Cubic Surfaces with Special Periods

We show that the vector of period ratios of a cubic surface is rational over $Q(\omega)$, where $\omega = \exp(2\pi i/3)$ if and only if the associate abelian variety is isogeneous to a product of Fermat elliptic curves. We also show how to construct cubic surfaces from a suitable totally real quintic number field $K_0$. The ring of rational endomorphisms of the associated abelian variety is $K = K_0(\omega)$.

The endomorphism ring of such a curve always contains the integer dilations z → nz. An elliptic curve with a larger endomorphism ring is said to have complex multiplication. The additional endomorphisms are of the form z → λz, and it is easy to see that λ lies in a purely imaginary quadratic extension K = Q( √ −d) of the rational numbers. This number field can be identified with End(E) ⊗ Q and is called the CM field of E. Conversely, if K is a purely imaginary quadratic extension of the rational numbers, an elliptic curve with CM field K can be constructed as C/O K , where O K is the ring of integers in K.
A theorem of Siegel (1932) tells us that if the coefficients g i are algebraic numbers, then at least one of the periods is transcendental, and in 1934, Schneider showed that the nonzero periods are always transcendental. For a specific example, consider the Fermat elliptic curve, with affine equation y 2 = 4(x 3 − 1). Then with e 1 , e 2 , e 3 = 1, ω, ω 2 , in that order, See [10, equation (10)]. Here, and throughout this paper, ω = e 2πi/3 , a primitive cube root of unity.
When E has complex multiplication, the period ratio lies in the CM field. In this case, the transcendence degree of the field the periods, Q(λ 1 , λ 2 ), is one. It is conjectured that in the non-CM case, the field generated by the periods is of transcendence degree two.
Consider next a cubic surface S. Because the Hodge structure on H 2 (S) is entirely of type (1,1), there are no periods of interest. However, there is an auxiliary Hodge structure of weight three associated to S which does have interesting periods and which behaves in many ways like the Hodge structure of an elliptic curve. This is the Hodge structure of a three-sheeted cover T of projective 3-space branched along S. The period domain for such Hodge structures is B 4 , the unit ball in complex four-space, which is the same as the complex hyperbolic space CH 4 . In this paper we use the notation B 4 rather than CH 4 .
There is a natural period map (2) P : M st −→ Γ\B 4 from the moduli space of stable cubic surfaces (those with at worst nodal singularities) to the indicated quotient of the ball by an arithmetic group. It is an isomorphism [2]. The aim of this paper is to explore the relation between cubic surfaces and their periods. The theory of elliptic curves serves as a model of what might be possible to establish, at least in part. In particular, we will investigate the rationality of period ratios. While the results presented in this direction are modest, they point to several interesting questions. We discuss them at the end of this paper.
We are indebted to Madhav Nori, who explained to us how to use the different ideal to check whether the Abelian variety defined by a number field is principally polarized. We would also like to express our thanks to the referee, whose comments greatly improved this paper.

Statement of results
We first review the main results of [2] and explain the mapping (2). We summarize the results we need, details can be found in [2] and in [4].
2.1. Standard Notation. The following notation will be used throughout this paper. If S ⊂ P 3 is a cubic surface defined by a cubic equation F (x) = 0, let T be the cubic threefold given by {y 3 = F (x)} ⊂ P 4 , where x ∈ C 4 and y ∈ C. Let ω denote the primitive cube root of unity ω = exp(2πi/3), and let σ be the automorphism of T over P 3 defined by σ(x, y) = (x, ωy), Finally, let θ = i √ 3 be the square root of −3 situated on the positive imaginary axis in the complex plane.
