Abstract
Abstract. We show that crystals with the properties of crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic p > 0, exhibit a remarkable similarity with the well known structure on the cohomology of complex Calabi-Yau threefolds near a boundary point of the moduli space with maximal unipotent local monodromy. In particular, there are canonical coordinates and an analogue of the prepotential of the Yukawa coupling. Moreover in Formulas (2.25) and (2.29) we show p-adic analogues of the integrality properties for the canonical coordinates and the prepotential of the Yukawa coupling, which have been observed in the examples of Mirror Symmetry.
Introduction
Calabi-Yau manifolds of dimensions 1 and 2, i.e. elliptic curves and K3-surfaces, have a long and successful tradition in geometry and number theory. In the 1980's, in connection with developments in string theory, Calabi-Yau manifolds of dimension 3 moved to the forefront. Emphasis has been on their complex and symplectic geometry, and in particular on the variation of the Hodge structure for Calabi-Yau threefolds near the so-called large complex structure limit. This is the first of a series of papers in which we want to describe certain aspects of the arithmetic geometry of families of ordinary Calabi-Yau threefolds and analogies with complex Calabi-Yau threefolds near the large complex structure limit. More specifically, this paper discusses the associated crystals, i.e. modules with an integrable connection, like the Gauss-Manin connection on the cohomology of a family of Calabi-Yau threefolds. Crystals which originate from geometry come with a Hodge filtration, which is not preserved by the connection but instead satisfies Griffiths transversality. Crystals which originate from geometry in characteristic p additionally carry an action of Frobenius operators behaving in an appropriate way with respect to the connection and the Hodge filtration: they are divisible Hodge F -crystals. For so-called divisible ordinary Hodge F -crystals the space of flat sections decomposes into eigenspaces for the Frobenius operators. The position of the Hodge filtration with respect to this eigenspace decomposition is the source for the canonical coordinates and the Yukawa coupling: see Theorems 1.8, 2.2, 2.4. The ordinariness requirement implies, exactly as the requirement of maximally unipotent local monodromy in the traditional analysis of complex Calabi-Yau threefolds near the large complex structure limit [8] , that there is a filtration which is opposite to the Hodge filtration and is invariant under the Gauss-Manin connection and whose associated graded is of prop.1.3.6, [5] §6).
Mathematics Subject Classification. [. c 0000 American Mathematical Society
The theory of ordinary Hodge F -crystals has its origins in Dwork's work on the variation of the zeta function in a family of varieties over a finite field [7] and in the work of Serre and Tate on the formal moduli of deformations of an ordinary elliptic curve over a field of positive characteristic. In [4] Deligne and Illusie present a general theory of ordinary Hodge F -crystals and apply it to investigate the formal moduli of ordinary abelian varieties and ordinary K3-surfaces. The general theory works equally well for ordinary CY3 crystals, i.e. crystals of the kind that arises as the crystalline cohomology of ordinary Calabi-Yau threefolds in characteristic p > 0. Thus, exactly as in [4] , there are canonical coordinates on the formal moduli space. In Section 1 we briefly recall the general theory of [4] . In Section 2 we take the same algebraic path as Bryant, Griffiths, Morrison and Deligne [2, 6, 8, 5] in their analysis of the variation of the Hodge structure of complex Calabi-Yau threefolds near a maximally degenerate (≈ maximally unipotent local monodromy ≈ large complex structure) boundary point of the moduli space. In particular we find for ordinary CY3 crystals an exact analog of the cubic form of Bryant-Griffiths (≈ Yukawa coupling). Interestingly, besides the standard characterization (2.21) of the prepotential of this cubic form we find a characterization (2.28) by means of the action of the canonical lift of Frobenius. The latter characterization remains valid if there are no parameters, i.e. if n = 0 in (1.1), while in that case the former description is vacuous. Also for the canonical coordinates the formalisms in the ordinary and the large complex structure situations match very well, but the match is not perfect: in the ordinary case the canonical coordinates take the value 1 at the origin of the deformation space (cf. (2.22)), whereas the canonical coordinates at the large complex structure limit point vanish.
