Search

For a family of G-fields K, we show an effective prime ideal theorem with probability 1. Namely, outside a density zero set, π(x, K)=Li(x)+O(x/(log x)2) for x≥(log dK)c for some c depending on G. Using the effective prime ideal theorem, we show that the exponents eK of the ideal class groups of a family of G-fields increase along with dK with probability 1. [ABSTRACT FROM PUBLISHER]

Let K be a number field and let E/K be an elliptic curve. If E has complex multiplication, we show that there is a positive lower bound for the canonical height of nontorsion points on E defined over the maximal abelian extension Kab of K. This is analogous to results of Amoroso, Dvornicich (2000) and Amoroso, Zannier (2000) for the multiplicative group. We also show that if E has nonintegral j-invariant (so that, in particular, E does not have complex multiplication), then there exists C > 0 such that there are only finitely many points P ∈ E(Kab) of canonical height less than C. This strengthens a result of Hindry and Silverman (1990). [ABSTRACT FROM AUTHOR]