2.3. The Hodge structure of T . Since H 3,0 (T ) = 0, the Hodge structure has the form H 2,1 ⊕ H 1,2 . The symmetry σ is an automorphism of H 3 = H 3 (T, C) with eigenvalues ω andω. Let H 3 = H 3 ω ⊕ H 3 ω be the eigenspace decomposition. The summands are in natural duality by the cup-product pairing and are interchanged by complex conjugation. Because σ is holomorphic, its action on cohomology is compatible with the Hodge decomposition. Thus one has complex Hodge structures where the summands have dimensions 4, 1, 1, 4 in the order listed. Equivalently, the same summands give the eigenspace decompositions is positive definite on H 2,1 , negative definite on H 1,2 and these two subspaces are h ′ -orthogonal. There are complex linear isomorphisms and the integral sub-module Z 10 ⊂ C 10 corresponding to H 3 (T, Z) ⊂ H 3 (T, C) projects to Eisenstein lattices E 4,1 , E 1,4 in each summand.
The Hodge structure on H 3 (T ) is determined by either subspace H 2,1 ω ⊂ H 3 ω or H 2,1 ω ⊂ H 3 ω , since each subspace determines the other by conjugation and orthogonality. Either collection constitutes a ball B 4 in a Grassmannian, namely the positive hyperplanes in C 4,1 , respectively the positive lines in C 1,4 . We choose the second description for the B 4 of (2) with the consequent description of Γ = P U (1, 4, E), the projectivized group of E-linear isometries of E 1,4 .
2.4. The period map. The period map P is defined on a covering space of the space of cubic forms. The points of this covering space corresponding to a way of choosing the isomorphisms in (5), which is called a framing in (3.9) of [2]. If φ is the chosen isomorphism, P(F, φ) = φ(H 2,1 ω ) ⊂ C 1,4 . It descends to the map P of (2).
Note that the choice of hermitian form in (2.4) of [2] is the negative of the standard choice (3). Consequently, we have to make all the obvious sign changes when quoting [2]. Note also that the isomorphisms (4) and (5) are complex linear, and incorporate the correction added in proof to (2.2) of [2]. Our choice is equivalent to Beauville's choice of a positive hyperplane in C 4,1 , and P is holomorphic.
2.5. The intermediate Jacobian. This period map determines J(T ), the intermediate Jacobian of T , which is a five-dimensional principally polarized abelian variety, thus classified by a point in the moduli space A 5 = Sp(10, Z)\H 5 , where H 5 is the Siegel upper half plane of genus 5. Taking a standard model (Z 10 , < , >) of a free abelian group of rank 10 with integral symplectic form < , > and defining hermitan form h ′ on Z 10 ⊗ C = C 10 = C 5,5 as above, H 5 is given as the space of positive Lagrangian subspaces. These subspaces of C 10 are 5-dimensional, < , >isotropic, and h ′ -positive. We obtain a map ι : B 4 −→ H 5 defined by ι(l) =l ⊥ ⊕ l ⊂ C 5,5 = C 4,1 ⊕ C 1,4 , see (9.2.1) of [2] and (4.2) of [4]. This gives a holomorphic, totally geodesic embedding ι : B 4 −→ H 5 which is equivariant with respect to the natural homomorphism λ : Γ → Sp(10, Z) given by λ(γ) =γ ⊕ γ = t γ −1 ⊕ γ in the decomposition (6) into spaces that are simultaneously dual and conjugate to each other (transposition is with respect to the complex bilinear form h 0 (z,w)). The image of λ is the centralizer of σ in Sp(10, Z). It follows that ι descends to an embedding ι : Γ\B 4 −→ A 5 , and the image of ι consists of those principally polarized abelian varieties that are isomorphic to one on which σ acts.
2.6. The hyperplane arrangement. The group Γ is generated by complex reflections of order 6 (hexaflections) on the hyperplanes v ⊥ , where v ∈ E 1,4 and h * 0 (v) = −1, where h * 0 = −h 0 is the standard form on C 1,4 , see §7 of [2]. Let H ⊂ B 4 be defined by This is a locally finite collection of hyperplanes and the group Γ acts transitively on its elements. Moreover, H is an orthogonal arrangement of hyperplanes, meaning that any two elements are either disjoint or meet at right angles, see (7.29) of [2]. In particular, given any x ∈ B 4 , there can be at most 4 hyperplanes in this collection that contain x.