The standard algorithms (cf. [3, 8] ) for computing canonical coordinates and Yukawa coupling for complex Calabi-Yau threefolds near the "maximally unipotent local monodromy boundary point" use special solutions of the Picard-Fuchs equations associated with the family of CY3's and a nowhere vanishing global 3-form. In Remark 3.4 we point out that in the ordinary case one has basically the same relation between canonical coordinates, Yukawa coupling and special solutions of the Picard-Fuchs equations (although explicit computations seem here out of reach).
The present paper is a survey of structures none which -except maybe (2.28) and (2.29) -is new by itself, but which appear in the literature in different contexts. So we hope that there is an inspiring and cross-fertilizing effect from putting them into one (con)text. For instance, being alerted that for the deformation theory of ordinary Calabi-Yau threefolds the cubic form of Bryant-Griffiths (Yukawa coupling) should be relevant, one may wonder how that shows up in connection with the Hodge-Tate decomposition of p-adicétale cohomology as in [1] p. 109.
The original motivation for our work was to find a general method of proof for the integrality conjectures in Mirror Symmetry by reducing them to known results or easy to check conditions on the crystalline cohomology for families of Calabi-Yau threefolds in positive characteristics: It has been observed in many examples in the literature on Mirror Symmetry for Calabi-Yau threefolds, that the coefficients in appropriate expansions of the canonical coordinates and of the prepotential of the Yukawa coupling are integers. In the famous example of the quintics in P 4 (see [3, 8] ) one has for instance q = t + 770 t 2 + 1014275 t 3 + 1703916750 t 4 + 3286569025625 t 5 + . . .
where t = ψ −5 is the parameter and q is the canonical coordinate of the family
the Yukawa coupling Y is given by
The prepotential Z is defined (up to terms of order ≤ 2 in log q) by Y = q d dq 3 Z and has an expansion
where Li 3 (x) = j≥1 x j j 3 is the trilogarithm function. According to the Mirror Symmetry Conjecture [3, 8] the number 5b n is the number of rational curves of degree n on a generic quintic hypersurface in P 4 . For the Integrality Conjecture one is interested in the less astonishing, but still highly non-trivial fact that these numbers are integers. Now one may note that the numbers 5b n are integers if and only if for every prime number p
where Z p is the ring of p-adic integers.
In this paper we show that p-adic analogues of the integrality conjectures, in particular an analogue of (0.1), hold true for ordinary CY3 crystals; see Formulas (2.25) and (2.29)
1 . Throughout this paper the prime p is fixed and the constructions involve choices which may depend on p. For the integrality conjectures for families of Calabi-Yau threefolds, like the above quintics, one must however deal with all primes. Thus one is confronted with the challenge to 'glue' or to 'synchronize' those choices for the various primes p. In [10] we will describe the beginnings of such a synchronized set-up for "ordinariness". The present paper starts at the fairly abstract heights of F -crystals and wants to find out from there whether the path of ordinariness leads to interesting vistas. In [10] we will start at a less abstract more geometric level.
