A Complete Hypergeometric Point Count Formula for Dwork Hypersurfaces
Abstract
We extend our previous work on hypergeometric point count formulas by proving that we can express the number of points on families of Dwork hypersurfaces $$X_{\lambda}^d: \hspace{.1in} x_1^d+x_2^d+\ldots+x_d^d=d\lambda x_1x_2\cdots x_d$$ over finite fields of order $q\equiv 1\pmod d$ in terms of Greene's finite field hypergeometric functions. We prove that when $d$ is odd, the number of points can be expressed as a sum of hypergeometric functions plus $(q^{d1}1)/(q1)$ and conjecture that this is also true when $d$ is even. The proof rests on a result that equates certain Gauss sum expressions with finite field hypergeometric functions. Furthermore, we discuss the types of hypergeometric terms that appear in the point count formula and give an explicit formula for Dwork threefolds.
 Publication:

arXiv eprints
 Pub Date:
 October 2016
 DOI:
 10.48550/arXiv.1610.09754
 arXiv:
 arXiv:1610.09754
 Bibcode:
 2016arXiv161009754G
 Keywords:

 Mathematics  Number Theory;
 11T24;
 11G25;
 33C20
 EPrint:
 Typos corrected and references updated. To appear in Journal of Number Theory, Volume 179, October 2017