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Differential equations, hypergeometric families, and Beilinson's conjectures

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Matthew Kerr
Washington University in St. Louis/MPIM
Wed, 26/07/2023 - 14:30 - 15:30
MPIM Lecture Hall
Parent event: 
Number theory lunch seminar

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.

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