Ordinary Hodge F -crystals
We recall the theory of ordinary Hodge F -crystals; the general reference is [4] . We work over a perfect field k of characteristic p > 0 and explicitly keep track of which results require passing to the algebraic closure k of k. Let W be the ring of Witt vectors of k and K the field of fractions of W . So W is a local ring with residue field k; its field of fractions K has characterisitic 0; its maximal ideal is pW and W is complete and separated for the p-adic topology:
Moreover we need the rings of formal power series
A crystal over A 0 is, by definition, a finitely generated free A-module H together with a connection
which is integrable and p-adically topologically nilpotent; this means that if we set D i = ( On the characteristic p rings k and A 0 one has the Frobenius endomorphism σ raising elements to their p-th power: σ(x) = x p . On the perfect field k this is an automorphism and it lifts canonically to an automorphism on W , also denoted by σ. There are many different lifts of Frobenius on A, i.e. ring endomorphisms φ of A which restrict to σ on W and reduce modulo p to σ on A 0 . If H is an A-module and φ : A → A is a lift of Frobenius we write
where A φ is A viewed as an A-A-bimodule with left structure via the identity map id : A → A and right structure via φ : A → A. The natural map
. Now let (H, ∇) be a crystal as above and let φ, ψ : A → A be two lifts of Frobenius. Since φ and ψ are equal modulo p the connection ∇ provides, via a kind of Taylor expansion, a canonical isomorphism of A-modules
more precisely, the map χ(φ, ψ)φ * : H → ψ * H is given by the formula
Note that for m ≥ 1 the rational number 
which is horizontal for the connection ∇, i.e. the square
is commutative, and such that for every pair of lifts of Frobenius φ, ψ :
If for one, and hence every, lift of Frobenius
Combining Formula (1.4) with the Taylor expansion formula for χ(φ, ψ) one gets
(1.5) The horizontality relation for F (φ) and ∇ (1.3) makes it possible to solve the problem of finding ∇-flat sections (i.e. essentially solving differential equations) by finding F (φ)φ * -fixed sections (i.e. essentially solving polynomial equations). Indeed, one has the following result. Lemma 1.2 Suppose e ∈ H is such that F (φ)φ * e = e for some lift of Frobenius φ : A → A. Then ∇e = 0 and F (ψ)ψ * e = e holds for every lift of Frobenius ψ.
Proof By definition there is for every a ∈ A an element a 1 ∈ A such that φ(a) = a p + pa 1 . From this one gets by induction for every a ∈ A and m ≥ 1 elements a 1 , . . . , a m ∈ A such that
Taking m → ∞ we see ∇e = 0. The equality F (ψ)ψ * e = e for any lift of Frobenius ψ now follows from (1.5).
Let H be an F -crystal 2 and H 0 = H ⊗ A A 0 . For a lift of Frobenius φ : A → A and an integer i we set 
(1.6) is called the conjugate filtration and (1.7) is called the Hodge filtration. The connection ∇ on H induces a connection ∇ 0 on H 0 . The conjugate filtration is horizontal for ∇ 0 and the Hodge filtration satisfies the Griffiths transversality condition, i.e. for all i:
Let gr • H 0 and gr • H 0 denote the graded modules associated with the conjugate and Hodge filtrations respectively. If gr • H 0 is a free A 0 -module, then so is gr
• H 0 and for every i the modules
(called the Hodge filtration on H) which lifts the Hodge filtration from H 0 and satisfies Griffiths transversality, i.e. for every i 
(1.10)
such that for every i
where 
As a consequence one has a decomposition
The above proposition gives a first result on the structure of ordinary Hodge Fcrystals. Stronger results (Theorem 1.8 below) can be obtained over the algebraic closure k of the base field k. Let W the ring of Witt vectors of k and
. One can base change an F -crystal H over A 0 to an F -crystal H over A 0 as follows. By tensoring with A one gets the free A-module with connection
A → A is an arbitrary lift of Frobenius, the connection ∇ provides an isomorphism χ(ψ, φ) : • T = (τ ij ) 0≤i,j≤N with τ ij an h i × h j -matrix, τ ij = 0 for j < i, τ ii =identity-matrix, the matrix τ ij (0) has entries in p j−i W .
• the matrix of the connection ∇ with respect to the basis {e im } is
• for every lift of Frobenius ψ the matrix of the map F (ψ)ψ * with respect to the basis {e im } is T −1 · P · ψ(T ) (1.13) where P is the diagonal matrix
with the entry p j repeated h j -times.
for every lift of Frobenius ψ : A → A. Lift the basis f i1 , . . . , f ih i of V i to a basis e i1 , . . . , e ih i of H i . This gives the basis {e im } 0≤i≤N,1≤m≤h i for H.
