Monodromy of Picard-Fuchs differential equations for Calabi-Yau threefolds

In this paper we are concerned with the monodromy of Picard-Fuch differential equations associated with one-parameter families of Calabi-Yau threefolds. Our results show that in the hypergeometric cases the matrix representations of monodromy relative to the Frobenius bases can be expressed in terms of the geometric invariants of the underlying Calabi-Yau threefolds. This phenomenon is also verified numerically for other families of Calabi-Yau threefolds in the paper. Furthermore, we discover that under a suitable change of bases the monodromy groups are contained in certain congruence subgroups of Sp(4,Z) of finite index whose levels are related to the geometric invariants of the Calabi-Yau threefolds


Introduction
Let M z be a family of Calabi-Yau n-folds parameterized by a complex variable z ∈ P 1 (C), and ω z be the unique holomorphic differential n-form on M z (up to a scalar). Then the standard theory of Gauss-Manin connections asserts that the periods γz ω z satisfy certain linear differential equations, called the Picard-Fuchs differential equations, where γ z are r-cycles on M z .
(Actually, it is the mirror partner of the quintic Calabi-Yau threefolds that has Hodge number h 2,1 = 1 and hence the Picard-Fuch differential equation is of order 4. But the mirror pair of Calabi-Yau threefolds share the same "principle periods". This means that the Picard-Fuchs differential equation of the original quintic Calabi-Yau threefold of order 204 contains the above order 4 equation as a factor and the factors corresponding to the remaining 200 "semiperiods". In this article we are concerned with the monodromy aspect of the Picard-Fuchs differential equations. Let L : r n (z)θ n + r n−1 (z)θ n−1 + · · · + r 0 (z), r i ∈ C(z), be a differential operator with regular singularities. Let z 0 be a singular point and S be the solution space of L at z 0 . Then analytic continuation along a closed curve γ circling z 0 gives rise to an automorphism of S, called monodromy. If a basis {f 1 , . . . , f n } of S is chosen, then we have a matrix representation of the monodromy. Suppose that f i becomes a i1 f 1 + · · · + a in f n after completing the loop γ, that is, if      then the matrix representation of the monodromy along γ relative to the basis {f i } is the matrix (a ij ). The group of all such matrices are referred to as the monodromy group relative to the basis {f i } of the differential equation. Clearly, two different choices of bases may result in two different matrix representations for the same monodromy. However, it is easily seen that they are connected by conjugation by the matrix of basis change. Thus, the monodromy group is defined up to conjugation. In the subsequent discussions, for the ease of exposition, we may often drop the phrase "up to conjugation" about the monodromy groups, when there is no danger of ambiguities. It is known that for one-parameter families of Calabi-Yau varieties of dimension one and two (i.e., elliptic curves and K3 surfaces, respectively), the monodromy groups are very often congruence subgroups of SL(2, R). For instance, the monodromy group of (1) is Γ(2), while those of (2) and (3) are Γ 0 (2)+ω 2 and Γ 0 (6)+ω 6 , respectively, where ω d denotes the Atkin-Lehner involution. (Technically speaking, the monodromy groups of (2) and (3) are subgroups of SL(3, R) since the order of the differential equations is 3. But because (2) and (3) are symmetric squares of second-order differential equations, we may describe the monodromy in terms of the second-order ones.) Moreover, suppose that y 0 (z) = 1 + · · · is the unique holomorphic solution at z = 0 and y 1 (z) = y 0 (z) log z + g(z) is the solution with logarithmic singularity. Set τ = cy 1 (z)/y 0 (z) for a suitable complex number c. Then z, as a function of τ , becomes a modular function, and y 0 (z(τ )) becomes a modular form of weight 1 for the order 2 cases and of weight 2 for the order 3 cases. For example, a classical result going back to Jacobi states that where θ 2 (τ ) = q 1/8 n∈Z q n(n+1)/2 , θ 3 (τ ) = n∈Z q n 2 /2 , q = e 2πiτ , or equivalently, that the modular form y(τ ) = θ 2 3 , as a function of z(τ ) = θ 4 2 /θ 4 3 , satisfies (1). Here 2 F 1 denotes the Gauss hypergeometric function.
