Metric heights are modified height functions on the non-zero algebraic numbers Q ∗ which can be used to define a metric on certain cosets of Q ∗ . They have been defined with a view to eventually applying geometric methods to the study of Q ∗ . In this paper we discuss the construction of metric heights in general. More specifically, we study in some detail the metric height obtained from the n...

2 Derived p-adic heights 2.1 Derived heights for cyclic groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5 2.2 Comparison of pairings . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 10 2.3 Compatibility of the derived heights . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 2.4 Derived p-adic heights . . . . . . . . . . . . . . . . . . . . . . . . . ....

These notes correspond to the ALGANT Graduate course given in Bordeaux during the academic year 2009-2010 (second semester). The course intends to be as self contained as possible. The notes (in French) of the graduate course Géométrie Différentielle Complexe (Niamey, December 2009), available on line at http://www.math.u-bordeaux1.fr/~yger/Cours-Niamey.pdf will help for some eventual perequisi...

In this paper several recurrences and formulas are presented leading to an upper bound and a lower bound, both logarithmic, for the expected height of a node in a heap. These bounds are of interest for algorithms that select the kth smallest element in a heap. CR categories: E.1, F.2.2, G.2.1.

In this article, we prove the equivalence of dynamical stability, preperiodicity, and canonical height 0, for algebraic families of rational maps ft : P(C)→ P(C), parameterized by t in a quasi-projective complex variety. We use this to prove one implication in the if-and-only-if statement of [BD2, Conjecture 1.10] on unlikely intersections in the moduli space of rational maps; we present the co...