Zeta functions of alternate mirror Calabi–Yau families

We prove that if two Calabi–Yau invertible pencils have the same dual weights, then they share a common factor in their zeta functions. By using Dwork cohomology, we demonstrate that this common factor is related to a hypergeometric Picard–Fuchs differential equation. The factor in the zeta function is defined over the rationals and has degree at least the order of the Picard–Fuchs equation. As an application, we relate several pencils of K3 surfaces to the Dwork pencil, obtaining new cases of arithmetic mirror symmetry.


Introduction
1.1. Motivation. For a variety X over a finite field F q , the zeta function of X is the exponential generating function for the number of F q r -rational points, given by In his study of the Weil conjectures, Dwork analyzed the way the zeta function varies for one-parameter deformations of Fermat hypersurfaces in projective space, like the pencil (1.1.1) x n+1 0 + · · · + x n+1 n − (n + 1)ψx 0 x 1 · · · x n = 0 in the parameter ψ. In his 1962 ICM address [Dwo62], Dwork constructed a family of endomorphisms whose characteristic polynomials determined the zeta functions of the hypersurfaces modulo p. Furthermore, he identified a power series in the deformation parameter with rational function coefficients that satisfies an ordinary differential equation with regular singular points. In fact, this differential equation is the Picard-Fuchs equation for the holomorphic differential form [Kat68]. The pencil (1.1.1) is a central example in both arithmetic and algebraic geometry [Kat09]; we label this family F n+1 . On the arithmetic side, Dwork [Dwo69] analyzed F 4 in detail to explore the relationship between the Picard-Fuchs differential equation satisfied by the holomorphic form on the family and the characteristic polynomial of Frobenius acting on middle-dimensional cohomology. Dwork identifies the reciprocal zeros of the zeta function for this family of K3 surfaces explicitly by studying p-adic solutions of the Picard-Fuchs equation. This analysis motivated Dwork's general study of p-adic periods.
On the algebraic side, the family of Calabi-Yau threefolds F 5 has been used to explore the deep geometric relationship known as mirror symmetry. Mirror symmetry is a duality from string theory that has shaped research in geometry and physics for the last quarter-century. Loosely defined, it predicts a duality where, given a Calabi-Yau variety X, there exists another Calabi-Yau variety Y , the mirror, so that various geometric and physical data is exchanged. For example, Candelas-de la Ossa-Green-Parkes [CDGP91] showed that the number of rational curves on quintic threefolds in projective space can be computed by studying the mirror family, realized via the Greene-Plesser mirror construction [GP90] as a resolution of a finite quotient of F 5 .
Combining both sides, Candelas, de la Ossa, and Rodriguez-Villegas used the Greene-Plesser mirror construction and techniques from toric varieties to compare the zeta function of fibers X ψ of F 5 and the mirror pencil of threefolds Y ψ [CDRV00, CDRV01,CD08]. They found that for general ψ, the zeta functions of X ψ and Y ψ share a common factor related to the period of the holomorphic form on X ψ . 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).
The Dwork pencil F 5 is not the only highly symmetric pencil that may be used to construct the mirror to quintic threefolds. In fact, there are six different pencils of projective Calabi-Yau threefolds, each admitting a different group action, that yield such a mirror: these pencils were studied by Doran-Greene-Judes [DGJ08] at the level of Picard-Fuchs equations. Bini-van Geemen-Kelly [BvGK12] then studied the Picard-Fuchs equations for alternate pencils in all dimensions.
A general mechanism for finding alternate mirrors is given by the framework of Berglund-Hübsch-Krawitz (BHK) duality. This framework identifies the mirrors of individual Calabi-Yau varieties given by invertible polynomials, or more generally of invertible pencils, the one-parameter monomial deformation of invertible polynomials; these notions are made precise in the next section. Aldi-Peruničić [AP15] have studied the arithmetic nature of invertible polynomials via D-modules.
In this paper, we show that invertible pencils whose mirrors have common properties share arithmetic similarities as well. Revisiting work of Gährs [Gäh11], we find that invertible pencils whose BHK mirrors are hypersurfaces in quotients of the same weighted projective space have the same Picard-Fuchs equation associated to their holomorphic form. In turn, we show that the Picard-Fuchs equations for the pencil dictate a factor of the zeta functions of the pencil. We then show that the factor of the zeta function is bounded by the degree of the Picard-Fuchs equation and the dimension of the piece of the middle cohomology that is invariant under the action of a finite group of symmetries fixing the holomorphic form.

Main theorem. An invertible polynomial is a polynomial of the form
where the matrix of exponents A = (a ij ) i,j is an (n + 1) × (n + 1) matrix with nonnegative integer entries, such that: • det(A) = 0, • there exist r 0 , . . . , r n ∈ Z >0 and d ∈ Z such that n j=0 r j a ij = d (i.e., the polynomial F A is quasi-homogeneous), and • the function F A : C n+1 → C has exactly one singular point at the origin.
We will be particularly interested in the case where F A is invertible and homogeneous of degree d = n + 1: then the hypersurface defined by F A = 0 defines a Calabi-Yau variety in P n .
These conditions are restrictive. In fact, Kreuzer-Skarke [KS92] proved that any invertible polynomial F A (x) can be written as a sum of polynomials, each of which belongs to one of three atomic types, known as Fermat, loop, and chain: Fermats : x a , loops : x a1 1 x 2 + x a2 2 x 3 + · · · + x am−1 m−1 x m + x am m x 1 , and chains : x a1 Let F A be an invertible polynomial. Inspired by Berglund-Hübsch-Krawitz (BHK) mirror symmetry [BH93,Kra09], we look at the polynomial obtained from the transposed matrix A T : Then F A T is again an invertible polynomial, quasihomogeneous with (possibly different) weights q 0 , . . . , q n for which we may assume gcd(q 0 , . . . , q n ) = 1, so that F A T = 0 defines a hypersurface X A T in the weighted-projective space W P n (q 0 , . . . , q n ). We call q 0 , . . . , q n the dual weights of F A . Let d T := i q i be the sum of the dual weights.
We define a one-parameter deformation of our invertible polynomial by Then X A,ψ : F A,ψ = 0 is a family of hypersurfaces in P n in the parameter ψ, which we call an invertible pencil. The Picard-Fuchs equation for the family X A,ψ is determined completely by the (n+1)-tuple of dual weights (q 0 , . . . , q n ) by work of Gährs [Gäh11, Theorem 3.6]. In particular, there is an explicit formula for the order D(q 0 , . . . , q n ) of this Picard-Fuchs equation that depends only on the dual weights: see Theorem 4.1.3 for details. We further observe that the Picard-Fuchs equation is a hypergeometric differential equation.
