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- hypergeometric functions over finite fields; algebraic curves
Classical hypergeometric functions and their relations to counting points on curves over finite fields have been investigated by mathematicians since the beginnings of 1900. In the mid 1980s, John Greene developed the theory of hypergeometric functions over finite fi elds. He explored the properties of these functions and found that they satisfy many summation and transformation formulas analogous to those satisfi ed by the classical functions. These similarities generated interest in finding connections that hypergeometric functions over finite fields may have with other objects. In recent years, connections between these functions and elliptic curves and other Calabi-Yau varieties have been investigated by mathematicians such as Ahlgren, Frechette, Fuselier, Koike, Ono and Papanikolas. A survey of these results is given at the beginning of this dissertation. We then introduce hypergeometric functions over finite fi elds and some of their properties. Next, we focus our attention on a particular family of curves and give an explicit relationship between the number of points on this family over Fq and sums of values of certain hypergeometric functions over Fq. Moreover, we show that these hypergeometric functions can be explicitly related to the roots of the zeta function of the curve over Fq in some particular cases. Based on numerical computations, we are able to state a conjecture relating these values in a more general setting, and advances toward the proof of this result are shown in the last chapter of this dissertation. We nish by giving various avenues for future study.