In this paper we will address the monodromy problem for Calabi-Yau threefolds. At first, given the experience with the elliptic curve and K3 surface cases, one may be tempted to guess that the monodromy group of such a differential equation will be the symmetric cube of some congruence subgroup of SL(2, R). After all, there is a result by Stiller [23] (see also [27]) asserting that if t(τ ) is a non-constant modular function and F (τ ) is a modular form of weight k on a subgroup of SL(2, R) commensurable with SL(2, Z), then F, τ F, . . . , τ k F , as functions of t, are solutions of a (k + 1)-st order linear differential equation with algebraic functions of t as coefficients. However, this is not the case in general. A quick way to see this is that the coefficients of the symmetric cube of a second order differential equation y ′′ + r 1 (t)y ′ + r 0 (t)y = 0 is completely determined by r 1 and r 0 , but the coefficients of the Picard-Fuchs differential equations, including (4), do not satisfy the required relations. (The exact relations can be computed using Maple's command symmetric power.) Nevertheless, in the subsequent discussion we will show that, with a suitable choice of bases, the monodromy groups for Calabi-Yau threefolds are contained in certain congruence subgroups of Sp(4, Z) whose levels are somehow described in terms of the geometric invariants of the manifolds in question. This is proved rigorously for the hypergeometric cases and verified numerically for other (e.g., non-hypergeometric) cases. Furthermore, our computation in the hypergeometric cases shows that the matrix representation of the monodromy around the finite singular point (different from the origin) relative to the Frobenius basis at the origin can be expressed completely using the geometric invariants of the associated Calabi-Yau threefolds. This phenomenon is also verified numerically in the non-hypergeometric cases. Although it is highly expected that geometric invariants will enter into the picture, in reality, geometry will dominate the entire picture in the sense that every entry of the matrix is expressed exclusively in terms of the geometric invariants.
The monodromy problem in general has been addressed by a number of authors. Papers relevant to our consideration include [6], [9], [11], [16], and [26], to name a few. In [6], Beukers and Heckman studied monodromy groups for the hypergeometric functions n F n−1 . They showed that the Zariski closure of the monodromy groups of (4) is Sp(4, C). The same is true for other Picard-Fuchs differential equations for Calabi-Yau threefolds that are hypergeometric. In [9], Candelas et al. obtained precise matrix representations of monodromy for (4). Then Klemm and Theisen [16] applied the same method as that of Candelas et al. to deduce monodromy groups for three other hypergeometric cases. In [11] Doran and Morgan determined the monodromy groups for all the hypergeometric cases. Their matrix representations also involve geometric invariants of the Calabi-Yau threefolds. For Picard-Fuchs differential equations of non-hypergeometric type, there is not much known in literature. In [26] van Enckevort and van Straten computed the monodromy matrices numerically for a large class of differential equations. In many cases, they are able to find bases such that the monodromy matrices have rational entries. We will discuss the above results in more detail in Sections 3-5.
Our motivations of this paper may be formulated as follows. Modular functions and modular forms have been extensively investigated over the years, and there are great body of literatures on these subjects. As we illustrated above, the monodromy groups of Picard-Fuchs differential equations for families of elliptic curves and K3 surfaces are congruence subgroups of SL(2, R). This modularity property can be used to study properties of the differential equations and the associated manifolds. For instance, in [18] Lian and Yau gave a uniform proof of the integrality of Fourier coefficients of the mirror maps for several families of K3 surfaces using the fact that the monodromy groups are congruence subgroups of SL 2 (R). For such an application, it is important to express monodromy groups in a proper way so that properties of the associated differential equations can be more easily discussed and obtained. Thus, the main motivation of our investigation is to find a good representation for monodromy groups from which further properties of Picard-Fuchs differential equations for Calabi-Yau threefolds can be derived.