For a smooth projective hypersurface X in P n , we have Our main result is as follows (for the notion of nondegenerate, see section 2).
Theorem 1.2.3: Let X A,ψ and X B,ψ be invertible pencils of Calabi-Yau (n − 1)-folds in P n . Suppose A and B have the same dual weights (q i ) i . Then for each ψ ∈ F q such that gcd(q, (n + 1)d T ) = 1 and the fibers X A,ψ and X B,ψ are nondegenerate and smooth, the polynomials P X A,ψ (T ) and P X B,ψ (T ) have a common factor We show that the common factor R ψ (T ) is attached to the holomorphic form on X A,ψ and X B,ψ , explaining the link to the Picard-Fuchs differential equation: it is given explicitly in terms of a hypergeometric series (4.1.8). For this reason, if we had an appropriate theorem for rigidity of hypergeometric motives, we could further conclude that there exists a factor of degree precisely D(q 0 , . . . , q n ) in Q Our proof of Theorem 1.2.3 uses the p-adic cohomology theory of Dwork, as developed by Adolphson-Sperber [AS89,AS08], relating the zeta function of a member of the family to the L-function of an exponential sum. Our main theorem then follows from a result of Dwork [Dwo89] on the uniqueness of the Frobenius structure on the differential equation and the fact that the Picard-Fuchs equations for the holomorphic forms of X A,ψ and X B,ψ coincide.
Theorem 1.2.3 overlaps work of Miyatani [Miy15,Theorem 3.7]. In our notation, his theorem states that if X A,ψ is an invertible pencil, q satisfies certain divisibility conditions depending on A, and ψ ∈ F × q is such that X A,ψ is smooth and ψ d T = 1, then P X A,ψ (T ) has a factor in Q[T ] that depends only on q and the dual weights (q i ) i . In particular, if A and B have the same dual weights, the zeta functions of X A,ψ and X B,ψ (for ψ satisfying these conditions) will have a common factor in Q[T ]. His factor [Miy15, (2.4), Remark 3.8(i)] divides the common factor appearing in Theorem 1.2.3. He uses finite-field versions of Gauss sums together with a combinatorial argument.
To compare these two theorems, we observe that Theorem 1.2.3 provides slightly more information about the common factor and places fewer restrictions on q: for arithmetic applications, it is essential for the result that it hold without congruence conditions on q. Our techniques are different, and are ruled by the powerful governing principle that factors of the zeta function are organized by Picard-Fuchs differential equations. For example, our method could extend to pencils for which the associated differential equation may not be hypergeometric.
1.3. Implications. Theorem 1.2.3 relates the zeta functions of many interesting Calabi-Yau varieties: for example, the dual weights are the same for any degree n + 1 invertible pencil composed of Fermats and loops. For specificity, we compare the zeta functions of the Dwork pencil F n and the generalized Klein-Mukai family F 1 L n , defined by the pencil The pencil takes its name from Klein's quartic curve, whose group of orientationpreserving automorphisms is isomorphic to the simple group of order 168, and the member of the family F 1 L 3 at ψ = 0, which appears as an extremal example during Mukai's classification of finite groups of automorphisms of K3 surfaces that preserve a holomorphic form (cf. [Lev99,Muk88,OZ02]). In this setting, we give a concrete proof of Theorem 1.2.3. We also consider a collection of five invertible pencils of K3 surfaces in P 4 , including F 4 and F 1 L 3 . The other three pencils, F 2 L 2 , L 2 L 2 , and L 4 , also have only Fermats and loops as atomic types; all five are described by matrices with the same dual weights (see Table (5.1.1) for defining polynomials). Let H be the Greene-Plesser mirror family of quartics in P 3 , which is obtained by taking the fiberwise quotient of F 4 by (Z/4Z) 2 and resolving singularities. A computation described by Kadir [Kad04,Chapter 6] shows that for odd primes and ψ ∈ F q such that ψ 4 = 1 (that is, such that H ψ is smooth), This calculation combined with Theorem 1.2.3 and properties of K3 surfaces yields the following corollary, exemplifying arithmetic mirror symmetry in these cases.
Then there exists r 0 ≥ 1 such that for all q = p r with r 0 | r and p = 2, 5, 7 and all ψ ∈ F q with ψ 4 = 1, we have Accordingly, we could say that the zeta functions Z(X ,ψ /F q , T ) and Z(H ψ /F q , T ) are potentially equal-i.e., they are equal after a finite extension of F q . (The explicit value of r 0 in Corollary 1.3.3 will be computed in future work [DKSSVW].) Finally, we remark on a simple relationship between the numbers of points of members of alternate mirror families over F q , reminiscent of the strong arithmetic mirror symmetry studied by Fu-Wan [FW06], Wan [Wan06], and Magyar-Whitcher [MW16]. Corollary 1.3.4: Let X A,ψ and X B,ψ be invertible pencils of Calabi-Yau (n − 1)-folds in P n such that A, B have the same dual weights. Then for all ψ ∈ F q , Corollary 1.3.4 is slightly more general than Theorem 1.2.3-there is no hypothesis on the characteristic or on the smoothness of the fiber-but it arrives at a weaker conclusion.
1.4. Plan of paper. In section 2, we introduce our cohomological setup. In section 3, we consider first the generalized Klein-Mukai family as a warmup to the main theorem, giving a detailed treatment in this case. In section 4, we prove the main result by recasting a result of Gährs [Gäh13] on Picard-Fuchs equations in hypergeometric terms, study the invariance under symmetry of the middle cohomology, and then apply Dwork cohomology. To conclude, in section 5, we specialize to the case of K3 surfaces and give some further details for several pencils of particular interest.
Acknowledgements. The authors heartily thank Marco Aldi, Amanda Francis, Xenia de la Ossa, Andrija Peruničić, and Noriko Yui for many interesting discussions, as well as Alan Adolphson, Remke Kloosterman, Yang Liping, Fernando Rodriguez-Villegas, Duco van Straten, and the anonymous referee for helpful comments. They thank the American Institute of Mathematics and its SQuaRE program, the Banff International Research Station, the Clay Mathematics Institute, MATRIX in Australia, and SageMath for facilitating their work together. Doran was supported by NSERC and the Campobassi Professorship at the University of Maryland. Kelly acknowledges that this material is based upon work supported by the NSF under Award No. DMS-1401446 and the EPSRC under EP/N004922/1. Voight was supported by an NSF CAREER Award (DMS-1151047).

