Matt Kerr

Algebraic K-theory of Toric Hypersurfaces

We describe how to use toric data to construct relative higher Chow cycles for families of Calabi-Yau (n-1)-folds. The regulator "periods" of such elements, which may be regarded as generalized "normal functions", satisfy an inhomogeneous Picard-Fuchs equation related to the Yukawa coupling. This setting leads to motivic proofs of irrationality of zeta(2) and zeta(3) and unusual series expressions for arithmetic constants. We will also explain relations to Mahler measure, modular forms, and local mirror symmetry. The first half of the talk will feature a detailed geometric example for an elliptic curve family, emphasizing the direct relationship between regulators on higher Chow cycles and Griffiths's Abel-Jacobi mapping for usual algebraic cycles.