Small height in fields generated by singular moduli

Abstract

We prove that some fields generated by j j -invariants of CM elliptic curves (of infinite dimension over Q \mathbb {Q} ) satisfy the Property (B). The singular moduli are chosen so as to have supersingular reduction simultaneously above a fixed prime q q , which provides strong q q -adic estimates leading to an explicit lower bound for the height.

Small totally p-adic algebraic numbers

“…We take the opportunity to use the last remark to remove a technical condition in a theorem of A. Galateau [7]. For any set S of primes, let L S be the compositum of the Hilbert class fields of Q( √ −p) for all p ∈ S. In [7], the set S has density 1 4 , since the proof requires the additional assumption that all primes in S are congruent 1 modulo 4.…”

“…Lemma 5.1. Let n ∈ N, then it is 7). In order to prove the first congruence, we consider f (2p − 1) modulo p 3 .…”

“…We take the opportunity to use the last remark to remove a technical condition in a theorem of A. Galateau [7]. For any set S of primes, let L S be the compositum of the Hilbert class fields of Q( √ −p) for all p ∈ S.…”

“…In [7], the set S has density 1 4 , since the proof requires the additional assumption that all primes in S are congruent 1 modulo 4.…”

On Multiplicative Dependence of Values of Rational Functions and a Generalization of the Northcott Theorem

“…It is easy to see that some of our results can be extended to any field with the Bogomolov property, that is, fields L ⊆ Q for which there exists a constant c L > 0, such that for any non-zero α ∈ L \ U we have h(α) ≥ c L . In particular, from [2, Theorem 1.2] it follows that K ab has the Bogomolov property, see [1,11,13,14] for non-abelian examples of such fields and further references.…”