Cohomological setup
We begin in this section by setting up notation and establishing a few basic results. In the cohomology theory of Dwork, following the approach for related exponential sums as developed by Adolphson-Sperber [AS89, AS08], we will define cohomology spaces endowed with a Frobenius operator with the property that the middle-dimensional primitive factor of the zeta function is realized as the characteristic polynomial of the Frobenius operator acting on non-vanishing cohomology. We refer to the work of Adolphson-Sperber for further reference and to Sperber-Voight [SV13] for an algorithmic framing.
Throughout the paper, let F q be a finite field with q elements and characteristic p, with q = p a . Let F q be an algebraic closure of F q .

Nondegeneracy and convenience. Let
be a nonconstant homogeneous polynomial, so that the vanishing of F (x) defines a projective hypersurface X ⊆ P n Fq . Using multi-index notation, we write Definition 2.1.1: We say F is nondegenerate (with respect to its Newton polyhedron Δ) if for all faces τ ⊆ Δ (including τ = Δ), the system of equations In this case, with F homogeneous, the definition employed by Adolphson and Sperber, that F is nondegenerate (with respect to Δ ∞ (F )) requires that the system of equations for every face τ ⊆ Δ, (including τ = Δ). Note that when the characteristic p does not divide the degree of F , the Euler relation ensures the two definitions are equivalent. Finally, we observe that if w is a new variable and we consider the form wF , then wF is nondegenerate with respect to Δ ∞ (wF ) if and only if F is nondegenerate with respect to its Newton polyhedron Δ.
In the calculations below we will make use of a certain positioning of coordinates. For a subset J ⊆ {x 0 , . . . , x n } of variables, we let F ¡ J be the polynomial obtained from F by setting the variables in J equal to zero.
Definition 2.1.4: We say that F is convenient with respect to a subset S ⊆ {x 0 , . . . , x n } provided that for all subsets J ⊆ S, we have 2.2. Dwork cohomology. Let G m be the multiplicative torus (so G m (F q ) = F × q ) and fix a nontrivial additive character Θ : F q → C × of F q . Denote by Tr F q r /Fq : F q r → F q the field trace. We will effectively study the important middle-dimensional factor of the zeta function by considering an appropriate exponential sum on G s m × A n+1−s and treating toric and affine variables somewhat differently. For r ∈ Z ≥1 , define where the sum runs over all n+1-tuples x = (x 0 , . . . , x n ) where x 0 , . . . , x s−1 ∈ F × q r and x s , . . . , x n ∈ F q r . Consider the L-function of the exponential sum associated to F defined by is a rational function in T with coefficients in the cyclotomic field Q(ζ p ), where ζ p is a primitive pth root of unity.
Theorem 2.2.1 ([AS89, Theorem 2.9, Corollary 2.19]): If F is nondegenerate and convenient with respect to S = {x s+1 , . . . , x n }, and dim Δ ∞ (F ) = n + 1, then the L-function is a polynomial in T with coefficients in Q(ζ p ) of degree given explicitly in terms This theorem also gives information about the p-adic size of the reciprocal zeros of L(T ) (−1) n+1 .
We now proceed to relate the L-function of such an exponential sum to the zeta function of the corresponding hypersurface. In general, we write representing the characteristic polynomial of Frobenius acting on the primitive middle-dimensional cohomology of X. Let Y ⊆ A n+1 be the affine hypersurface defined by the vanishing of F , the cone over X. Let w be a new variable. A standard argument with character sums shows that On the other hand, one has So putting these together we have By combining Equations (2.2.2) and (2.2.5), we have Finally, splitting the domain for the variable w as In the special case where F is nondegenerate with respect to Δ ∞ (F ) and convenient with respect to {x 0 , . . . , x n }, Theorem 2.2.1 applies. Under these hypotheses, Adolphson-Sperber [AS89, Section 6] prove the following: there exists a p-adic cohomology complex Ω • such that the trace formula and finally For more details, see also Adolphson-Sperber [AS08, Corollary 6.23] and Sperber-Voight [SV13, Section 1 and pages 31-32]. In particular, the formula (2.2.10) gives a fairly direct way to compute P (T ) in the case of the Dwork family of hypersurfaces, since the defining polynomial F is convenient with respect to the full set of variables {x 0 , . . . , x n }.
2.3. Unit roots. For convenience, we conclude this section by recalling the relationship between Hodge numbers and the p-adic absolute values of the reciprocal zeros and poles of the zeta function.
The following is a consequence of the Katz conjecture proved in full generality by Mazur [Maz72]. In the present context, it follows directly from Adolphson-Sperber [AS89, Theorem 3.10]. We now apply this to our invertible pencils, as defined in (1.2.1). In particular, we have X A,ψ a smooth projective hypersurface in P n defined by a polynomial F A,ψ of degree n + 1, so X A,ψ is a Calabi-Yau variety of dimension n − 1. By a standard calculation, the first Hodge number of X A,ψ is h 0,n−1 = 1. Therefore the Hodge polygon of middle-dimensional primitive cohomology starts with a segment of slope zero having length 1. By Proposition 2.3.1, there is at most one reciprocal root of the polynomial P ,ψ,q (T ) that is a p-adic unit: we call this reciprocal root when it occurs a unit root.
Example 2.3.2: If n = 3 and deg F = 4, and X is smooth, then X is a quartic K3 surface, and so the Newton polygon of P ,ψ,q (T ) lies over the Hodge polygon (i.e., the Newton polygon of ( There is a polynomial defined over F p depending on A, called the Hasse invariant, with the property that H A (ψ) = 0 for a smooth fiber ψ ∈ F × q if and only if there is a unique unit root. In this case, we call X A,ψ ordinary, otherwise we say X A,ψ is supersingular. The polynomial H A is nonzero as the monomial x 0 x 1 . . . x n appears in F A,ψ [AS16, (1.9), Example 1] (in their notation, we have μ = 0). 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]).
These p-adic estimates can be seen explicitly in Dwork cohomology, as follows. By (2.2.10), for the hypersurface X A,ψ we are interested in the action of q −1 Frob on the cohomology group H n+2 (Ω • ).
Lemma 2.3.3: The operator q −1 Frob acting on H n+2 (Ω • ) reduced modulo p has rank at most 1 and has rank exactly 1 if and only if X A,ψ is ordinary.