Equations (1.15) and Griffiths transversality show that the connection matrix with respect to this basis has the following shape The entries of η i,i+1 are closed 1-forms and hence exact, by the Poincaré lemma. So there is a matrix τ i,i+1 of size
(1.18) Using the equations in (1.17) and induction on j − i one sees that for i + 2 ≤ j ≤ N there exist matrices τ ij of size
For j < i we set τ ij = h i × h j -zero-matrix and for j = i we set τ ii = h i × h i -identitymatrix. We collect the matrices τ ij into one (block structured) matrix
Then (1.18) and (1.19) are equivalent with
(1.20)
We fix the constants of integration by taking the constant term T (0) of T as in ( 1.22) Therefore the matrix M φ of the φ-linear map F (φ)φ * : H → H has the shape
with P as in (1.14) and L an uppertriangular matrix with entries in A and 1's along the diagonal. Let L(0) denote the constant term of L. After these preparations we define T to be the unique matrix which satisfies the differential equation (1.20) and the initial condition 
This shows that (1.13) holds for the particular lift of Frobenius φ. In order to check it for an arbitrary lift of Frobenius we first define the new basis {ε im } 0≤i≤N,1≤m≤h i for H⊗K u [[t 1 , . . . , t n ]] so that T −1 is the matrix whose columns give the coordinates of the new basis vectors with respect to the old basis {e im } 0≤i≤N,1≤m≤h i :
The triangular shape of the matrix T implies
for all i, m.
(1.24)
The connection matrix with respect to the new basis is
The matrix of F (φ)φ * with respect to the new basis is T M φ φ(T )
This holds for the special lift of Frobenius φ (1.21). If ψ is any lift of Frobenius (1.5) and (1.25) show F (ψ)ψ * ε im = F (φ)φ * ε im . Thus for any lift of Frobenius ψ we have
The matrix M ψ of F (ψ)ψ * with respect to the basis {e im } is therefore
as claimed in (1.13). To finish the proof of Theorem 1.8 we note that (1.22) implies that the matrix τ ij (0) has entries in p j−i W for j > i.
Ordinary CY3-crystals over an algebraically closed field
We now specialize the general result of Theorem 2.2 to crystals of the type that appears as the third crystalline cohomology of an ordinary Calabi-Yau threefold over the algebraically closed field k. More precisely, we want to see what happens, if we combine the hypotheses in Theorem 2.2 with the hypotheses for the variation of Hodge structure of Calabi-Yau threefolds, which are explicitly stated in [2] §2. So from now on p > 3 and H is a divisible ordinary Hodge F -crystal over A 0 of level 3 with filtrations . Moreover we assume that there is given a non-degenerate alternating bilinear form
such that for all x, y ∈ H and for every lift of Frobenius φ :
We also require the Riemann bilinear relations:
Definition 2.1 An ordinary CY3 crystal over A 0 is a divisible ordinary Hodge F -crystal of level 3 over A 0 equipped with a non-degenerate alternating bilinear form , such that (2.1), (2.2), (2.3) are satisfied.
In view of the Griffiths tranversality condition (1.9) the connection ∇ induces an A-linear map
where 
and such that the Gramm matrix of the form , with respect to this basis is
)
Moreover there is a matrix T such that the matrix of the connection ∇ with respect to the basis e 0 , . . . , e 2h+1 is
and such that for every lift of Frobenius ψ the matrix of the map F (ψ)ψ * with respect to the basis e 0 , . . . , e 2h+1 is
This matrix T factors as
where Z, τ 23 , τ 13 , τ 12 are matrices with entries in
Assume in addition to the preceding assumptions that the map in (2.4) is an isomorphism, and hence
denote the components of the h × 1-matrix τ 23 :
Then the matrix ∂τ i ∂t j 1≤i,j≤h (2.10)
and
Proof Take any A-basis of H and matrix for which the statements in Theorem 1.8 hold and denote these as e 
Also take the basis ε
as in (1.24), related to the previous basis by the transformation with the matrix T ′ ; see (1.23). Then ∇ε
. Now take the (block structured) matrix with entries in
This matrix is invertible because the form , is non-degenerate. We use it for the basis transformations (e 0 , e 1 , . . . , e h , e h+1 , . . . , e 2h , e 2h+1 ) := (e
For the new basis ε 0 , . . . , ε 2h+1 , the (block structured) Gramm matrix is
(2.13)
so that we have the basis relation (e 0 , e 1 , . . . , e h , e h+1 , . . . , e 2h , e 2h+1 ) = (ε 0 , ε 1 , . . . , ε h , ε h+1 , . . . , ε 2h , ε 2h+1 )T .