The terminology "modularity" has been used for many different things. One aspect of the modularity that we would like to address is the modularity question of the Galois representations attached to Calabi-Yau threefolds, assuming that Calabi-Yau threefolds in question are defined over Q. Let X be a Calabi-Yau threefold defined over Q. We consider the L-series associated to the thirdétale cohomology group of X. It is expected that the L-series should be determined by some modular (automorphic) forms. The examples of Calabi-Yau threefolds we treat in this paper are those with the third Betti number equal to 4. It appears that Calabi-Yau threefolds with this property are rather scarce. Batyrev and Straten [4] considered 13 examples of Calabi-Yau threefolds with Picard number h 1,1 = 1. Then their mirror partners will fulfill this requirement. (We note that more examples of such Calabi-Yau threefolds were found by Borcea [8].) All these 13 Calabi-Yau threefolds are defined as complete intersections of hypersurfaces in weighted projective spaces, and they have defining equations defined over Q.
To address the modularity, we ought to have some "modular groups", and this paper offers candidates for appropriate modular groups via the monodromy group of the associated Picard-Fuchs differential equation (of order 4). In these cases, we expect that modular forms of more variables, e.g., Siegel modular forms associated to the modular groups for our congruence subgroups would enter the scene.
In general, the third Betti numbers of Calabi-Yau threefolds are rather large, and consequently, the dimension of the associated Galois representations would be rather high. To remedy this situation, we first decompose Calabi-Yau threefolds into motives, and then consider the motivic Galois representations and their modularity. Especially, when the principal motives (e.g., the motives that are invariant under the mirror maps) are of dimension 4, the modularity question for such motives should be accessible using the method developed for the examples discussed in this paper.
The modularity questions will be treated in subsequent papers.

Statements of results
To state our first result, let us recall that among all the Picard-Fuchs differential equations for Calabi-Yau threefolds, there are 14 equations that are hypergeometric of the form . Their geometric descriptions and references are given in the following Table 1. Some comments might be in order for the notations in the table. We employ the notations of van Enckevort and van Straten [26]. X(d 1 , d 2 , . . . , d k ) ⊂ P n (w 0 , . . . , w n ) stands for a complete intersection of k hypersurfaces of degree d 1 , . . . , d k in the weighted projective space with weight (w 0 , · · · , w n ). For instance, X(3, 3) ⊂ P 5 is a complete intersection of two cubics in the ordinary projective 5-space P 5 defined by Slightly more generally, X(4, 4) ⊂ P 5 (1, 1, 2, 1, 1, 2) denotes a complete intersection of two quartics in the weighted projective 5-space P 5 (1, 1, 2, 1, 1, 2) and may be defined by the equations  [1].
In [9], using analytic properties of hypergeometric functions, Candelas et al. proved that with respect to a certain basis, the monodromy matrices around z = 0 and z = 1/3125 for the quintic threefold case (Equation 1 from Table 1)  respectively. (Note that these two matrices are both in Sp(4, Z).) Applying the same method as that of Candelas et al., Klemm and Theisen [16] also obtained the monodromy of the one-parameter families of Calabi-Yau threefolds for Equations 2, 7, and 8. Presumably, their method should also work for several other hypergeometric cases. However, the method fails when the indicial equation of the singularity ∞ has repeated roots. To be more precise, it does not work for Equations 3-6, 10, 13 and 14. Moreover, the method uses the explicit knowledge that the singular point z = 1/C is of conifold type. (Note that in geometric terms, a conical singularity is a regular singular point whose neighborhood looks like a cone with a certain base. For instance, a 3-dimensional conifold singularity is locally isomorphic to XY − ZT = 0 or equivalently, to X 2 + Y 2 + Z 2 + T 2 = 0. Reflecting to the Picard-Fuchs differential equations, this means that the local monodromy is unipotent of index 1.) Thus, it can not be applied immediately to study monodromy of general hypergeometric differential equations. In [11] Doran and Morgan proved that if the characteristic polynomial of the monodromy around ∞ is then there is a basis such that the monodromy matrices around z = 0 and z = 1/C are respectively. It turns out that these numbers d and k both have geometric interpretation. Namely, the number d = H 3 is the degree of the associated threefolds and k = c 2 · H/12 + H 3 /6 is the dimension of the linear system |H|. Doran and Morgan's representation has the advantage that the geometric invariants can be read off from the matrices directly (although there is no way to extract the Euler number c 3 from the matrices), but has the disadvantage that the matrices are no longer in the symplectic group (in the strict sense).