Proof. The cohomology group H n+2 (Ω • ) has a basis of monomials By the Calabi-Yau condition, the unique monomial with |μ|/d = 1 is so (2.3.5) implies that the matrix q −1 A has at most one nonzero row modulo p, so its reduced rank is at most 1; and this rank is equal to 1 if and only if the fiber X A,ψ is ordinary, which occurs if and only if where ν = (1, 1, . . . , 1) corresponds to ω 0 . The reduced polynomialā ν,ν is thus the Hasse invariant for the given family.

Generalized Klein-Mukai family
As a warm-up to the main theorem, we now consider in detail the generalized Klein-Mukai family F 1 L n of Calabi-Yau n-folds. We give a proof of the existence of a common factor-realizing these as alternate mirrors, from the point of view of p-adic cohomology. Since it is of particular interest, and has rather special features, along the way we provide further explicit details about this family.
3.1. Basic properties. For n ≥ 1, let and define X ψ ⊆ P n to be the generalized Klein-Mukai family of hypersurfaces over Z defined by the vanishing of F ψ . The polynomial (3.1.1) of degree n + 1 in n + 1 variables may be described as consisting of a single Fermat term together with a single loop of length n, so we will also refer to it by the symbol F 1 L n . Throughout, let m := n n + (−1) n+1 . Note (n + 1) | m. Let k be a field and ζ ∈ k a primitive mth root of unity.
The subgroup acting trivially on X ψ is cyclic of order n + 1, and the quotient acting faithfully on X ψ is generated by z n+1 .
Proof. This statement follows from a direct computation.
Lemma 3.1.3: Suppose p m. Then for all ψ ∈ F q such that ψ n+1 = 1, the hypersurface defined by F ψ (x) is smooth, nondegenerate, and convenient with respect to {x n }.
Proof. The statement on convenience is immediate. We begin with the full face Δ, where nondegeneracy (using the Euler relation) is equivalent to smoothness. We compute for i = 0, . . . , n − 1 that with indices taken modulo n, and Setting these partials to zero and subtracting (3.1.5) from (3.1.4), we obtain the n × (n + 1)-matrix equation The absolute value of the determinant of the left n × n block of the matrix in (3.1.6) is m = n n + (−1) n+1 , so by our assumption on p the full matrix has rank n over F q . By homogeneity, the vector (1, . . . , 1) t therefore generates the kernel of the full matrix; the solution vector lies in this kernel, so we conclude , by scaling we may assume x n = 1. Thus x n i−1 x i = 1 for i = 1, . . . , n − 1; taking the product of these gives (x 0 · · · x n−1 ) n+1 = 1.
Since ψx 0 · · · x n = 1 as well, we conclude ψ n+1 = 1; and these are precisely the excluded values. Now suppose that τ Δ is a proper face of Δ. Then clearly (1, 1, . . . , 1) does not belong to τ . If τ contains (0, . . . , 0, n + 1), then by restricting (3.1.5) to τ , we see that a zero of To overcome the fact that the generalized Klein-Mukai pencil is only convenient with respect to {x n } (as opposed to the case of the Dwork pencil, which is convenient with respect to the full set of variables {x 0 , . . . , x n }), we prove the following lemma.
Lemma 3.1.7: We have The point of this combinatorial lemma is that one obtains the same value of the exponential sum when changing affine coordinates to toric coordinates, so that Theorem 2.2.1 applies.
for the linear subspace defined by the vanishing of Let r ∈ Z ≥0 . A standard inclusion-exclusion argument gives We claim, in fact, that every summand on the right-hand side of (3.1. To this end, suppose that J = ∅; then at least one coordinate is sent to zero in F ¡ J (x), and the deforming monomial x 0 · · · x n is set to zero.
with t = #J c . We then compute that Summing the innermost sum on the right side of (3.1.10) over x j ∈ F q r counts with multiplicity q r the number of zeros of wx n j−1 with w ∈ F × q r , where x j−1 ∈ F q r is fixed. If x j−1 = 0, then there are no such zeros and the inner sum is zero. Therefore, letting J = J ∪ {j − 1, j} (with indices taken modulo n), Let S be the set of variables of F appearing in the Fermat (diagonal form) piece of the defining polynomial F in either case. Then F is convenient with respect to S. Suppose ψ ∈ F q is such that F ψ (x) is nondegenerate with respect to Δ ∞ (F ). Therefore, we have a p-adic complex Ω • such that (2.2.8)-(2.2.10) hold.
We prove that for each fiber, the zeta functions in these two families have middle-dimensional cohomology with a common factor of degree n determined by action of the Frobenius on the (∂/∂ψ)-stable subspace containing the unique holomorphic nonvanishing differential n-form. In both cases, the monomial wx 0 x 1 · · · x n ∈ Ω n+2 corresponds to this n-form. For q = p r , let Q q be the unramified extension of Q p of degree r.
Proposition 3.2.2: If p (n + 1)d T and ψ ∈ F × q is a smooth, nondegenerate fiber, then the polynomials P ,ψ (T ) where ∈ {F n+1 , F 1 L n } have a common factor R ψ (T ) ∈ Q q [T ] of degree n.
Proof. Viewed over a ring with derivation ∂/∂ψ, for all i the cohomology H i (Ω • ) has an action by the connection where γ 0 is an appropriate p-adic constant. The monomial wx 0 x 1 · · · x n then spans an (∂/∂ψ)-stable subspace of H n+2 (Ω • ), denoted Σ . In both cases ∈ {F n+1 , F 1 L n }, we have a Frobenius map Frob • acting as a chain map on the complex Ω • and stable on Σ . As a consequence, we conclude that Let Φ (ψ) represent the Frobenius map Frob restricted to Σ . We appeal to work of Dwork [Dwo69]. We find that in the sense of Dwork, there are two Frobenius structures, both of which are strong Frobenius structures as a function of the parameter ψ on the hypergeometric differential equation, corresponding to the two values of . The hypergeometric differential equation (over C p , or any field of characteristic zero) is irreducible because none of the numerator parameters {1/(n + 1), . . . , n/(n + 1)} differs from the denominator parameter {1} by an integer [Beu08, Corollary 1.2.2]. As a consequence, the hypotheses of a lemma of Dwork [Dwo89, Lemma, pp. 89-90] are satisfied, and we have that the two Frobenius structures agree up to a multiplicative constant c ∈ C × p ; in terms of matrices, We now show that c = 1. Let ψ 0 ∈ F q be such that ψ n+1 0 = 1. Then the fiber for each family at ψ = ψ 0 satisfies F ,ψ0 (x) = 0 and the defining polynomial wF ,ψ0 (x) is nondegenerate. Let ψ 0 be the Teichmüller lift of ψ 0 . We recall section 2.3. Suppose that ψ 0 is an ordinary fiber for both families. Then Without loss of generality, we may assume that c is a p-adic integer. Since the two families have the same Picard-Fuchs differential equation, we obtain p-adic analytic formulas for the unique unit root of Tr(Φ Fn+1 ( ψ 0 )) by Jeng-Daw Yu [Yu08], and for the unique unit root of Tr(Φ F1Ln ( ψ 0 )) by work of Adolphson-Sperber [AS16] (also proven by Miyatani [Miy15]). These formulas are given in terms of the unique holomorphic solution of Picard-Fuchs (at ∞) so that the formulas are the same, so the unique unit roots for the two families agree, and this forces c ≡ 1 (mod q). Repeating this argument over all extensions F q r with r ≥ 1, we conclude similarly that c r ≡ 1 (mod q r ). Taking r coprime to p, by binomial expansion we conclude c = 1 as desired.