2.7.
Explicit formula for period map. To get an explicit formula for the period map, let γ = {γ 0 , . . . , γ 4 } be a basis, as a Z[ω]-module, for the dual space H 3 (T, Z) of H 3 (T, Z), for which the dual hermitian form is diagonal with diagonal entries (−1, +1, +1, +1, +1). (This choice is equivalent to a framing as in (3.9) of [2].) Let Φ = Φ(F ) be the nonzero element Ω/F 4/3 of H 2,1 ω associated to the cubic form F as in (3) of Theorem (6.5) of [2]. Then the period vector associated to these choices is the vector More explicitly, let f (x, y, z) = 0 be an affine equation for the cubic surface in one of the standard affine open sets of P 3 . Then w 3 = f (x, y, z) is an affine equation for the cubic threefold, and the natural generator Φ(F ) for This is the analogue of the abelian differential dx/y for an elliptic curve.
• The map P of (7) descends to a map P : M sm −→ Γ\B 4 of the quotient spaces, and extends to the map P : M st −→ Γ\B 4 of (2), which is an isomorphism.
• Moreover, if M sm is the moduli space of smooth cubic surfaces, then its restriction to M sm gives an isomorphism • Finally, if M nod is the moduli space of nodal cubic surfaces, then M st = M sm ∪ M nod and the restriction of the period map to M nod gives an isomorphism 2.9. The abelian varieties. We need one final result that is not explicitly stated in [2] but follows easily from that paper. In fact, it is the underlying philosophy of proofs of the results just quoted.
if and only if it has an automorphism σ of order 3 of type (4, 1), meaning that the induced action on H 1,0 (A) has eigenvalue ω of multiplicity 4 andω of multiplicity 1.

if and only if
A is reducible and contains at least one irreducible factor isomorphic to a Fermat elliptic curve.
The proof of the first statement follows from the last statement of (2.5) and the fact that all automorphisms of type (4, 1) are conjugate under Sp(10, Z). The second and third statements follows from the fact that the intermediate Jacobian of a non-singular cubic threefold is irreducible, while each nodal surface gives rise to an A 2 -singularity in the associated threefold, which is turn gives a summand of the Hodge structure of a Fermat elliptic curve. See Lemmas (5.4), (5.7) of [2] for the last point.
2.10. New results. Given this background, we now come to the results of this paper. If f has coefficients in a field K, then so does Φ -it is a Krational differential. Consequently Φ is well-defined up to a nonzero element of K. Therefore questions about the rationality of periods, referred to such a K-rational differential, make sense for any field L extending K.
There is a philosophy that periods of K-rational differentials are almost always transcendental; this is, however, difficult to prove in any concrete instance. Thus our focus will therefore be on period ratios such as ω 2 /ω 1 for cubic curves or v i = γ i Φ γ 0 Φ for cubic surfaces. In the last ratio, we may rescale Φ so that v 0 = 1. The resulting periods v i , which are now relative to a differential with unknown rationality properties, should be thought of as period ratios. This will often be our point of view in what follows.
Let us write the period vector as ). When a = 1, we say that the period vector is normalized. In this case, since (a, b) is a positive vector vector for h * 0 , we have |a| 2 − |b| 2 > 0, so b is a vector in C 4 of length at most 1. The corresponding Hodge structure of level one and genus five is determined by a 5 × 10 matrix of periods P = (A, B), where P represents ι(a, b). In the usual terminology, P is normalized in the case that A is the identity (in which case B ∈ H 5 , that is, B is symmetric with positive-definite imaginary part). Section (2.9) tells us that, in the presence of the action by σ, the period vector and the period matrix determine each other, up to natural equivalences (action of Γ and Sp(10, Z)).