Before we state our Theorem 1, let us recall the definition of Frobenius basis. Since the only solution of the indicial equation at z = 0 for each of the cases is 0 with multiplicity 4, the monodromy around z = 0 is maximally unipotent. (See [20] for more detail.) Then the standard method of Frobenius implies that at z = 0 there are four solutions y j , j = 0, . . . ,, with the property that y 0 = 1 + · · · , y 1 = y 0 log z + g 1 , where g i are all functions holomorphic and vanishing at z = 0. We remark that these solutions satisfy the relation and therefore the monodromy matrices relative to the ordered basis {y 0 , y 2 , y 3 , y 1 } are in Sp(4, C), as predicted by [6]. Now we can present our first theorem.
be one of the 14 hypergeometric equations, and H 3 , c 2 · H, and c 3 be geometric invariants of the associated Calabi-Yau threefolds given in the table above. Let y j , j = 0, . . . , 3, be the Frobenius basis specified by (6). Then with respect to the ordered basis {y 3 /(2πi) 3 , y 2 /(2πi) 2 , y 1 /(2πi), y 0 }, the monodromy matrices around z = 0 and z = 1/C are Remark 1. We remark that by conjugating by the matrix  we do recover Doran and Morgan's representation (5). Thus, our Theorem 1 strenthens the results of Doran and Morgan [11]. Although the referee suggested that Theorem 1 might be a reformulation of the results of Doran and Morgan. However, we do not believe that is the case. For one thing, the argument of Doran-Morgan is purely based on Linear Algebra. It might be possible to derive our Theorem 1 combining the results of Doran-Morgan and those of Kontsevich; we will not address this question here, but left to future investigations.
The appearance of the geometric invariants c 2 , c 3 , H and d is not so surprising. In [9], it was shown that the conifold period, defined up to a constant as the holomorphic solution f (z) = a 1 (z − 1/C) + a 2 (z − 1/C) 2 + · · · at z = 1/C that appears in the unique solution f (z) log(z − 1/C) + g(z) with logarithmic singularity at z = 1/C, is asymptotically [15].) Therefore, it is expected that the entries of the monodromy matrices should contain the invariants. However, it is still quite remarkable that the matrix is determined completely by the invariants alone. We have numerically verified the phenomenon for other families of Calabi-Yau threefolds, and also for general differential equations of Calabi-Yau type. (See [1] for the definition of a differential equation of Calabi-Yau type. See also Section 5 below.) It appears that if the differential equation has at least one singularity with exponents 0, 1, 1, 2, then there is always a singularity whose monodromy relative to the Frobenius basis is of the form stated in the theorem. Thus, this gives a numerical method to identify the possible geometric origin of a differential equation of Calabi-Yau type.
We emphasize that our proof of Theorem 1 is merely a verification. That is, we can prove it, but unfortunately it does not give any geometric insight why the matrices are in this special form.
Acutally, the referee has pointed out that such a geometric interpretation seems to exist by Kontsevich. In the framework of "homological mirror symmetry" of Kontsevich, the first matrix in Theorem 1 would be the matrix associated to tensoring by the hyperplane line bundle in the bounded derived category of sheaves on the Calabi-Yau variety. In general, the matrices in Theorem 1 describe the cohomology action of certain Fourier-Mukai functors. In particular, this explains why the matrices are determined by topological invariants of the underlying Calabi-Yau manifolds. The paper of van Enckevort and van Straten [26] addressed monodromy calculations of fourth order equations of Calabi-Yau type based on homological mirror symmetry. The reader is referred to the article [26] for full details about geometric interpretations of matrices. We wonder, though, if the Kontsevich's results fully explain why there are no "non-geometric" numbers in the second matrices. To be more precise, here is our question. Since the second matrix M is unipotent of rank 1, we know that the rows of M − Id are all scalar multiples of a fixed row vector. We probably can deduce from Kontsevich's result that the fourth row is (−d, 0, −b, −a), but why the first three rows of M − Id are −a/d, −b/a, and 0 times this vector (but not other "non-geometric" scalars?