In the next section, we generalize this result and also prove that R ψ (T ) ∈ Q[T ].

Proof of the main result
We now prove the main theorem in the general setting of families of alternate mirrors.
4.1. Hypergeometric Picard-Fuchs equations. To begin, we study the Picard-Fuchs equation for the holomorphic form of an invertible pencil. We use the structure of the Picard-Fuchs equation to identify a factor of the zeta function associated to the holomorphic form, establishing a version of our main theorem with coefficients defined over a number field. By work of Gährs [Gäh13,Gäh11], we know that if two invertible pencils have the same dual weights, then their Picard-Fuchs equations are the same. We now state her result and recast it in a hypergeometric setting.
Let F A be an invertible polynomial, where q i are its dual weights and d T := i q i is the weighted degree of the transposed polynomial F A T . For each ψ, let H n (X A,ψ ) be the de Rham cohomology of the holomorphic nforms on the complement P n \ X A,ψ , and write the usual holomorphic form on P n as Ω 0 = n i=0 (−1) i x i dx 0 ∧ · · · ∧ dx i−1 ∧ dx i+1 ∧ · · · ∧ dx n . Then one may use the Griffiths residue map Res : H n (X A,ψ ) → H n−1 (X A,ψ , C), whose image is primitive cohomology, to realize the holomorphic form on X A,ψ as Res(Ω 0 /F A,ψ ). Systematically taking derivatives of the holomorphic form establishes the Picard-Fuchs differential equation associated to the holomorphic form that Gährs computes via a combinatorial formulation of the Griffiths-Dwork technique. We now state her result.
We first define the rational numbers The elements of the multiset α α α have no repetition, so we can think of α α α as a set. Take the intersection I = α α α ∩ β β β. Note that all of these sets depend only on the dual weights q i . Let δ = ψ d dψ . As β i0 = 0 ∈ β β β for all i, we have 0 ∈ β β β I, hence the Picard-Fuchs equation is a hypergeometric differential equation. In particular, a solution is given by the (generalized) hypergeometric function where D = D(q 0 , . . . , q n ) and I ∪ ∪ ∪ {0} is the multiset obtained by adjoining 0 to I. Proof. This differential equation has parameters such that α i − β jk ∈ Z for all i, j, k, for the following reason: the elements of α α α and β β β are already in [0, 1), so two differ by an integer if and only if they are equal; and whenever two coincide, they are taken away by the set I (noting the elements of α α α are distinct). Therefore, the differential equation is irreducible [Beu08, Corollary 1.2.2].

Group invariance.
In this section, we show that the subspace of cohomology associated to the Picard-Fuchs equation for the holomorphic form is contained in the subspace fixed by the action of a finite group. This group arises naturally in the context of Berglund-Hübsch-Krawitz mirror symmetry. Throughout, we work over C.
We begin by establishing three groups that are useful when studying invertible potentials and prove a result about the invariant pieces of cohomology associated to them. Let F A be an invertible polynomial. First, consider the elements of the maximal torus G n+1 m acting diagonally on P n and leaving the polynomial F A invariant: Write A −1 = (b ij ) i,j ∈ GL n+1 (Q) and for j = 0, . . . , n let ρ j = (exp(2πib 0j ), . . . , exp(2πib nj )); then ρ 0 , . . . , ρ n generate Aut(F A ).
Next, we consider the subgroup acting invariantly on the holomorphic form, and the subgroup (4.2.3) J FA := ρ 0 · · · ρ n obtained as the cyclic subgroup of Aut(F A ) generated by the product of the generators ρ j . Then J FA is the subgroup of Aut(F A ) that acts trivially on X A . We now describe Berglund-Hübsch-Krawitz mirrors explicitly. Consider a group G such that J FA ⊆ G ⊆ SL(F A ). Then we have a Calabi-Yau orbifold Z A,G := X A /(G/J FA ). The mirror is given by looking at the polynomial F A T obtained from the transposed matrix A T and the hypersurface X A T ⊂ W P n (q 0 , . . . , q n ), where q i are the dual weights.
We define the dual group to G to be Thus, we obtain a Calabi-Yau orbifold Berglund-Hübsch-Krawitz duality states that Z A,G and Z A T ,G T are mirrors.
Proposition 4.2.5: Let X A,ψ be an invertible pencil of Calabi-Yau (n − 1)folds determined by the integer matrix A. Then for all ψ such that X A,ψ , we have In certain cases, we have equality. We can compute dim C H n−1 prim (X A , C) SL(FA) in the following way. Let be the defining polynomial for the Fermat hypersurface X A ⊆ P n . Here, SL(F A ) = (Z/(n + 1)Z) n where an element (ξ 1 , . . . , ξ n ) ∈ SL(F A ) acts by Note that in order for i x ai i ∈ Q FA to be SL(F A )-invariant, it must satisfy the equalities a 0 − a i ≡ 0 (mod n + 1) for all i. Thus the only SL(F A )-invariant elements of the Milnor ring are the n elements (x 0 · · · x n ) a for 0 ≤ a < n. Note that n = d T − #I, or the order of the Picard-Fuchs equation for this example, so equality holds in (4.2.6).

4.3.
Frobenius structure for the subspace associated to the holomorphic form. In this section, we will study a subspace W ψ ⊂ H n+2 (Ω • ,ψ ) generated by the connection acting on the holomorphic form. The dimension of this subspace is equal to the order of the Picard-Fuchs equation. It will in the end correspond to a factor of the zeta function for X A,ψ0 , where X A,ψ0 is a nondegenerate and smooth member of the pencil. We prove in this section that there is a Frobenius structure on W ψ by examination of the unit root.