Finally, recall that an abelian variety over C is said to be of CM-type if (a) it is isogeneous to a product A 1 × · · · × A r of simple abelian varieties If the fields K i are equal, we say that K = K i is the CM field of the abelian variety. See Mumford [8, p. 347]. We can now state our first result. Theorem 1. Let S be a cubic surface and let J be its abelian variety. The following are equivalent: (a) one (and hence all) normalized period vectors of S have coefficients in Q(ω); (b) one (and hence all) normalized period matrices of S have coefficients in Q(ω), (c) J is isogeneous to a product of Fermat elliptic curves.
It follows from the theorem that for a cubic surface with period vector in Q(ω), End(J) ⊗ Z Q is the ring of 5 × 5 matrices with coefficients in Q(ω); as a consequence of the main theorem, we see that cubic surfaces with period vector in Q(ω) are of CM-type with CM field Q(ω). In particular, we get the following well-known fact: Because the period map (2) is surjective, there is a dense set of smooth cubic surfaces with periods in Q(ω). As noted in [2, Theorem 11.6, 11.9], some explicit surfaces with periods of this kind are known. A period vector (not normalized) of the Fermat cubic surface x 3 + y 3 + z 3 + w 3 = 0 is and that of the diagonal cubic surface 1, 1, 1, 1).
Again, it is more convenient to write this vector in non-normalized form.
The rationality properties of the differential form which gives the periods are unknown. Consequently the periods should be thought of as period ratios.
It is natural to ask how to characterize cubic surfaces with periods in Q(ω) in purely geometric terms. We do not know the answer, even conjecturally.
In a different direction, we state our last result, which concerns the opposite extreme of complex multiplication, namely cubic surfaces whose CM algebra is a field: Theorem 2. There exist simple, principally polarized abelian varieties A of dimension five with three-fold symmetry σ of type (4, 1) and with rational endomorphism ring End(A) ⊗ Q isomorphic to a CM-field K of the form where K 0 is a totally real field of degree five. The Hodge structure of A is rational over K, and it is defined by a period vector rational over the same field. By (2.9), A is the abelian varietly of a smooth cubic surface.
Remark. We study extreme cases of CM types. In [1, 6.2], Achter classifies, up to isogeny, all possible CM types, endomorphism algebras, and Mumford-Tate groups of abelian five-folds which admit an action of Z[ω].

Cubic surfaces with period vector rational over Q(ω)
The key to the proof of the first theorem is an analysis of the periods of the most singular stable cubic surface. We begin with the following: Proposition 1. Any normalized period vector of the Cayley cubic surface, Proof. The Cayley cubic surface is a cubic surface with 4 nodes. It is classically known to be the surface with the maximum possible number of nodes. Thus, by the last part of (2.8), it lies in the intersection of 4 distinct hyperplanes in H. One such point in (1, 0, 0, 0, 0), which is the intersection of (0, 1, 0, 0, 0) ⊥ , . . . , (0, 0, 0, 0, 1) ⊥ . It is easy to see that these are precisely the points in v ∈ E 1,4 with h * 0 (v) = 1 and that all such points are equivalent under Γ. Thus the period of the Cayley cubic is the Γ-orbit of (1, 0, 0, 0, 0).
Note that this argument implies the well-know classical fact that all 4nodal cubic surfaces are isomorphic over the complex numbers. Also (2.8) combined with (2.6) reproves the classical fact quoted in this proof that 4 is the maximum number of nodes.
As we show in a moment, the formula we will derive for the period matrix in terms of the period vector will give: where I is the identity matrix.Thus the intermediate Jacobian of the Cayley cubic surface is the product of five Fermat elliptic curves.
