20 september, Charles Favre (Université Denis Diderot): Equidistribution of points of small heights (joint work with Juan Rivera-Letelier)

Abstract. Let R be a rational map with complex coefficients of degree at least 2. For any point z on the Rieman sphere (with at most two exceptions), the sequence of measures equidistributed on the preimages of z under R^n converges to a measure m_R which is independent on z. Our aim is to present an arithmetic analog of this result. Suppose R has rational coefficients. It is then possible to construct a so-called height function h_R on the algebraic closure of the field of rational numbers. Our main result states that for any sequence of points z_n such that h_R(z_n) tends to 0, the sequence of measures equidistributed on the Galois conjugates of z_n converges to the measure m_R. The proof of this result relies on basic properties of the Laplace operator on the complex plane, as well as on the affine line associated to the p-adic field C_p.