A lower bound for the Mordell--Weil rank of the generic fiber of an ellipric K3 surface given by the ramified double cover of a particular rational elliptic surface

Abstract

K3 surfaces are a very relevant field of research, being in the intersection between Complex Geometry, Algebraic Geometry and Arithmeric Geometry. They also appear in some research works in String Theory in Physics. K3 surfaces are Calabi--Yau varieties of dimension 2 and a natural analog of Elliptic Curves in dimension 2. Some of their algebro-geometrical properties are notably difficult to compute, in particular their Picard numbers and the behavior of the Picard numbers of familoes of K3 surfaces. K3 surfaces have an interesting relationship with elliptic curves. In particular, every K3 surface with Picard number at least 5 has an elliptic fibration, known as an elliptic K3 surface. Shioda--Tate formula shows an outstanding relation between the arithmetic of elliptic curves and the geometry of elliptic K3 surfaces, and is an important tool to study the Picard of these surfaces. In this work we study a special case of K3 surfaces, and a lower bound for its Picard number is computed using an elliptic fibration.