A result on the orders of the reductions of an element of the group of S-units of a number field is obtained by investigating three height functions for groups of S-units of number fields which are analogous to local, global and canonical height functions for elliptic curves.
@article{bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-1-4, author = {Stefan Bara\'nczuk}, title = {Height Functions for Groups of S-units of Number Fields and Reductions Modulo Prime Ideals}, journal = {Bulletin of the Polish Academy of Sciences. Mathematics}, volume = {63}, year = {2015}, pages = {25-31}, zbl = {06488954}, language = {en}, url = {http://dml.mathdoc.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-1-4} }
Stefan Barańczuk. Height Functions for Groups of S-units of Number Fields and Reductions Modulo Prime Ideals. Bulletin of the Polish Academy of Sciences. Mathematics, Tome 63 (2015) pp. 25-31. http://gdmltest.u-ga.fr/item/bwmeta1.element.bwnjournal-article-doi-10_4064-ba63-1-4/