Torsion points in families of Drinfeld modules (Q2868207)

The Manin-Mumford conjecture states that the torsion points of a semiabelian variety \(G\) defined over \({\mathbb C}\) is not Zariski dense in a subvariety \(V\) of \(G\), unless \(V\) is the translate of an algebraic subgroup of \(G\) by a torsion point. In their study of the Manin-Mumford conjecture for elliptic curves, [\textit{D. Masser} and \textit{U. Zannier}, Am. J. Math. 132, No. 6, 1677--1691 (2010; Zbl 1225.11078); Math. Ann. 352, No. 2, 453--484 (2012; Zbl 1306.11047)] proved the following result: For each \(\lambda \in {\mathbb C}\setminus \{0,1\}\), let \(E_{\lambda}\) be the elliptic curve given by the equation \(y^2=x(x-1)(x-\lambda)\). Let \(P_{\lambda}\) and \(Q_{\lambda}\) be two families of points on \(E_{\lambda}\) depending algebraically on \(\lambda\). If there exist infinitely many \(\lambda\in{\mathbb C}\) such that both \(P_{\lambda}\) and \(Q_{\lambda}\) are torsion points for \(E_{\lambda}\), then the points \(P_{\lambda}\) and \(Q_{\lambda}\) are linearly dependent over \({\mathbb Z}\) on the generic fiber of \(E_{\lambda}\).NEWLINENEWLINEIn this paper the authors consider the similar question for Drinfeld modules. Drinfeld modules are similar to elliptic curves, particularly Drinfeld modules of rank \(2\). Consider a finite extension \(K\) of the rational function field \({\mathbb F}_q(t)\) and let \(K(z)\) be the rational function field in the variable \(z\). Let \(\Phi: {\mathbb F}_q[t]\to \roman{End}_{K(z)}({\mathbb G}_a)\) be a Drinfeld module defined over \(K(z)\) and let \(\Phi^{\lambda}\) be the Drinfeld module obtained letting \(z=\lambda\in K^{\text{sep}}\). As a consequence of one of the main results of the paper (Theorem 2.6), the authors show that if \(\Phi\) is the family of modules \(\Phi_t(x)=tx+\sum_{i=1}^{r-1}g_i(z)x^{q^i}+x^{q^r}\) and if \(f,h\in K[z]\) satisfy that there exist infinitely many \(\lambda \in K^{\text{sep}}\) such that \(f(\lambda)\) and \(h(\lambda)\) are torsion points for \(\Phi^{\lambda}\), then at least one of the following properties holds: (1) \(f\) or \(h\) is a torsion for \(\Phi\); (2) For each \(\lambda\in\bar{K}\), \(f(\lambda)\) is torsion for \(\Phi^{\lambda}\) if and only if \(h(\lambda)\) is torsion for \(\Phi^{\lambda}\).NEWLINENEWLINEThe other two main results of this paper are the following: Let \(r\geq 2\) and let \(K\) be a finitely generated extension of \({\mathbb F}_q(t)\) such that \({\mathbb F}_q\) is algebraically closed in \(K\) (resp. \(K=\bar{{\mathbb F}_q}(t)\)). Let \({\mathbf{a}}, {\mathbf{b}} \in K\) and let \(\Phi: {\mathbb F}_q[t]\to \text{End}_{K(z)}({\mathbb G}_a)\) be the family of Drinfeld modules given by \(\Phi_t(x)=tx+zx^q+ x^{q^r}\). If there exist infinitely many \(\lambda\in K^{\text{sep}}\) such that both \({\mathbf{a}}\) and \({\mathbf{b}}\) are torsion points for \(\Phi^{\lambda}\), then \({\mathbf{a}}\) and \({\mathbf{b}}\) are linearly dependent over \({\mathbb F}_q\). Moreover, for \(\lambda\in \bar{K}\), \({\mathbf{a}}\) is a torsion point for \(\Phi^{\lambda}\) if and only if \({\mathbf{b}}\) is a torsion point for \(\Phi^{\lambda}\).NEWLINENEWLINEThe problem in this paper is analogous to the one of \textit{M. Baker} and \textit{L. DeMarco} [Duke Math. J. 159, No. 1, 1--29 (2011; Zbl 1242.37062)] in the setting of Drinfeld modules. The results and proofs follow closely the paper of Baker and DeMarco with several technical details different, since now we are in positive characteristic.