Now conjugating the matrices by we can bring the matrices into the symplectic group Sp(4, Z). The results are  for z = 0 and z = 1/C, respectively, where k = 2b + d/6. Since the monodromy group is generated by these two matrices, we see that the group is contained in the congruence subgroup Γ(d, gcd(d, k)), where the notation Γ(d 1 , d 2 ) with d 2 |d 1 represents We remark that the entries of the matrices in Γ(d 1 , d 2 ) satisfy certain congruence relations inferred from the symplecticity of the matrices. To be more explicit, let us recall that the symplectic group is characterized by the property that where A, B, C, and D are n × n blocks, if and only if We now summarize our finding in the following theorem.

Theorem 2. Let
be one of the 14 hypergeometric equations. Let y j , j = 0, . . . , 3 be the Frobenius basis. Then relative to the ordered basis the monodromy matrices around z = 0 and z = 1/C are Remark.
We remark that what we show in Theorem 2 is merely the fact that the monodromy groups are contained in the congruence subgroups Γ(d 1 , d 2 ). Although the congruence subgroups Γ(d 1 , d 2 ) are of finite index in Sp(4, Z) (see the appendix by Cord Erdenberger for the index formula), the monodromy groups themselves may not be so. At this moment, we cannot say anything definite about these groups, e.g., their finite indexness. In fact, there are two opposing speculations (one by the authors, and the other by Zudilin) about these groups. We believe, based on a result (Theorem 13.3) of Sullivan [24], that it might be justified to claim that the monodromy group is an arithmetic subgroup of the congruence subgroup Γ(d 1 , d 2 ), and hence is of finite index. (Andrey Todorov pointed out to us Sullivan's theorem, though we must confess that we do not fully understand the paper of Sullivan.) As opposed to our belief, Zudilin has indicated to us via e-mail that a heuristic argument suggests that the monodromy groups are too "thin" to be of finite index.
It would not be of much significance if the hypergeometric equations are the only cases where the monodromy groups are contained in congruence subgroups. Our numerical computation suggests that there are a number of further examples where the monodromy groups continue to be contained in congruence subgroups of Sp(4, Z). However, the general picture is not as simple as that for the hypergeometric cases.
As mentioned earlier, our numerical data suggest that the Picard-Fuchs differential equations for Calabi-Yau threefolds known in literature all have bases relative to which the monodromy matrices around the origin and some singular points of conifolds take the form (7) described in Theorem 1. Thus, with respect to the basis given in Theorem 2, the matrices around the origin and the conifold points again have the form (10). However, with this basis change, the monodromy matrices around other singularities may not be in Sp(4, Z), but in Sp(4, Q) instead, although the entries still satisfy certain congruence relations. Furthermore, in most cases, we can still realize the monodromy groups in congruence subgroups of Sp(4, Z), by a suitable conjugation. Example 1. Consider the differential equation In [4] it is shown that this is the Picard-Fuchs differential equation for the Calabi-Yau threefolds defined as the complete intersection of three hypersurfaces of degree (1, 1, 1) in P 2 × P 2 × P 2 . The invariants are H 3 = 90, c 2 ·H = 108, and c 3 = −90. There are 6 singularities 0, 1/27, ±i/ √ 27, 5/9, and ∞ for the differential equation. Among them, the local exponents at z = 5/9 are 0, 1, 3, 4 and we find that the monodromy around z = 5/9 is the identity. For others, our numerical computation shows that relative to the basis in Theorem 2 the monodromy matrices are From these, we see that the monodromy group is contained in the following group a 21 , a 31 , a 41 , a 32 , a 34 ≡ 0 mod 18, a 11 , a 33 ≡ 1 mod 6, we find that the monodromy group can be brought into the congruence subgroup Γ(6, 3).
Example 2. Consider the differential equation.