As a general principle, the period vector in our context determines the period matrix and conversely: Let us now compute the period matrix P associated to a general cubic surface S with equation F = 0, that is, the period matrix of the intermediate Jacobian of the associated cyclic cubic threefold T as in (2.1). To this end, recall from (2.7) that γ = {γ 0 , · · · , γ 4 } is a unitary basis of H 3 (T, Z) as a hermitian Eisenstein module. Let γ ′ = {γ ′ 0 , · · · , γ ′ 4 } = {σ −1 γ 0 , σγ 1 , · · · , σγ 4 }. The the homology basis γ ∪ γ ′ is a sympletic basis with respect to the standard form where I is the identity matrix. Choose a basis {Φ 0 , . . . , Φ 4 } for H 2,1 where Φ 0 ∈ H 2,1 ω and Φ i ∈ H 2,1 ω for i > 0. Then the period matrix takes the form (11) P = (A, B) where The change of variable formula in the calculus coupled with the fact that the Φ i are eigenvectors of σ imply that where λ ∈ {ω,ω}. We may choose the basis {Φ 1 , . . . , Φ 4 } so that A ij = δ ij for i, j ∈ {1, . . . , 4}. The first Riemann bilinear relation determines the first column of A in terms of the second, while (12) determines B in terms of A. We conclude that In the case of the Cayley cubic surface, Proposition (1) gives that the parameter vector b is zero and the period matrix is where I is the identity matrix. It follows that the abelian variety of the Cayley cubic surface is the product of five Fermat elliptic curves. This proves Corollary 2.
The map ψ(b) = Z, where Z = BA −1 and A, B are as in (11), gives a matrix formula for the imbedding ι of (2.5) of the unit ball in the Siegel upper half space of genus five. It can be written more concisely as To understand better the location of Z(b) in the Siegel upper half space, note the quantity δ = det where Z is the normalized matrix of B-periods, a symmetric matrix with positive definite imaginary part. The inverse of A is given by the following expression: Re-writing the matrix Z, we obtain the following proposition: be a a row vector (thus |b| < 1), let t b be the corresponding column vector, and let b ⊗ b be the matrix whose ij-th entry is a function of b that maps points of the unit ball to points of the Siegel upper half space, with the origin of the ball mapped to the normalized period matrix of product of five Fermat elliptic curves. It gives a matrix formula for the imbedding ι : B 4 −→ H 5 defined in (2.5).
Proof of Theorem 1.
The proof proceeds as follows. (1) The isogeny class of an abelian variety with lattice Λ ⊂ C n is the isomorphism class of the embedding Λ ⊗ Q ⊂ C n (isomorphism by complex linear maps of C n ). (2) If Λ 0 is the lattice of E 5 , where E is the Fermat elliptic curve, then Λ 0 ⊗ Q = Q(ω) 5 ⊂ C 5 . (3) If the period vector b is in Q(ω), then formula (13) for the period matrix shows the columns of the matrix, which give a basis for the lattice Λ, have entries in Q(ω). Thus Λ ⊗ Q is isomorphic to Λ 0 ⊗ Q as rational subspaces of C 5 .

Proof of the second theorem
As we have just seen, one can distinguish certain points in the ball quotient, e.g., those whose period vector is rational over Q(ω). In that case the Abelian variety is isogeneous to a product of Fermat elliptic curves.
We now seek special points in the ball quotient where the corresponding abelian variety is simple and where the rational endomorphism ring is a number field. To this end, consider first a totally real number field K 0 of degree n, and a purely imaginary quadratic extension which we may write as K = K 0 ( √ δ) for some element δ in K 0 . The field K is a CM field. It has 2n distinct embeddings τ i in the complex numbers. A CM type for K is a choice Φ = (τ 1 , . . . , τ n ), where τ i =τ j for any i, j. Let O K be the ring of integers in K. Then one can form the complex torus A(K, Φ) = C n /Φ(O K ). Note that the ring O K acts by endomorphisms on A(K, Φ), so that If A(K, Φ) is simple, then the rational endomorphism ring is a division algebra of dimension at most 2n over Q. But dim Q K = 2n, so Thus the rational endomorphism ring of such a torus is the field K. Given such an element, the expression defines a positive Hermitian form on K. Its associated skew form is Since α is an element of the field K, the trace form takes rational values.
