Variation of the unit root along the Dwork family of Calabi–Yau varieties

Abstract

We study the variation of the unit roots of members of the Dwork families of Calabi-Yau varieties over a finite field by the method of Dwork-Katz and also from the point of view of formal group laws. A p-adic analytic formula for the unit roots away from the Hasse locus is obtained.

Frobenius polynomials for Calabi–Yau equations

“…, j. For the Apery sequence (case b) this was also conjectured in [22]. (It follows from [10] that A, B, C, D satisfy the Dwork congruences).…”

“…(see [8,13,20,22] 1. An F -crystal over W (k) is a free W (k)-module H of finite rank with a σ-linear endomorphism…”

Monomial deformations of certain hypersurfaces and two hypergeometric functions

“…The first half of the theorem follows from Lemma 2.2.1 because a ≡ F 1,1 (λ −α ) (mod p). The proof of the second half goes exactly as in [Yu,p.76,proof of Theorem 4.3 (2)].…”

Zeta functions of alternate mirror Calabi–Yau families

“…In turn, they related the other nontrivial factors of Z(X ψ , T ) to the action of discrete scaling symmetries of the Dwork pencil F 5 on homogeneous monomials. In related work (but in a somewhat different direction), Jeng-Daw Yu [Yu08] showed that the unique unit root for the middle-dimensional factor of the zeta function for the Dwork family in dimension n can be expressed in terms of a ratio of holomorphic solutions of a hypergeometric Picard-Fuchs equation (evaluated at certain values).…”