A general approach
Let y (n) + r n−1 y (n−1) + · · · + r 1 y ′ + r 0 y = 0, r i ∈ C(z), be a linear differential equation with regular singularities. Then the monodromy around a singular point z 0 with respect to the local Frobenius basis at z 0 is actually very easy to describe, as we shall see in the following discussion. Consider the simplest cases where the indicial equation at z 0 has n distinct roots λ 1 , . . . , λ n such that λ i − λ j ∈ Z for all i = j. In this case, the Frobenius basis consists of y j (z) = (z − z 0 ) λj f j (z), j = 1, . . . , n, where f j (z) are holomorphic near z 0 and have non-vanishing constant terms. It is easy to see that the matrix of the monodromy around z 0 with respect to {y j } is simply  Now assume that the indicial equation at z 0 has λ 1 , . . . , λ k , with multiplicities e 1 , . . . , e k , as solutions, where e 1 + · · · + e k = n and λ i − λ j ∈ Z for all i = j. Then for each λ j , there are e j linearly independent solutions where f j,h are holomorphic near z = z 0 and satisfy f j,0 (z 0 ) = 1 and f j,h (z 0 ) = 0 for h > 0. Since f j,h are all holomorphic near z 0 , the analytic continuation along a small closed curve circling z 0 does not change f j,h . For other factors, circling z 0 once in the counterclockwise direction results in Thus, the behaviors of y j,h under the monodromy around z 0 are governed by  where ω j = e 2πiλj . When the indicial equation of z 0 has distinct roots λ i and λ j such that λ i − λ j ∈ Z, there are many possibilities for the monodromy matrix relative to the Frobenius basis, but in any case, the matrix still consists of blocks of entries that take the same form as above.
From the above discussion we see that monodromy matrices with respect to the local Frobenius bases are very easy to describe. Therefore, to find monodromy matrices uniformly with respect to a given fixed basis, it suffices to find the matrix of basis change between the fixed basis and the Frobenius basis at each singularity. When the differential equation is hypergeometric, this can be done using the (refined) standard analytic method, in which we first express the Frobenius basis at z = 0 as integrals of Barnes-Mellin type and then use contour integration to obtain the analytic continuation to a neighborhood of z = ∞. This gives us the monodromy matrices around z = 0 and z = ∞. Since the monodromy group is generated by these two matrices, the group is determined.
When the differential equation is not hypergeometric, we are unable to determine the matrices of basis change precisely. To obtain the matrices numerically we use the following idea. Let z 1 and z 2 be two singularities and {y i } and {ỹ j }, i, j = 1, . . . , n, be their Frobenius bases. Observe that if y i = a i1ỹ1 + · · · + a inỹn , then we have  Thus, to determine the matrix (a ij ) it suffices to evaluate y at a common point. To do it numerically, we expand the Frobenius bases into power series and assume that the domains of convergence for the power series have a common point z 0 . We then truncate and evaluate the series at z 0 . This gives us approximation of the matrices of basis changes. We will discuss some practical issues of this method in Section 5.

The hypergeometric cases
Throughout this section, we fix the branch cut of log z to be (−∞, 0] so that the argument of a complex variable z is between −π and π. Recall that a hypergeometric function p F p−1 (α 1 , . . . , α p ; β 1 , . . . , β p−1 ; z) is defined for β i = 0, −1, −2, . . . by It satisfies the differential equation Moreover, it has an integral representation 1 2πi Then, for j = 0, . . . , m, the functions are solutions of (11). Moreover, if | arg(−z)| < π and h is a small quantity such that α k + h are not zero or negative integers, then F (h, z) has the integral representation where C is any path from −i∞ to i∞ such that the integers 0, 1, 2, . . . lies on the right of C and the poles of Γ(s + a k + h) lie on the left of C.
Proof. The first part of the lemma is just a specialization of the Frobenius method (see [14]) to the hypergeometric cases. We have If the number of 1's among β k is m, then the first non-vanishing term of the Taylor expansion in h of the last expression is h m+1 . Consequently, we see that are solutions of (11) for j = 0, . . . , m.
The proof of the second part about the integral representation is standard. We refer the reader to Chapter 5 of [21].
We now prove Theorem 1. Here we will only discuss the cases representing the four classes whose indicial equations at z = ∞ have one root with multiplicity 4, two distinct roots, each of which has multiplicity 2, one repeated root and two other distinct roots, and four distinct roots, respectively. The other cases can be proved in the same fashion. where C = 11664. In fact, by considering the contribution of the first term, we see that these four functions make up the Frobenius basis at z = 0.
Proof of the case (A, B) = (1/3, 1/3). Let z be a complex number with −π < arg z < 0. By the same argument as before. We find that the Frobenius basis {y j } at z = 0 can be expressed as π sin πs e πis z s ds.