To ensure that the values of the form are integers, we choose α to be in O ∨ K , the lattice dual to O K .
To show the existence of the element α in ( * ), Mumford argues as follows. Let δ be an element of K 0 such that K = K 0 ( √ δ). Since K is a totally imaginary extension of the totally real field K 0 , for each i we can write where γ i is a nonzero real number. One can also find an element η ∈ K 0 such that τ i (η) and γ i have the same sign for all i. Then There remains the question of whether this polarization, which is determined by a suitable element α ∈ K, can be chosen to be principal. If α ∈ O ∨ K and in addtion,  In the case that K = K 0 ( √ −3), one can restate the criterion ( * ) in terms of the different of K 0 . To this end we introduce the following notion. Definition 1. Let β be an element of a totally real field K 0 of degree n over Q. Let τ i , i = 1..n be the imbeddings of K 0 in the real numbers. Let ǫ be an n-vector with entries ±1. Then β is ǫ-positive if In the case that ǫ i = +1 for all i, ǫ-positivity the same as total positivity. Proposition 5. Fix a vector ǫ = (ǫ 1 , . . . , ǫ n ) with ǫ i = ±1. Let Φ be a CMtype for K which extends a set of embeddings τ i : K 0 −→ R to embeddings Suppose that there is an ǫ-positive element β of K 0 whose inverse generates the different of K 0 . Then α = −βθ −1 defines a principal polarization of A(K, Φ).
Proof. The element α = −βθ −1 is the product of inverses of generators of the different for K 0 and Q( √ −3). By the multiplicativity of the different, this product is an inverse of a generator of the different of K, that is, a generator of O ∨ K . See [9,Prop. 2.2,p. 195]. Now The quantity in parenthesis is positive if ǫ i τ i (β) is positive, Thus both (19) and ( * ) are satisfied.
There remains the question of whether the conditions of ǫ-positivity for a generator of the different of K 0 can ever be satisfied. The next result sets forth a criterion for its satisfaction which can sometimes be verified by computation. We will carry out such a computation in the next section. Proposition 6. Let Φ and β be as in the preceding proposition. Suppose that there is a unit η of O K 0 such that τ i (η) has the same sign as ǫ i τ i (β). Then the element βη is ǫ-positive.
Let us return to the case of K = K 0 ( √ −3), where K 0 is a totally real quintic field, and consider the problem of whether there are choices so that A(K, Φ, α) is a principally polarized abelian variety which has an automorphism of order three, and such that the eigenvalues of that automorphism on the space of abelian differentials are ω with multiplicity four andω with multiplicity one.
It is the totally really quintic field of smallest discriminant, namely 11 4 = 14641. It is not hard to see that Q(ζ) is a CM field and that one can construct a simple Abelian variety from it. See [5, p. 24]. What we need, however, is a simple abelian variety with a suitable action of ω. To this end, we establish the following: . Let Φ be the CM type extended from a vector of embedding (τ 1 , . . . τ 5 ) of K 0 as in equation (20). Then A(K, Φ) is a simple, principally polarized abelian variety. It therefore corresponds to a smooth cubic surface.
Proof. The class number of K 0 is one, so that all ideals in O K 0 are principal. The different of O K 0 is the ideal generated by where r = −ρ generates K 0 . This element is not totally positive. The group of units of O K 0 is isomorphic to C 2 × Z 4 . Generators for the free abelian part of this group are so that an arbitrary unit has the form ±u j 1 u k 2 u ℓ 3 u m 4 . One finds that element δ = u 4 δ 0 , which also generates the different ideal, is ǫ-positive for ǫ = (+, +, −, +, +). The element β = δ −1 is also ǫ-positive, and it is the element we use to define the principal polarization. The criterion for ǫ-positivity is satisfied, and the abelian variety A(K, Φ, α) is principally polarized.
