Frobenius Map for Quintic Threefolds

Abstract

Abstract. We calculate the matrix of the Frobenius map on the middle dimensional cohomology of the one parameter family that is related by mirror symmetry to the family of all quintic threefolds.

Frobenius map on local Calabi–Yau manifolds

“…If ω is a holomorphic nowhere vanishing three-form on CY as above, then an explicit computation in the Dwork model for the cohomology shows that (10) ∇ λ∂ λ ω ∈ H 1,0 (C)…”

“…This extra data was obtained in [11] using the theory of motives. In [10] this analysis was performed explicitly for the case of the mirror quintic.…”

“…We point out however that the problem of calculating the Frobenius map at the boundary point is a separate issue that is not at all trivial. One can attempt to use explicit computations as was done in [10], or perhaps the theory of motives can not be avoided if we ask for a general enough answer.…”

“…Finally, in the appendix we provide a brief discussion of the modifications of this method necessary to handle the projective case that arises even in the present, seemingly affine, situation. The discussion is our interpretation of the Griffiths-Dwork method ( [3] and references therein) that we use in [10].…”

Introduction to Arithmetic Mirror Symmetry

“…which is a truncation of (3). In fact, the number of F p -rational points on X ψ can be written as a modulo p 5 expression (24) involving all of the solutions of (2), as well as an additional term arising from a so-called semi-period. Arithmetic of varieties appearing in the context of mirror symmetry can also be studied through their zeta functions, defined for a variety X over a finite field F q with q = p n elements by…”