Genus two curves with everywhere good reduction over quadratic fields

We address the question of existence of absolutely simple abelian varieties of dimension 2 with everywhere good reduction over quadratic fields. The emphasis will be given to the construction of pairs (K, C), where K is a quadratic number field and C is a genus 2 curve with everywhere good reduction over K. We provide the first infinite sequence of pairs (K, C) where K is a real (complex) quadratic field and C has everywhere good reduction over K. Moreover, we show that the Jacobian of C is an absolutely simple abelian variety.