Arithmetic geometric approach for period integrals

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We study period integrals of algebraic varieties from the view point of algebraic correspondence, algebraic cycles and special functions such as automorphic functions. By applying the method of bar construction to algebraic varieties over positive characteristic base field, we describe their p-fundamental group, and establish a basic tools for Hodge realization functors using Deligne differential graded algebras. We also prove some part of Beilinson conjectures concerning mixed elliptic motives. Concerning automorphic functions, we stablish several new Thomae type theorems related to K3 surfaces, etc.