The block structure of the matrix T is
(2.14)
The statements contained in formulas (1.25), (1.26), (1.12), (1.13) are still valid with respect to the new bases. The Gramm matrix of the form , with respect to the basis e 0 , . . . , e 2h+1 is
Now assume that the map in (2.4) is an isomorphism. The partial derivations ∂ ∂tj (j = 1, . . . , h) constitute a natural basis for T A/W . A natural basis for Hom(H 3 , H 2 ) consists of the maps e 2h+1 → e h+i (i = 1, . . . , h). With respect to these bases the map (2.4) is given by the matrix (2.20) This matrix is therefore invertible. Equation (2.11) is an obvious restatement of (2.19), while Equation (2.12) follows from (2.18) and (2.11).
In the preceding proof we have essentially followed the same algebraic path as Bryant and Griffiths in their analysis of the variation of the Hodge structure of (complex) Calabi-Yau threefolds [2, 6] . Formulas (2.17), (2.11), (2.12) are familiar from the theory of Bryant and Griffiths. The following computation shows that the equally familiar formula for the cubic form of Bryant and Griffiths (also known as Yukawa coupling) holds in the present situation as well:
Corollary 2.3 In the situation of Theorem 2.2 one has
All hypotheses and notations of Theorem 2.2 remain in force. The divisibility hypothesis (1.11) and Formula (2.7) imply that for every lift of Frobenius ψ the matrix T −1 · P · ψ(T ) · P −1 has entries in A. This means in particular
From Theorem 1.8 we also know that τ i (0) ∈ pW for i = 1, . . . , h. By a lemma of Dwork this implies (see [4] cor.1.4.5) that, if we set
then q i is well defined and (2.22) [4] cor.1.4.7 now yields that p, q 1 − 1, . . . , q h − 1 is a regular system of parameters for the ring A i.e.
The elements q 1 , . . . , q h are called the canonical coordinates. There is an associated canonical lift of Frobenius
For every lift of Frobenius ψ the matrix of the map F (ψ)ψ * with respect to the basis e 0 , . . . , e 2h+1 is given by Formula (2.7). A straightforward computation makes the entries of this matrix explicit. One then sees that exactly for the canonical lift of Frobenius ϕ this matrix takes the following elegant form:
For the canonical lift of Frobenius ϕ the matrix of F (ϕ)ϕ * with respect to the basis e 0 , . . . , e 2h+1 is 
Proof (2.28) and (2.29) follow directly from (2.23) and the divisibility hypothesis (1.11).
3 Ordinary CY3-crystals over a perfect field Having a good hold on the structures when the base field is algebraically closed we now turn to a base field which is just a perfect field k. 
Proof Letω ∈ V 3 andê 2h+1 ∈ V 3 denote the respective images of ω 0 and e 2h+1 under the isomorphisms Fil
On the other hand, (2.7) implies F (φ)φ * e 2h+1 = p 3 e 2h+1 and hence in the (untwisted) unit F -crystal V 3 we have
We set for j = 1, . . . , h
where
On the other hand, Thenω 0 ∈ H 0 and ω 0 , ω 0 = −1. Soω 0 lies in fact in H 0 and gives a basis dual to the basis ω 0 of H 3 . We summarize the preceding discussion in the following proposition. Remark 3.4 Up to now we have discussed all structures in terms of a connection, to be thought of as the Gauss-Manin connection of a family of Calabi-Yau threefolds. In the literature descriptions of canonical coordinates and Yukawa coupling for complex Calabi-Yau threefolds near the large complex structure limit usually start from the Picard-Fuchs equations and their solutions. We want to point out how such a description can also be seen in our discusssion of ordinary CY3 crystals. The hypotheses and notations are as before in this section.

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