Torsion points on elliptic curves over fileds of low degree.
In recent papers we proved a special case of a variant of Pink’s Conjecture for a variety inside a semiabelian scheme: namely for any curve inside anything isogenous to a product of two elliptic schemes. Here we go beyond the elliptic situation by settling the crucial case of any simple abelian surface scheme defined over the field of algebraic numbers, thus confirming an earlier conjecture of Shou-Wu Zhang. This is of particular relevance in the topic, also in view of very recent counterexamples...
Let be a Henselian discrete valuation ring with field of fractions . If is a smooth variety over and a torus over , then we consider -torsors under . If is a model of then, using a result of Brahm, we show that -torsors under extend to -torsors under a Néron model of if is split by a tamely ramified extension of . It follows that the evaluation map associated to such a torsor factors through reduction to the special fibre. In this way we can use the geometry of the special...
1 Supported in part by the Norwegian Research Council for Science and the Humanities. It is a pleasure for this author to thank the Department of Mathematics of the University of Sofia for organizing the remarkable conference in Zlatograd during the period August 28-September 2, 1995. It is also a pleasure to thank the M.I.T. Department of Mathematics for its hospitality from January 1 to July 31, 1993, when this work was started. 2Supported in part by NSF grant 9400918-DMS.We introduce and study...
Based on the results of the first two parts to this paper, we prove that the canonical bundle of a minimal Kähler threefold (i.e. is nef) is good,i.e.its Kodaira dimension equals the numerical Kodaira dimension, (in particular some multiple of is generated by global sections); unless is simple. “Simple“ means that there is no compact subvariety through the very general point of and not Kummer. Moreover we show that a compact Kähler threefold with only terminal singularities whose canonical...