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

Abstract

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.