A complete hypergeometric point count formula for Dwork hypersurfaces

Abstract

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 hypersurfacesin 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 d−1 − 1)/(q − 1) 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.

The number of -points on Dwork hypersurfaces and hypergeometric functions

“…When d | q − 1 it is possible express the results of Koblitz, and those in this paper, in terms of hypergeometric functions over finite fields, as defined by Greene [8], or using a normalized version defined by the author [14]. For example, see [7,16,17] for related results. To extend these results beyond q ≡ 1 (mod d) it is necessary to move to the padic setting as we have done in this paper.…”

Hypergeometric properties of genus 3 generalized Legendre curves

“…This phenomenon also occurs for families of curves not expressible in Legendre form over Q (see, for example, [7,8,15]). In fact, this phenomenon seems to extend to some higher dimensional Calabi-Yau manifolds as is shown in [1,9,10,17,21] leading us to wonder if this will be the case for a large class of algebraic varieties.…”

“…The classical and finite field hypergeometric expressions -including the two that are not covered by Theorem 2.2 -for the generalized Legendre curves "match" in the same way that period and trace of Frobenius expressions match for elliptic curves: we replace the fraction a b with a character of order b raised to the ath power. This phenomenon also seems to extend to some other curves (see [19]) and to higher dimensional Calabi-Yau manifolds (see, for example, [9,10,13,17,20]. By testing values in Sage [6], we know that it is not the case that congruences exist between arbitrary (matching) truncated hypergeometric series and finite field hypergeometric functions.…”

“…In [1], Ahlgren and Ono gave a formula for the number of F p points on a modular Calabi-Yau threefold. We extended this work in [9,10] by showing that the number of points on Dwork hypersurfaces over finite fields can be expressed in terms of Greene's finite field hypergeometric functions.…”