Proof of the case (A, B) = (1/2, 1/2). Let z be a complex number such that −π < arg z < 0. We find that the Frobenius basis {y j } at z = 0 can be expressed as Cz)), where C = 256 and π sin πs e πis z s ds.
Here h is assumed to be a small real number and C denotes the vertical line Re s = −1/4. The integrand has quadruple poles at s = −k − 1/2 − h for non-positive integers k. Moving the line of integration to Re s = −3/4 and computing the residue at s = −1/2 − h, we see that B n (h) n! z −1/2 (log z) n + (higher order terms in 1/z), where where µ = 16 log 2 + πi. Let 3 (log 3 z/6 + g 1 (1/z) log 2 z/2 + g 2 (1/z) log z + g 3 (1/z))f 0 (z) be the Frobenius basis at z = ∞ with g n (0) = 0. Using the evaluation we can find the analytic continuation of the Frobenius at z = 0 in terms off n (z). Now the monodromy around ∞ relative to the basis From this we can determine the monodromy matrix around z = 1/C with respect to the Frobenius basis at z = 0. We find that the result agrees with the general pattern depicted in Theorem 1, although the detailed computation is too complicated to be presented here.
Of course, there is no reason why our approach should be applicable only to order 4 cases. Consider the hypergeometric differential equations of the form (12) L :  [12,13] to construct series representations for 1/π 2 . Applying the above method, we determine the monodromy of these differential equations in the following theorem whose proof will be omitted.
In [1] a fourth order linear differential equation satisfying all conditions except (c) is said to be of Calabi-Yau type. Using various techniques, Almkvist and etc. found more than 300 such equations. (See Section 5 of [2] for an overview of strategies of finding Calabi-Yau equations. The paper also contains a "superseeker" that tabulates the known Calabi-Yau equations, sorted according to the instanton numbers.) Among them, there are 178 equations that have singularities with exponents 0, 1, 1, 2. It is speculated that all such equations should have geometric origin.
In [26] van Enckevort and van Straten numerically determined the monodromy for these 178 equations. They were able to find rational bases for 145 of them, among which there are 64 cases that are integral. Their method goes as follows. Let z 1 , . . . , z k be the singularities of a Calabi-Yau differential equation. They first chose a reference point p and piecewise linear loops each of which starts from p and encircles exactly one of z i . Then the problem of determining analytic continuation becomes that of solving several initial value problems in sequences. This was done numerically using the dsolve function in Maple. Then they used the crucial observation that the Jordan form for the monodromy around a conifold singularity is unipotent of index one to find a rational basis. Finally, assuming that (5) and (8) hold for general differential equations of Calabi-Yau type, conjectural values of geometric invariants can be read off.
Here we present a different method of computing monodromy based on the approach described in Section 3. Let 0 = z 0 , z 1 , . . . , z n be the singular points of a Calabi-Yau differential equation, and assume that f i,k , i = 0, . . . , n, k = 1, . . . , 4 form the Frobenius bases at z i . According to Section 3, to find the matrix of basis change between {f i,k } and {f j,k }, we only need to evaluate f (m) i,k and f (m) j,k at a common point ζ where the power series expansions of the functions involved all converge. In practice, the choice of ζ is important in order to achieve required precision in a reasonable amount of time.
Let R i denote the radius of convergence of the power series expansions of the Frobenius basis at z i . In general, R i is equal to the distance from z i to the nearest singularity z j = z i , meaning that if we truncate the power series expansion of f i,k at the nth term, the resulting error is Of course, the O-constants depend on the differential equation and z i . Since we do not have any control over them, in practice we just choose ζ in a way such that If this does not yield needed precision, we simply replace n by a larger integer and do the computation again.
The singularities are z 0 = 0, z 1 = 1/3125, and z 2 = ∞. The radii of convergence for the Frobenius bases at 0 and 1/3125 are both 1/3125. Thus, to find the matrix of basis change, we expand the Frobenius bases, say, for 30 terms, and evaluate the Frobenius bases and their derivatives at ζ = 1/6250. Then we use the idea in Section 3 to compute the monodromy matrix around z 1 with respective to the Frobenius basis at 0. We find that the computation agrees with (7) in Theorem 1 up to 7 digits.
