Differential equations, hypergeometric families, and Beilinson's conjectures

We discuss two approaches to solving inhomogeneous equations of the form L(.)=t^{1/d}, where L is a hypergeometric differential operator attached to a family of CY varieties.  The first is by elementary complex analysis, using so-called Frobenius deformations, and gives an explicit series solution. The second is via normal functions attached to algebraic cycles (both "classical" and "higher") on a base-change of the family.

I will briefly review regulator maps, their relation to inhomogeneous Picard-Fuchs equations, and the relevant cases of the Beilinson conjecture.  Turning then to the CY3 examples classified by Doran-Morgan, I will explain how to identify which types of cycles arise (viz., K_0, K_2, or K_4 classes), and how to use the Frobenius deformations to make the conjecture more explicit.  This is joint work with Vasily Golyshev.