To conclude that A(K, Φ, α) corresponds to a smooth cubic surface, following 2.9, it only remains to show that it is irreducible. It will be enough to show that it is simple. To that end, consider the Galois group G of K/Q. It is a group of order ten. The subgroup Gal(K 0 /Q) is normal. Since it is of order five, it is cyclic. The subgroup Gal(Q(ω)/Q) is also normal, and since it is of order two, it is cyclic. A group of order ten with these structural features must be cyclic.
The subfields of K are the fixed sets of subgroups of the Galois group. Since a cyclic group of order ten has only two non-trivial subgroups, the field K has only two nontrivial subfields, namely, K 0 and Q(ω). Once we know the subfields of K, Mumford's criterion for simplicity of A(K, Φ, α) [7, p. 213-14] is easy to apply. If suffices to show that there are τ i and τ j such that τ i |Q(ω) = τ j |Q(ω). The condition on the τ 's holds by construction. See equation (20).
We have a construction If Z ij is rational over K then so is b. The converse also comes from formula (14). The period vector could be rational over K 0 and still correspond to a period matrix rational over K; it cannot, however, be rational over Q or Q(ω).

Computations and experiments
Above we described a method to show that the totally real quintic field of discriminant 14641 defines a principally polarized abelian variety with three-fold symmetry of the correct type. The same method can be used to produce lists of quintic fields with this property. Using the Sage code below, for example, we show that (1) there are 414 totally real quintic fields of discriminant less than 10 6 ; (2) none of these have class number bigger than 1; (3) there are 412 fields which satisfy the hypotheses of Proposition 5 (4) the density of such fields in the indicated range is about 0.995. Below is the data for the fields of discriminant < 10 5 . The second column is the discriminant of the field. The third column answers the question: Is there a cubic surface with the given discriminant ?   discr  result  1  14641  True  2  24217  True  3  36497  True  4  38569  True  5  65657  True  6  70601  True  7  81509  True  8  81589  True  9 89417 True The test for ǫ-positivity is carried out by the function below: def epsilon_positive(x): pos_minus_neg = sum( sgn(tx) for tx in x.complex_embeddings() ) return ( pos_minus_neg == 3 ) Finally, we enumerate the totally real quintic fields of discriminant less than N, applying the above test to each, and collecting various statistics. nNonUFD = nNonUFD + 1 ratio = nPass/(1.0*n) (9) above, are found by symmetry considerations. The Fermat and Clebsch surfaces are uniquely determined by their automorphism group and this group is generated by reflections. This suggests an easier sub-problem: • Find values of the period map for all surfaces with symmetries These surfaces are totally classified. One special family of such surfaces that should be more tractable is the following: • Find the values of the period map for all cyclic cubic surfaces. Express them as explicit functions of the periods of the corresponding cubic curves. A cyclic cubic surface means a surface S ⊂ P 3 given by an equation is a non-singular cubic form, in other words, S is the cyclic cover of P 2 branched along the cubic curve f = 0.
• Study special values of the period map. We mean the situation when the abelian variety has complex multiplication.
• Find an explicit inverse to the period map. We mean: given b ∈ B 4 − H, find an explicit cubic form F so that the class of P(F, γ) is b. In principle this can be done using suitable theta functions. Allcock and Freitag [3] and Matsumoto and Terasoma [6] have given projective embeddings of Γ\B 4 with image an algebraic embedding of (a covering of) M st . In this way one obtains formulas for algebraic invariants of F in terms of b. but we are not aware of explicit formulas for a cubic form.
• Study the algebraic and arithmetic nature of the period map. Both the domain and target of P : M st −→ Γ\B 4 are algebraic varieties and P, in spite of its transcendental definition, is an algebraic map. See [1] for more information and for progress on this question.