Poncelet’s Theorem and Curves of Genus Two With Real Multiplication of Δ=5

Curves of genus two having real multiplication by the quadratic order of discriminant 5 are discussed, on which a completely elementary theory is established, including the classical results of G. Humbert. Namely we study algebraic correspondences on hyperelliptic curves which are the lifts of algebraic correspondences on a conic in ℙ² associated with Poncelet’s pentagon. Our main results are simple concrete description of the correspondences in the geometry of conics, and a proof that they induce the endomorphism ∅ on the jacobian satisfying ∅²+∅-1=0. We next study the modular equation of Humbert for the case of discriminant 5, and prove the rationality of the hypersurface defined by this equation. We also study Mestre’s family of curves with real multiplication by the quadratic order of discriminant 5 in detail, and prove the versality of it as well as its new derivation.