Hypergeometric Functions and Arithmetic Properties of Algebraic Varieties

Abstract

In this thesis, we investigate the relationship between special functions and arithmetic properties of algebraic varieties. More specifically, we use Greene's finite field hypergeometric functions to give point count formulas for families of algebraic varieties over finite fields. We demonstrate that this is possible for a family of genus 3 curves and for families of higher dimensional varieties called Dwork hypersurfaces. We work out the calculations in great detail for Dwork K3 surfaces over fields whose order is congruent to 1 modulo 4. Furthermore, for K3 surfaces, we also give point count formulas in terms of finite field hypergeometric functions defined by McCarthy. This allows us to give formulas that hold for all primes. Inspired by a result of Manin for curves, we study the relationship between certain period integrals and the trace of Frobenius of these varieties. We show that these can be expressed in terms of ``matching" classical and finite field hypergeometric functions. Through congruences between classical and finite field hypergeometric functions that we prove, we show that the fundamental period is congruent to the trace of Frobenius for Dwork K3 surfaces and conjecture that this is true for higher dimensional Dwork hypersurfaces.