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Elliptic Fibrations on Covers of the Elliptic Modular Surface of Level 5

2018; Springer International Publishing; Linguagem: Inglês

10.1007/978-3-319-74998-3_9

2364-5733

Francesca Balestrieri, Julie Desjardins, Alice Garbagnati, Céline Maistret, Cecília Salgado, Isabel Vogt,

Cryptography and Residue Arithmetic

We consider the K3 surfaces that arise as double covers of the elliptic modular surface of level 5, $R_{5,5}$. Such surfaces have a natural elliptic fibration induced by the fibration on $R_{5,5}$. Moreover, they admit several other elliptic fibrations. We describe such fibrations in terms of linear systems of curves on $R_{5,5}$. This has a major advantage over other methods of classification of elliptic fibrations, namely, a simple algorithm that has as input equations of linear systems of curves in the projective plane yields a Weierstrass equation for each elliptic fibration. We deal in detail with the cases for which the double cover is branched over the two reducible fibers of type $I_5$ and for which it is branched over two smooth fibers, giving a complete list of elliptic fibrations for these two scenarios.