Hypergeometric Motives from Euler Integral Representations

We revisit certain one-parameter families of affine covers arising naturally from Euler’s integral representation of hypergeometric functions. We introduce a partial compactification of this family. We show that the zeta function of the fibers in the family can be written as an explicit product of $L$-series attached to nondegenerate hypergeometric motives and zeta functions of tori, twisted by Hecke Grossencharacters. This permits a combinatorial algorithm for computing the Hodge numbers of the family.

💡 Research Summary

The paper revisits a family of affine cyclic covers that arise naturally from Euler’s integral representation of hypergeometric functions. Starting with integers (n,m\ge1) and parameter vectors (\mathbf a=(a_1,\dots,a_n)), (\mathbf b=(b_1,\dots,b_n)), the authors define the affine variety
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