SÉMINAIRE DE GÉOMÉTRIE

Vendredi, 23 Juin 2000 10:30 – 11:30 Salle 2 , Laboratoire MPB

Dino LORENZINI, University of Georgia

“ Thue equations and the method of Chabauty-Coleman ”

Résumé :

This talk is a report on joint work with Thomas Tucker. Let F (x, y) ∈ ℤ[x, y] be a homogenous polynomial of degree n, assumed to have unit content. Let h ∈ ℤ be such that the polynomial hz^{n} − F (x, y) is irreducible in \overline{ℚ}[x, y, z]. Let X_{F,h}/ℚ be the nonsingular complete model of the projective plane curve defined by the equation hz^{n} − F (x, y) = 0. Let N(F, h) denote the cardinality of the set {(x, y) ∈ ℤ^{2} | F (x, y) = h and gcd(x, y) = 1}. Let r(X_{F,h}) denote the Mordell-Weil rank over ℚ of the jacobian of X_{F,h}/ℚ. Assume that the genus g of X_{F,h} is at least 2. We show that, if r(X_{F,h}) < g, then N(F, h) ≤ 2n^{3} −2n−3. To prove this theorem, we suitably refine the method of Chabauty-Coleman so that it can be applied to any regular model of X_{F,h}/ℚ_{p} with p > n.