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

Banff International Research Station for Mathematical Innovation and Discovery

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 an extremal rational elliptic surface with a unique reducible fiber. This K3 surface will have a non-symplectic involution Ï fixing two smooth Galois-conjugate genus 1 curves. In this joint work we provide a list of all elliptic fibrations on those K3 surfaces together with the degree of a field extension over which each genus one fibration is defined and admits a section. We show that the latter depends, in general, on the action of the cover involution Ï on the fibers of the genus 1 fibration. This is a joint work with Alice Garbagnati, Cecília Salgado, Antonela Trbovíc and Rosa Winter.

Mathematics; Number theory; Algebraic geometry; Arithmetic number theory

video/mp4

Author affiliation: Katholieke Universiteit Leuven

2021-03-02

Attribution-NonCommercial-NoDerivatives 4.0 International

http://hdl.handle.net/2429/77419

Unreviewed

http://creativecommons.org/licenses/by-nc-nd/4.0/