give a criterion for this density to be positive, based on \textit{H. W. Lenstra} jun.'s approach [Invent. Math. 42, 201--224 (1977; Zbl 0362.12012)].
scientific article; zbMATH DE number 5655001