Fields of definition of elliptic fibrations on covers of certain extremal rational elliptic surfaces

K3 surfaces have been extensively studied over the past decades for several reasons. For once, they have a rich and yet tractable geometry and they are the playground for several open arithmetic questions. Moreover, they form the only class which might admit more than one elliptic fibration with section. A natural question is to ask if one can classify such fibrations, and indeed that has been done by several authors, among them Nishiyama, Garbagnati and Salgado. The particular setting that we were interested in studying is when a K3 surface arises as a double cover of a extremal rational elliptic surface with a unique reducible fiber. This K3 surface will have a non-symplectic involution τ fixing two smooth Galois-conjugate genus 1 curves. In this joint project we determine the fields of definition of the different fibrations on this K3 surface, depending on τ. Moreover, we determine as well the field of definition of the Mordell–Weil group of each fibration. This is a joint work with A. Garbagnati, C. Salgado, A. Trbovi ́c and R. Winter.