Note that in our case, working with a cyclic basis C 0 (ψ) is a solution of the scalar differential equation  Proof. This proposition is proven by Sabbah [Sab05, Theorem 2.4]; for completeness, we provide an argument here. Suppose M * 0 were such a nonzero, proper ∇ * -stable differential submodule of dimension r, 0 < r < N. We will show that if such a proper submodule existed, then the Picard-Fuchs operator L(D) has a proper factorization in the noncommutative polynomial ring K [D] and the Picard-Fuchs equation would necessarily be reducible, contradicting Proposition 4.1.11.
Without loss of generality we may assume M * 0 has a cyclic basis {γ 0 , ∇γ 0 , . . . , ∇ r−1 γ 0 , δ r , . . . , δ N −1 } is the dual basis for W ψ . Then we can write the connection matrix for W * ψ in the form We consider a horizontal section So the entries {B 0 , . . . , B r−1 } are dependent over K. We now can rewrite this horizontal section in terms of our original dual basis for some A i ∈ K. Note that A 0 must be a solution of the Picard-Fuchs differential equation. There exists some nonsingular matrix A over K so that Using this change of basis, we can see that . . . , B N −1 )A = (A 0 , . . . , A N −1 ) where This gives a non-trivial homogeneous relation among A 0 , . . . , D N −1 A 0 ; thus, A 0 satisfies a lower order differential equation defined over K. Using the usual argument via the division algorithm in the noncommutative ring K [D] we conclude that the Picard-Fuchs operator has a non-trivial right factor in K[D] which contradicts the irreducibility of the Picard-Fuchs equation.
Lemma 4.3.2: Let ψ ∈ P 1 be such that X A,ψ is nondegenerate and smooth. Then there exists a strong Frobenius structure on W * ψ .
Proof. We recall section 2.3. Suppose that X ψ is ordinary, a condition that holds for all but finitely many ψ ∈ F p . Then there is a unique unit root of the characteristic polynomial of Frobenius acting on H n+2 (Ω • ,ψ ), and this yields a unique unit root eigenvector η 0 up to scaling. The same holds for the dual space ,ψ ) with unique unit eigenvector η * 0 . We claim that η * 0 ∈ W * ψ . Assume for the purposes of contradiction that η * 0 ∈ W * ψ . Then we may take as a basis for K ψ a set containing η * 0 and the cyclic basis . Let A * be the matrix of q −1 -Frobenius in this basis. Since η * 0 is a unit eigenvector, the diagonal coefficient of A * corresponding to η * 0 is nonzero modulo p (and the other coefficients of this column are zero). But by (2.3.7), the diagonal coefficient of A * for ω * 0 is nonzero modulo p because X ψ is ordinary. Therefore A * has rank at least 2 modulo p, and this contradicts Lemma 2.3.3.
So now let η * 0 ∈ W * ψ be the unit root eigenvector, unique up to scaling and defined on the ordinary locus U ⊆ P 1 where U is the complement of the union of {0, 1, ∞} and the supersingular locus for the given pencil X A,ψ . Then writing Frob for q −1 -Frobenius where u ∈ K is a unit on the locus U . Frobenius commutes with the connection ∇ * , so which implies that Frobenius is stable on the submodule that is generated by the cyclic basis given by {(∇ * ) i η * 0 | i ∈ Z ≥0 }, but this is W * ψ by Proposition 4.3.1. Hence, for each choice of pencil indexed by the Picard-Fuchs equation has a strong Frobenius structure in the sense of Dwork [Dwo89].

4.4.
Proof of main result. In this section, we prove our main result. We will make use of the following lemma.
Lemma 4.4.1: Let X be a projective variety over F q and let G be a finite group of automorphisms of X = X × Fq F q stable under Gal(F q /F q ). Then the following statements hold: (a) The quotient X/G exists as a projective variety over F q . (b) Let = p be prime and suppose gcd(#G, ) = 1. Then for all i, the natural map Proof. See Harder-Narasimhan [HN75, Proposition 3.2.1] (with some extra descent).
Our main result (slightly stronger than Theorem 1.2.3) is as follows: Theorem 4.4.2: Let X A,ψ and X B,ψ be invertible pencils of Calabi-Yau (n − 1)-folds in P n . Suppose A and B have the same dual weights (q i ) i . Then for each ψ ∈ F q such that gcd(q, (n + 1)d T ) = 1 and the fibers X A,ψ and X B,ψ are nondegenerate and smooth, there exists a polynomial R ψ (T ) ∈ Q[T ] with Proof. Let F ,ψ (x) be invertible pencils, corresponding to matrices = A, B with the same weights. Then by Theorem 4.1.3, the Picard-Fuchs equations of order D(q 1 , . . . , q n ) are the same. Suppose that the two pencils have a common smooth fiber ψ ∈ F q .
We follow the construction of cohomology in Adolphson-Sperber [AS08], with a few minor modifications. We assume their base field Λ 1 is enlarged to treat ψ as a variable over Q p (ζ p ) with (unit) p-adic absolute value, so that Λ 1 has ∂/∂ψ as a nontrivial derivation. Then the construction of the complex Ω • ψ is unchanged as are the cohomology spaces H i (Ω • ψ ). Then [AS08, Theorem 6.4, Corollary 6.5] where ψ 0 is the Teichmüller lift of ψ 0 .
The connection . By work of Katz [Kat68], the associated differential equation is the Picard-Fuchs equation.
For each invertible pencil determined by a choice of , as in section 4.3, we have a subspace W ψ obtained by repeatedly applying the connection to the monomial wx 0 x 1 · · · x n corresponding to the holomorphic form. By Lemma 4.3.2, we obtain a strong Frobenius structure on this differential module. By construction, the associated differential equation is the hypergeometric Picard-Fuchs equation, and this equation is independent of by Theorem 4.1.3. By Proposition 4.1.11, this differential equation is irreducible. Under the hypothesis that p (n + 1)d T , there is a p-integral solution to this differential equation. Then by a result of Dwork [Dwo89, Lemma, pp. 89-90], the respective Frobenius matrices Φ ,ψ0 acting on W differ by a p-adic constant. As in the proof of Proposition 3.2.2, the same unique unit root at a smooth specialization implies that this constant is 1.
