Points of low height on P over number fields and bounds for torsion in class groups

Let K be a number field and ` a positive integer. The main theorem of [3] gives an upper bound for the order of the `-torsion subgroup in the ideal class group ClK of K under the Generalized Riemann Hypothesis, which can be made unconditional in certain special cases (e.g. when K/Q is a quadratic extension and ` = 3.) The main idea is to show that there are many ideal classes which are not `-torsion. This is accomplished by showing that there are many ideals I1, I2, . . . , Is of small height (in a sense which will be made precise below) but that there are no principal ideals of small height; this implies that `I1 − `I2, which also has small height, is non-principal, which shows that I1, I2, . . . , Is represent distinct classes of ClK/`ClK . This shows that ClK [`] cannot be too large. The aim of this note is to make the observation that the bounds of [3] could be improved if one had good bounds on the number of principal ideals of height at most X, when X is large enough that this number is nonzero. We are led to precise questions about the distribution of points of low height on P(K), which do not seem to have been well-investigated either theoretically or experimentally. Any nontrivial progress on these questions would lead to an improvement of the results of [3]. We remark that one expects to have |ClK [`]| d, ∆ K , so one should be aware that the results proved here, which bound the order of ClK [`] below a positive power of ∆K , are not expected to be anywhere close to sharp.