2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/149756We survey some aspects of the theory of elliptic surfaces and give some results aimed at determining the Picard number of such a surface. For the surfaces considered, this will be equivalent to determining the Mordell-Weil rank of an elliptic curve defined over a function field in one variable. An interesting conjecture concerning Galois actions on the relative de~Rham cohomology of these surfaces is discussed.12 pages, Plain TeXAlgebraic Geometrytext