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Abstract:
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Hypergeometric functions seem to be ubiquitous in mathematics . In this document , we present a couple of ways in which hypergeometric functions appear in arithmetic geometry . First , we show that the number of points over a finite field [mathematical symbol] on a certain family of hypersurfaces , [mathematical symbol] ([lamda] ) , is a linear combination of hypergeometric functions . We use results by Koblitz and Gross to find explicit relationships , which could be useful for computing Zeta functions in the future . We then study more geometric aspects of the same families . A construction of Dwork's gives a vector bundle of deRham cohomologies equipped with a connection . This connection gives rise to a differential equation which is known to be hypergeometric . We developed an algorithm which computes the parameters of the hypergeometric equations given the family of hypersurfaces . |