I‑AIMS Institute for Artificial Intelligence
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Genus 2 curves that admit a degree 5 map to an elliptic curve

Kay Magaard, Tanush Shaska, Helmut Völklein

I-AIMS paper 2005-1 · Published in Forum Math. 21 (2009), 547–566 · 2009

We continue our study of genus 2 curves C that admit a cover C → E to a genus 1 curve E of prime degree n. These curves C form an irreducible 2-dimensional subvariety ℒₙ of the moduli space ℳ₂ of genus 2 curves. Here we study the case n = 5. This extends earlier work for degree 2 and 3, aimed at illuminating the theory for general n. We compute a normal form for the curves in the locus ℒ₅ and its three distinguished subloci. Further, we compute the equation of the elliptic subcover in all cases, give a birational parametrization of the subloci of ℒ₅ as subvarieties of ℳ₂ and classify all curves in these loci which have extra automorphisms.

MSC classes: Primary 14H10, 14H30; Secondary 14H45, 14H35

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