The above method works quite well if the singularities of a differential equation are reasonably well spaced. However, it occurs quite often that a Calabi-Yau differential equation has a cluster of singular points near 0, and a couple of singular points that are far away. For example, consider Equation #19 The singularities are z 0 = 0, z 1 = 1/54, z 2 = (11 − 5 √ 5)/2 = −0.090 . . ., z 3 = −23/34, and z 4 = (11 + 5 √ 5)/2 = 11.09 . . .. In order to determine the monodromy matrix around z 4 , we need to compute the matrix of basis change between the Frobenius basis at 1/54 and that at z 4 . The radius of convergence for the Frobenius basis at 1/54 is 1/54, while that at z 4 is z 4 − 1/54 = 11.07 . . .. Even if we choose ζ optimally, we still need to expand the Frobenius bases for thousands of terms in order to achieve a precision of a few digits. In such situations, we can choose several points lying between the two singularities, compute bases for each of them, and then use the same idea as before to determine the matrices of basis change.
Take Equation 19 above as an example. We choose w k = (1 + 3 k )/54 and ζ k = (1 + 3 k /2)/54 for k = 0, . . . , 5. The radius of convergence for the basis at w k is 3 k /54. Thus, evaluating the first n terms of the power series expansions at ζ k and ζ k+1 will result in an error of which is good enough in practice.
Using the above ideas we computed the monodromy groups of the differential equations of Calabi-Yau type that have at least one conifold singularity. 1 Our result shows that if a differential equation comes from geometry, then the monodromy matrix around one of the conifold singularities with respect to the Frobenius basis at the origin takes the form (7). We then conjugate the monodromy matrices by the matrix (9) and find that the other matrices are also in Sp(4, Q). We now tabulate the results for the equations coming from geometry in the following table. Note that the notations Γ(d 1 , d 2 ) and Γ(d 1 , d 2 , d 3 ), d 2 , d 3 |d 1 , represent the congruence subgroups   In the second table we list a few equations whose monodromy matrices with respect to our bases have integers as entries. Note that the numbers H 3 , c 2 · H, and c 3 are all conjectural, obtained from evaluation of the monodromy around a singularity of conifold type. Note, again, that the matrix    

Acknowledgments
The second author (Yifan Yang) would like to thank Wadim Zudilin for drawing his attention to the monodromy problems and for many interesting and fruitful discussions. This whole research project started out as the second author and Zudilin's attempt to give a rigorous and uniform proof of Guillera's 1/π 2 formulas [12,13] in a way analogous to the modular-function approach in [10]. (See also [27].) For this purpose, it was natural to consider the monodromy of the fifthorder hypergeometric differential equations, and hence it led the second author to consider monodromy of general differential equations of Calabi-Yau type. The second author would also like to thank Duco van Straten for his interest in this project and for clarifying some questions about the differential equations.
During the preparation of this paper (at the final stage), the third author (N. Yui) was a visiting researcher at Max-Planck-Institut für Mathematik Bonn in May and June 2006. Her visit was supported by Max-Planck-Institut. She thanks Don Zagier, Andrey Todorov and Wadim Zudilin for their interest in this project and discussions and suggestions on the topic discussed in this paper. She is especially indebted to Cord Erdenberger of University of Hannover for his supplying the index calculation for the congruence subgroup Γ(d 1 , d 2 ) in Sp(4, Z).
The final version was prepared at IHES in January 2007 where the third author was a visiting member. For the preparation of the fnial version, the comments and suggestions of the referee were very helpful as well as discussions with Maxim Kontsevich. We thank them for their help.
We will from now on restrict to the case relevant to this paper, namely d 2 |d 1 . in SL 2 (Z/d 1 Z). An easy calculation shows that this index is equal to (1 − p −2 ).
We summarize the above calculation to obtain Theorem. The group Γ(d 1 , d 2 ) is a congruence subgroup in Sp(4, Z) and its index is given by (1 − p −2 ).
In fact, we can do a similiar calculation without the assumption that d 2 |d 1 and obtain the same formula as given above where one has to replace d 1 with the least common multiple of d 1 and d 2 .