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.