At the same time, the subspace is stable under the connection and has an action of Frobenius. The group SL(F A ) preserves the holomorphic form, so W ψ ⊆ Σ ,ψ . Let We have shown that , as it is a factor of the zeta function, we know immediately that R ψ0 (T ) ∈ K[T ] for K a number field, which we may assume is Galois over Q by enlarging. Next, we apply Lemma 4.4.1: the characteristic polynomial of Frobenius via the Galois action on H n−1 et (X A,ψ0 , Q ) SL(FA) is equal to S ,ψ0 (qT ). Therefore S ,ψ0 (qT ) ∈ Q [T ] for all but finitely many , and so is independent of and it also belongs to Q[T ]. Now let be the least common multiple of the polynomials obtained by applying Gal(K/Q) to the coefficients of R ψ0 . Then R ψ0 (T ) is still independent of , by Galois theory R ψ0 (T ) ∈ Q[T ], and R ψ0 (T ) | S ,ψ0 (T ) | P ,ψ0 (T ) is a factor of the zeta function and Remark 4.4.5: It is also possible to argue for a descent to Q[T ] of a common factor of degree d T − #I purely in terms of hypergeometric motives-without involving the group action-as follows. First, we need to ensure that the trace of Frobenius on the subspace of p-adic cohomology cut out by the hypergeometric Picard-Fuchs equation is given by an appropriately normalized finite field hypergeometric sum: this is implicit in work of Katz [Kat90,§8.2] and should be implied by rigidity [Kat90,§8.10], but we could not find a theorem that would allow us to conclude this purely in terms of the differential equation.

Remark 4.4.8:
There is yet a third way to observe a common factor purely in terms of group invariance using a common cover by a Fermat pencil (of larger degree): see recent work of Kloosterman [Kl17].
4.5. Unit roots and point counts. If X is a smooth Calabi-Yau variety, the polynomial P X (T ) appearing in the zeta function of X has at most one root that is a p-adic unit. This root is called the unit root. We have already used the unit root implicitly to compare zeta functions. We may also use the unit root directly to extract arithmetic information about an invertible pencil from A T . This yields a simple arithmetic relationship between different invertible pencils with the same dual weights. Remark 4.5.2: In the case of non-smooth, non-supersingular fibers, Adolphson-Sperber [AS16] describe what is meant here by the unit root and show that then the unit root is given by the same formula as in the smooth case. Dwork noted the possibility of a meaningful unit root formula for varieties that are not smooth [Dwo62].
Proof. In the case where p divides d T we replace d T ψ in the given families by ψ in order to obtain a nontrivial pencil. Adolphson-Sperber [AS16] provide a formula for the unit root using A-hypergeometric functions. The lattice of relations used to compute the A-hypergeometric functions is determined by the dual weights, and the character vector is the same in both families. Thus, the unit root formula is the same in both cases. More precisely, in the case of smooth fibers, the middle dimensional factor has a unique unit root which occurs in the common factor R ψ (T ) described above. It is given by a p-adic analytic formula in terms of the series defined above. The Hasse invariant is determined by the reduction of the A-hypergeometric series solution mod p. This proves the identity of the supersingular locus in cases where the weights agree.
Remark 4.5.3: In the case that ψ ∈ F × q yields a smooth member of the pencil X A,ψ , the result of Proposition 4.5.1 can also be obtained from Miyatani [Miy15, Theorem 2.9], where the unit root is nontrivial precisely when a formal power series defined using the hypergeometric parameters appearing in Equation 4.1.8 is nonzero. Miyatani also gives a formula for the unit root when it exists and X A,ψ is smooth, in terms of the same hypergeometric power series. As we have already observed, the hypergeometric parameters depend only on the weights of A T or B T . Proposition 4.5.1 implies a relationship between point counts for alternate mirrors, reminiscent of Wan's strong arithmetic mirror symmetry [FW06], [Wan06].
Corollary 4.5.4: Let F A (x) and F B (x) be invertible polynomials in n + 1 variables satisfying the Calabi-Yau condition. Suppose A T and B T have the same weights. Then for any fixed ψ ∈ F q and in all characteristics (including p | d T ) the F q -rational point counts for fibers X A,ψ and X B,ψ are congruent as follows: Proof. The formula is true vacuously when the fiber is supersingular (there is no unit root). Otherwise, the unit root controls the point count modulo q.
The congruence result given here is weaker of course for smooth fibers than the result given earlier on common factors, Theorem 4.4.2 above. It is possible that the common factor result for the piece of middle-dimensional cohomology invariant under the respective group actions does extend meaningfully to fibers that are not smooth as well. Computations in [Kad04,Kad06,CDRV01] show that a factor of the zeta function associated to the holomorphic form can be identified for singular fibers of the Dwork pencils of quartics and quintics, as well as for a certain family of octic Calabi-Yau threefolds in a weighted projective space. We expect there will be a common factor (for families with the same dual weights) for singular fibers in the case of K3 surfaces, since the unit root in this case should govern the relevant factor (using the functional equation and the fact that the determinant of Frobenius is constant).

Quartic K3 surfaces
We now specialize to the case of n = 3, i.e., K3 surfaces realized as a smooth quartic hypersurface in P 3 . 5.1. Pencils of K3 surfaces. The invertible pencils in P 3 whose Berglund-Hübsch-Krawitz mirrors are hypersurfaces in finite quotients of P 3 are listed in Table (5.1.1). We list the group of symplectic symmetries SL(F A )/J FA , which act nontrivially on each projective hypersurface and fix its holomorphic form, in the third column. Equation for X A,ψ Symmetries Recalling Example 4.1.9, we observe that each of these five pencils has the same degree three Picard-Fuchs equation for the holomorphic form, and that after a change of variables, this equation is the differential equation satisfied by the classical hypergeometric function The main result of this section is the following theorem.
Theorem 5.1.3: Let ∈ F = {F 4 , F 2 L 2 , F 1 L 3 , L 2 L 2 , L 4 } signify one of the five K3 families in Table (5.1.1). Let q = p r be a prime power with p = 2, 5, 7 and let ψ ∈ F q be such that ψ 4 = 1. Then X ,ψ is a smooth, nondegenerate fiber of the family . Let P ,ψ,q (T ) ∈ 1 + T Z[T ] be the nontrivial factor of Z(X ,ψ /F q , T ) of degree 21. Then the following statements hold: The reciprocal roots of Q ,ψ,q (T ) are of the form q times a root of 1. (c) The polynomial R ψ,q (T ) is independent of ∈ calF .
Remark 5.1.4: In future work [DKSSVW], we study these families in more detail: we describe a further factorization of Q ,ψ,q (T ) related to the action of each group, and we identify each of these additional factors as hypergeometric.
