Monomial deformations of certain hypersurfaces and two hypergeometric functions

Abstract

The purpose of this article is to give an explicit description, in terms of hypergeometric functions over finite fields, of zeta function of a certain type of smooth hypersurfaces that generalizes Dwork family. The point here is that we count the number of rational points employing both character sums and the theory of weights, which enables us to enlighten the calculation of the zeta function.

Counting points with Berglund-Huebsch-Krawitz mirror symmetry

“…where the parameters α i and β j depend only on the dual weights q i . This follows from results of Miyatani in [Miy15] when X A,ψ is smooth and ψ = 0, or Adolphson and Sperber in [AS16], in general. Thus, there is a truncated hypergeometric formula for #X ψ (F q ) (mod q), generalizing the results of Igusa for the Legendre family.…”

A complete hypergeometric point count formula for Dwork hypersurfaces

“…Note that this conjecture has been verified for d = 4, i.e. for Dwork K3 surfaces (see [10]), and should follow more generally from McCarthy's work in [23] and Miyatani's work in [27]. Furthermore, in Section 4.2 we discuss when certain types of hypergeometric terms will appear in the point count formulas.…”

Zeta functions of alternate mirror Calabi–Yau families

“…Therefore, the ordinary sublocus of P 1 {0, 1, ∞} is a nonempty Zariski open subset. This unit root has seen much study: for the Dwork family, it was investigated by Jeng-Daw Yu [Yu08], and in this generality by Adolphson-Sperber [AS16, Proposition 1.8] (see also work of Miyatani [Miy15]).…”