On Realizations of Hypergeometric Motives

In this thesis, we investigate some of the arithmetic and geometry underlying hypergeometric functions and their finite field analogues. Motivated by their appearance as periods of families of varieties and as traces of Frobenius in étale cohomology, we study the interplay between hypergeometric functions, and point-counting over finite fields. In Chapter 1, we prove that under mild assumptions, the complex hypergeometric local system coincides with the top weight variation of a family of hypersurfaces in a torus, thereby realizing classical hypergeometric functions as periods. In Chapter 2, we generalize point-counting results of Beukers, Cohen, and Mellit to a broader class of non-primitive toric hypersurfaces, expressing their counts in terms of finite hypergeometric functions. Finally, Chapter 3 formulates a conjecture on the complex variation of these families, and we verify it in some explicit example. These results yield new examples of L-functions as well as variations of Hodge structure and expand the dictionary between arithmetic geometry and hypergeometric phenomena.