On a versal family of curves of genus two with p 2-multiplication

This note is a summary of our study on curves of genus two having real multiplication by the quadratic order of discriminant 8. We give an elementary and concrete description of the family of such curves, including the classical results of G. Humbert. While Humbert’s work were based on theta functions and the theory of Kummer surfaces, our study is based on algebraic correspondences on hyperelliptic curves which are the lifts of algebraic correspondences on a conic in P 2 associated with Poncelet’s quadrangle. Our main results are simple concrete description of the correspondences in the geometry of conics, and a proof that they induce the endomorphism ϕ on the jacobian satisfying ϕ 2 2 = 0. We also give a versal family of genus two curves having p 2-multiplication. The details of the proofs, as well as some applications, will appear elsewhere.