Normal Functions, Picard-fuchs Equations, and Elliptic Fibrations on K3 Surfaces

Using Gauss-Manin derivatives of generalized normal functions, we arrive at results on the non-triviality of the transcendental regulator for Km of a very general projective algebraic manifold. Our strongest results are for the transcendental regulator for K1 of a very general K3 surface and its self-product. We also construct an explicit family of K1 cycles on H ⊕ E8 ⊕ E8-polarized K3 surfaces, and show they are indecomposable by a direct evaluation of the real regulator. Critical use is made of natural elliptic fibrations, hypersurface normal forms, and an explicit parametrization by modular functions.