Arithmetic discriminants and quadratic points on curves

Abstract

Let C be a curve defined over a number field k. Let h(P) denote the height of an algebraic point P ∈ C $P \in C(\overline k )$ relative to some fixed divisor of degree 1. For a number field F, let

$$ d(F) = \frac{{\log \left| {{D_{{F/Q}}}} \right|}}{{\left[ {F:Q} \right]}} $$

, and for an algebraic point P on C, let

$$ d(P) = d(k(P)) $$

.

Supported by the Miller Institute for Basic Research in Science