Zeta Functions of Monomial Deformations of Delsarte Hypersurfaces

Abstract

Abstract. Let X λ and X λ be monomial deformations of two Delsarte hypersurfaces in weighted projective spaces. In this paper we give a sufficient condition so that their zeta functions have a common factor. This generalises results by Doran, Kelly, Salerno, Sperber, Voight and Whitcher [arXiv:1612.09249], where they showed this for a particular monomial deformation of a Calabi-Yau invertible polynomial. It turns out that our factor can be of higher degree than the factor found in [arXiv:1612.09249].

Zeta functions of alternate mirror Calabi–Yau families

“…Theorem 5.1.3 implies that the subspace in cohomology cut out by the Picard-Fuchs equation is contained in the SL(F A )-invariant subspace and it contains H 2,0 . Consequently, as observed by Kloosterman [Kl17], this implies that the SL(F A )-invariant subspace in H 2 et (X A,ψ ) contains the transcendental subspace: indeed, one definition of the transcendental lattice of a K3 surface is as the minimal primitive sub-Q-Hodge structure containing H 2,0 [Huy16, Definition 3.2.5].…”

“…The complete Néron-Severi lattice of rank 19 for the case of the Dwork pencil F 4 is worked out via transcendental techniques by Bini-Garbagnati [BG14,§4]. It would be interesting to compute the full Néron-Severi lattices for the remaining four plus two families; Kloosterman [Kl17] has made some recent progress on this question and in particular has also shown (by a count of divisors) that the generic Néron-Severi rank is 16 for the C 2 F 2 and C 2 L 2 pencils.…”

Counting points with Berglund-Huebsch-Krawitz mirror symmetry

“…In Section 4, we use intuition from mirror symmetry to identify common factors in zeta functions of different Calabi-Yau pencils. Our exposition follows [DKSSVW18], though we also discuss work of Kloosterman in [Klo18]. We focus on a specific set of K3 surface examples in Section 5; we exploit the hypergeometric structure of Picard-Fuchs equations and point counts to describe an explicit motivic deconstruction for these families, following [DKSSVW20].…”

“…Kloosterman showed in [Klo18] that one can extend Theorem 4.1 to a broader class of hypersurfaces: he allows for invertible polynomials A and B that do not necessarily satisfy the Calabi-Yau condition, and permits more general one-parameter monomial deformations. The resulting common factor of P X A,ψ (T ) and P X B,ψ (T ) may be of larger degree than the common factor identified in Theorem 4.1.…”

Hasse--Witt matrices and mirror toric pencils

“…In particular, [DKSSVW18] uses BHK mirrors to identify pencils of Calabi-Yau varieties in P n that share common Picard-Fuchs equations and common factors in their zeta functions. Kloosterman further explored this phenomenon in [Klo18], using a geometric approach. Both [Kad06] and [CDRV01] use the Batyrev mirror construction and techniques of toric varieties for a more detailed analysis of the Greene-Plesser mirror.…”