The polynomials P ,ψ,q (T ) have degree 21 and all of their reciprocal roots α satisfy |α| = q, by the Weil conjectures. By a direct calculation in the computer algebra system Magma [BCP97], when p = 2, 5, 7 and ψ 4 = 1, the fiber X ,ψ is smooth and nondegenerate. Parts (a) and (c) of Theorem 5.1.3 now follow from Theorem 1.2.3 and the Picard-Fuchs differential equation computed in Example 4.1.9.
We now prove Theorem 5. 1.3(b). For all ∈ F, the trace formula (2.2.10) asserts that We now analyze the unit root. In section 2.3, we saw that there is at most one unit root of P ,ψ,q (T ). If there is no unit root, then the K3 surface X ,ψ is supersingular over F q , and Theorem 5.1.3(b) follows by the Tate conjecture for K3 surfaces. Thus, we need only analyze the case where there is a unit root.
Proposition 5.1.5: Suppose P ,ψ,q (T ) has a unit root u(ψ). Then the reciprocal zeros β = β of P ,ψ,q (T ) other than u(ψ) and the root q 2 /u(ψ) all have the form β = qζ where ζ is a root of unity.
Proof. We know that β is an algebraic integer which, by Deligne's proof of the Riemann hypothesis, has the form β = qζ with ζ an algebraic number with complex absolute value |ζ| ∞ = 1. By the functional equation ββ = q 2 , so that for any prime = p, we have that β (and ζ) are -adic units. Since we are considering now only ordinary fibers ψ, the first slope of Newton agrees with the first slope of Hodge. It then follows for every β a reciprocal zero of P (t) other than the unit root u(ψ), we have ord q (β) ≥ 1. As a consequence, ζ is a p-adic integer. This proves ζ is an algebraic integer. From the product formula |ζ| p = 1. We have shown that |ζ| v = 1 for all places v of Q. By Dirichlet's theorem, this implies ζ is a root of unity.
Before concluding this section, we consider the remaining invertible quartic pencils in P 3 . We may use methods similar to the analysis of Theorem 5.1.3 to relate two pencils of K3 surfaces whose equations incorporate chains. (5.1.6)

Family
Equation for X A,ψ Symmetries C 2 F 2 x 3 0 x 1 + x 4 1 + x 4 2 + x 4 3 − 12ψx 0 x 1 x 2 x 3 Z/4Z C 2 L 2 x 3 0 x 1 + x 4 1 + x 3 2 x 3 + x 3 3 x 2 − 12ψx 0 x 1 x 2 x 3 Z/2Z Let ♣ ∈ G = {C 2 F 2 , C 2 L 2 } signify one of the two K3 families in Table (5.1.6). The dual weights for these families are (4, 2, 3, 3). Let X ♣,ψ be a smooth member of ♣, and assume gcd(q, 6) = 1. Let P ♣,ψ (T ) ∈ 1 + T Z[T ] be the nontrivial factor of Z(X ♣,ψ , T ) of degree 21 as in (1.2.2). Then by Theorem 4.4.2 we have a factorization (5.1.7) P ♣,ψ (T ) = Q ♣,ψ (T )R ψ (T ) in Z[T ] with 6 ≤ deg R ψ ≤ 7 and R ψ (T ) is independent of ♣ ∈ G. However, we pin this down in the next subsection, and show in fact that deg R ψ = 6 (as expected), with deg Q ♣,ψ = 15. The reciprocal roots of Q ♣,ψ (T ) are of the form q times a root of 1 from a similar argument as in Proposition 5.1.5. Together, Theorem 5.1.3 and Equation (5.1.7) give a complete description of the implications of Theorem 1.2.3 for invertible pencils of K3 hypersurfaces in P 3 ; the remaining three pencils, classified for example by Doran-Garavuso [DG11], are each described by matrices with distinct sets of dual weights. 5.2. Discussion and applications. By Tate's conjecture, a theorem due to work of Charles [Cha13], Madapusi Pera [Per15], and Kim-Madapusi Pera [KP16], the Néron-Severi rank of a K3 surface X over F q is equal to one plus the multiplicity of q as a reciprocal root of P (T ) [vanL07, Corollary 2.3], and this rank is even. (The extra "one" corresponds to the hyperplane section, already factored in.) Thus Theorem 5.1.3(b) implies that each X ,ψ has Néron-Severi rank over the algebraic closure F q at least 18+1 = 19, so at least 20 because it is even. Similarly, each X ♣,ψ has Néron-Severi rank over F q at least 14 + 1 = 15, thus 16 because it is even.
By comparison, in characteristic 0 we can inspect the Néron-Severi ranks as follows. 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].
For the five pencils in Table (5.1.1) with dual weights (1, 1, 1, 1), we conclude that the generic Néron-Severi rank is at least 22 − 3 = 19; but it cannot be 20, because then the family would be isotrivial, so it is equal to 19. Similarly, for the two pencils in Table (5.1.6), the generic Néron-Severi rank ρ is at least 22 − 7 = 15: but the divisor defined by for either choice of i 2 = −1 is SL(F A )-invariant, so the generic Néron-Severi rank ρ is in fact at least 16. Now a specialization result due to Charles [Cha14] shows that the rank over F q is always at least ρ and is infinitely often equal to ρ if the rank is even and infinitely often ρ + 1 if the rank is odd. By the first paragraph of this section, we conclude that the generic Néron-Severi rank of these two pencils is exactly 16. 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.
We conclude by a discussion of some applications of Theorem 5.1.3 in the context of mirror symmetry. Let Y ψ be the pencil of K3 surfaces mirror to quartics in P 3 obtained by taking the quotient of F 4 by (Z/4Z) 2 and resolving singularities. It can be viewed as the minimal resolution of the complete intersection [NS01,dAMS03] Z(xyz(x + y + z − 4ψw) + w 4 ) ⊆ P 4 .
This calculation combined with Theorem 5.1.3 yields the following corollary.
Corollary 5.2.2: There exists r 0 ≥ 1 such that for all q = p r with r 0 | r and p = 2, 5, 7 and all ψ ∈ F q with ψ 4 = 1, we have In other words, for all ψ ∈ F q with ψ 4 = 1, not only do we have the strong mirror relationship #X ,ψ (F q r ) ≡ #Y ψ (F q r ) (mod q r ) for all ∈ F and r ≥ 1 (see Wan [Wan06]), but in fact we have equality #X ,ψ (F q r ) = #Y ψ (F q r ) for all r divisible by r 0 .
Accordingly, we say that the zeta functions Z(X ,ψ /F q , T ) for all ∈ F and Z(Y ψ /F q , T ) are potentially equal, that is, equal after a finite extension.