Genus 2 curves with (3,3)-split Jacobian and large automorphism group

Tanush Shaska

I-AIMS paper 2002-1 · Published in ANTS, Lecture Notes in Comput. Sci. 2369, Springer (2002), 205–218 · 2002

Let C be a genus 2 curve defined over k, char(k) = 0. If C has a (3,3)-split Jacobian then we show that the automorphism group Aut(C) is isomorphic to one of the following: ℤ₂, V₄, D₈, or D₁₂. There are exactly six ℂ-isomorphism classes of genus two curves C with Aut(C) isomorphic to D₈ (resp., D₁₂) and with (3,3)-split Jacobian. We show that exactly four (resp., three) of these classes with group D₈ (resp., D₁₂) have representatives defined over ℚ. We discuss some of these curves in detail and find their rational points.

MSC classes: Primary 14H37; Secondary 14H45

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