Lower bounds of linear forms in logarithms for Drinfeld modules
Let \(k=\mathbb F_q(T)\) be the field of rational functions over the finite field \(\mathbb F_q\). Put \(k_\infty=\mathbb F_q((1/T))\) and denote by \(C\) the completion of the algebraic closure of \(k_\infty\) and by \(K \subset C\) a finite extension of \(k\) of degree \(D\). The author considers \(n\) Drinfeld modules \((G_a,\Phi_i)\), \(1\leq i\leq n\), defined over \(K\) and of positive rank, with exponentials \(e_{\Phi_i}\). He considers a linear form of logarithms: \[ \Lambda =\beta _0+\beta _1u_1+\ldots+\beta _nu_n, \] where \(\beta _0\), \dots, \(\beta _n\in K\), and \(u_1\), \dots, \(u_n\in C\) are such that \(e_{\Phi_i}(u_i)\in K\) for \(i=1\), \dots, \(n\) (in other words \(u_1\), \dots, \(u_n\) are ``logarithms''). When \(\Lambda \not=0\) he obtains a lower bound for \(\Lambda \) in terms of the degree \(D\), the heights of the \(\beta \)'s, the absolute values of \(u_1\), \dots, \(u_n\), the heights of the \(e_{\Phi_i}(u_i)\) and the heights of the Drinfeld modules \((G_a,\Phi_i)\). This result is effective and explicit, except for some computable constant. This is the analogue for Drinfeld modules of results obtained in the case of complex algebraic groups by N. Hirata and S. David. The estimate is almost the best possible which could be expected when compared to the complex case.
scientific article; zbMATH DE number 1275047
scientific article