Norm form equations. V: Degenerate modules
[For parts III, IV, cf. the preceding review and Mathematika (to appear).] This is the fifth paper in a series of the author's studies on the solutions in a module of diophantine equations of norm form over a function field, and herein established is a necessary and sufficient condition on the module, in the case of characteristic zero, for the solutions to be always finite in number. Thus, let K denote a finite extension of the rational function field k(z) over an algebraically closed field of characteristic zero. Let L be a field intermediate between K and k(z), and let \({\mathcal O}_ L\) denote the integral closure of the polynomial ring k[z] in L. We define \(u_ L\) to be equal to \(rank_{{\mathbb{Z}}}({\mathcal O}^*_ L/k^*)\), the largest number of units in \({\mathcal O}^*_ L\) multiplicatively independent over \(k^*\). Also, for a free \({\mathcal O}_ L\)-module \({\mathcal M}\) in K and for any field J intermediate between K and L \[ {\mathcal M}^ J=\{m\in {\mathcal M}: mJ\subseteq mL\}. \] Now, the main theorem of this paper reads as follows. If the number of solutions x in such a module \({\mathcal M}\) of the equation \[ (*)\quad Norm_{K/L} (x)=c \] is finite for each c in L, then \({\mathcal M}^ J=\{0\}\) whenever \(u_ J>u_ L\). Conversely, if \({\mathcal M}^ J=\{0\}\) whenever \(u_ J>u_ L\), then (*) has only finitely many solutions x in \({\mathcal M}\) for each c in L, and if in addition \(k={\bar {\mathbb{Q}}}\) then these solutions may be determined effectively. The proof depends partly upon the author's previous study of norm form equations [J. Number Theory 22, 190-207 (1986; Zbl 0578.10021)], and partly upon some results on lower bounds for the Néron-Tate height on an abelian variety, due to \textit{D. W. Masser} [Compos. Math. 63, 153-169 (1984; Zbl 0551.14015)].
scientific article; zbMATH DE number 3970830