Multiplicative relations of points on algebraic groups

Yuval Z. Flicker¹ ; Piotr Krasoń²

¹ Ariel University Ariel 40700, Israel and The Ohio State University Columbus, OH 43210, U.S.A.
² University of Szczecin Wielkopolska 15 70-451 Szczecin, Poland

Bulletin of the Polish Academy of Sciences. Mathematics, Tome 65 (2017) no. 2, pp. 125-138.

Résumé

Our aim here is to restructure the area of multiplicative relations on points and congruences, by proposing a novel conjecture in the context of general reductive linear algebraic groups. To support our conjecture we check it in a few elementary but new cases, and claim this extends classical work in number theory on multiplicative relations on points and congruences, initiated by Skolem and Schinzel, which we rephrase group-theoretically as Hasse principles on commutative linear algebraic groups, or tori, so that a part of it becomes the abelian case of our conjecture. Our conjecture can then be viewed as an extension to general—not necessarily commutative—reductive linear algebraic groups of a part of Schinzel’s result. We relate it to the Erdős support problem. To motivate our conjecture from another perspective we note that analogues have been extensively developed for abelian varieties. We give a short account of this, and state a question on the “detecting linear dependence” problem.

Détail

DOI : 10.4064/ba8104-8-2017

Keywords: here restructure area multiplicative relations points congruences proposing novel conjecture context general reductive linear algebraic groups support conjecture check few elementary cases claim extends classical work number theory multiplicative relations points congruences initiated skolem schinzel which rephrase group theoretically hasse principles commutative linear algebraic groups tori part becomes abelian conjecture conjecture viewed extension general necessarily commutative reductive linear algebraic groups part schinzel result relate erd support problem motivate conjecture another perspective note analogues have extensively developed abelian varieties short account state question detecting linear dependence problem