On the $S$-Euclidean minimum of an ideal class
Tom 171 / 2015
Acta Arithmetica 171 (2015), 125-144
MSC: Primary 11H50, 13F07, 22D40; Secondary 11R04, 11H55, 54H20.
DOI: 10.4064/aa171-2-2
Streszczenie
We show that the $S$-Euclidean minimum of an ideal class is a rational number, generalizing a result of Cerri. In the proof, we actually obtain a slight refinement of this and give some corollaries which explain the relationship of our results with Lenstra's notion of a norm-Euclidean ideal class and the conjecture of Barnes and Swinnerton-Dyer on quadratic forms. In particular, we resolve a conjecture of Lenstra except when the $S$-units have rank one. The proof is self-contained but uses ideas from ergodic theory and topological dynamics, particularly those of Berend.
