# Elliptic fibrations on covers of the elliptic modular surface of level 5

## TL;DR

This paper classifies elliptic fibrations on K3 surfaces obtained as double covers of the elliptic modular surface of level 5, providing an algorithmic approach to find Weierstrass equations for these fibrations.

## Contribution

It introduces a simple algorithm to classify elliptic fibrations on these K3 surfaces using linear systems, with explicit cases analyzed.

## Key findings

- Complete list of elliptic fibrations for specific branch cases.
- Algorithm for deriving Weierstrass equations from linear systems.
- Detailed classification of fibrations on double covers of $R_{5,5}$.

## Abstract

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.