13th December: Alexandros Constantinou

Title: Elliptic surfaces and the Néron-Severi group

Abstract: Elliptic surfaces are ubiquitous in the study of algebraic surfaces. They arise naturally when studying families of elliptic curves, or equivalently varieties with defining equation E_t: Y^2=X^3+a_4(t)X+a_6(t), where the coefficients a_4 and a_6 depend on some parameter t which varies over some base curve. This can equivalently be regarded as a single elliptic curve over the function field of the base curve, and also as an elliptic surface admitting a map to the base curve. The main theme of this talk is to study the interaction between these distinct points of view, while giving an informal overview of the main results related to their arithmetic properties. Examples will be discussed whenever they become available.