p-ADIC CONSTRUCTION OF THE HODGE CYCLE - p-ADIC HODGE THEORY AND BORCHERDS PRODUCTS

We construct the algebraic cycle attached to a Hodge class. For divisor Hodge loci we use Borcherds products where the Frobenius unit satisfies u=1. In general we use integrality, Kedlaya unit-root extension, Crew, Fontaine-Laffaille, rigid maximum principle and norm descent to obtain ord_D(u)=0. Combined with pλ=λ⇒λ=0 and weight rigidity this yields flatness and algebraicity via G-functions.