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Period integral of open Fermat surfaces and special values of hypergeometric functions

In the previous paper by Asakura-Otsubo-Terasoma, we prove that the special values of the hypergeometric function 3F2 at 1 are linear combinations of logarithms of algebraic numbers and 1 over algebraic numbers, if exponents are rational numbers satisfying a certain arithmetic condition. Aoki and Shioda completely classified these sets of rational numbers satisfying this condition in connection with Hodge cycles on Fermat surfaces. In this paper, we give an explicit expression of special values of hypergoemetricy 3F2 which does not belong to exceptional characters.

Here, Q + Q log(Q × ) denotes the Q linear hull of log(λ) with λ ∈ Q × and 1.
As is explained in the next subsection, the special value of hypergeometric function of the above type is related to the extension class of mixed K-mixed Hodge structures (1.1.4) 0 → H 1 Hg (B, K)(χ α ) → H 2 Hg (X m , B, K)(χ α ) → H 2 Hg (X m , K)(χ α ) → 0 arising from the relative cohomologies of Fermat surface X m and its divisor B. Theorem 1.1 is a consequence of the fact that the extension class of (1.1.4) is actually a Hodge realization of a mixed Tate motives if the condition (1.1.2) is satisfied. In the proof of Theorem 1.1 in [AOT], we use Lefschetz-Hodge theorem and the fact that an algebraic cycles on algebraic variety defined over Q are defined on Q. Thus the method in [AOT] does not give an closed formulas for the special value (1.1.1). In this paper, we give an explicit formula for the above theorem for non-exceptional cases (see Theorem 1.2 for the definition of exceptional cases).
Let us explain the outline of the paper. In Section 1, we recall the result of [AOT] by introducing period integral of an open Fermat surfaces, or dually period for the relative cohomology Fermat surfaces with their divisors. Let χ be a character of a group G m acting on X m . If the χ-part of the cohomology of Fermat surface is generated by algebraic cycles, the corresponding periods can be written using the logarithmic function evaluated at algebraic numbers. We recall the result by Aoki and Shioda on algebraic cycles on Fermat surfaces. We also recall that except for finitely many characters, they are obtained by four types of characters. We give an explicit formula belonging to four types.
In Section 2, we explain that the existence of algebraic cycles implies the exactness for certain rational differential forms. The equation of algebraic cycles are key to find differential forms which bound the given differential forms. Actually, if one find such differential form, the story is independent of the existence of algebraic cycles.
From Section 3 to Section 5, we compute the integral and show that the integral expressing the extension class actually are expressed by simpler integration, which gives the explicit expression by logarithmic functions. Here we use Stokes formula for the variety obtained by blowing up C 2 .
In the first proof of [AOT], we used regulator maps for a symbols. If the symbol can be written explicitly, we also have a closed formula of the special value (1.1.1). In [AY], they obtained explicit formulas for the special values for some cases using this method. The one form obtained in Section 2 gives a key to find an explicit expression of related K-group via symbols. This topics will be left to a future research.
Acknowledgement This paper was first considered as a continuation of the paper [AOT]. The author express his acknowledgement to M. Asakura and N. Otsubo for discussion and private communication. He also thank them to let him know the relationship with regulator maps in K-theory and hypergeometric functions, which is a driving force to let him go into this subject.
1.2. Algebraic cycles on Fermat surfaces. A proof of the above theorem is based on the fact that the value (1.1.3) is considered as a period integral of certain relative cohomology of Fermat surface. We recall some properties of cohomologies of Fermat surfaces. Let X m be an affine Fermat surfaces defined by and µ m be the group of m-th roots of unities in C × and set K = Q(µ m ). Then the group G m = (µ m ) 3 acts on X m by where a i = mα i . The χ a -part of the singular cohomology H 2 B (X m , K) and algebraic de Rham cohomology H 2 dR (X m /Q) over Q of X m are denoted by H 2 B (X m , K)(χ a ) and H 2 dR (X m /Q)(χ a ), respectively. Then we have dim(H 2 dR (X m /Q)(χ a )) = dim(H 2 B (X m , K)(χ a )) = 1. The condition (1.1.2) is equivalent to the following condition (see [S]): (1.2.7) The space H 2 B (X m , K)(χ a ) is generated by Hodge cycles. The complete classification of the set of indices (a 0 , a 1 , a 2 , a 3 ) satisfying the condition (1.2.7) is conjectured in [S] and proved by [A].
The characters listed in (3) of Theorem 1.2, which is not appeared in (1), (2) is called the exceptional characters. There are 101 Galois orbits in the exceptional characters and the list of orbits is given in Appendix (Compare for the list in [S]).
1.3. Extension of mixed Tate motives.
1.3.1. Tate Hodge structure. Let m ≥ 2 be an integer and K be the field generated by µ m = e(1/m) over Q. A mixed K-Hodge structure V Hg consists of triple of filtered vector space over C, K and a comparison isomorphism compatible with W such that (F, F , W ) is oposit filtration defined in [D]. For mixed K-Hodge structures V 1,Hg , V 2,Hg , a homomorphisms from ϕ : V 1,Hg → V 1,Hg is a pair of homomorphisms ϕ B : V 1,B → V 2,B and ϕ dR : V 1,dR → V 2,dR preserving comparison isomorphisms c and filtrations W, F . By proposition in [D], the category of mixed K-Hodge structures whose morphisms are morphism of mixed K-Hodge structures becomes an abelian category. For a mixed Hodge structure V Hg = (V B , V dR , c) over Q, the tensor product V Hg ⊗ Q K = (V B ⊗ Q K, V dR , c) becomes a mixed K-Hodge structure in a natural way.
Tate Hodge structure Q(1) Hg is defined by the triple (Q(1) dR , Q(1) B , c), where Q(1) dR and Q(1) B are one dimensional vector spaces over C and Q generated by ξ dR and ξ B with be an exact sequence of mixed K-Hodge structures. Let c : K(−1) B ⊗ K C → K(−1) dR be the comparison isomorphism and s B and s dR be elements in Then we have an isomorphism cl : Ext 1 M HS(K) (K(−1), K) = C/2πiK by setting cl(e Hg ) = 2πic(s B ) − s dR ∈ 2πiK dR = Cξ 0 dR mod 2πiK B = 2πiKξ 0 B . The lifting satisfying the above condition (2) is called the Hodge canonical lift.
1.3.3. Extensions arising from relative cohomologies of Fermat surfaces. Let (X, B) a pair of varieties such that B ⊂ X. Let H i B (X, B, K) and H i dR (X, B/C) be the singular and de Rham cohomologies of the pair (X, B) with the coefficient in K and de Rham cohomology with the coefficient in C. By the natural comparison map c, we have a mixed K-Hodge structure (H B (X, B, K), H dR (X, B), c) Let G → Aut(X, B) be a group action of G on (X, B) and χ be a character of G with the value in K × . Then H Hg (X, B, K)(χ) = (H B (X, B, K)(χ), H dR (X, B)(χ), c) becomes a mixed K-Hodge structure.
We define a subvariety B of Fermat sufrace X m by The the variety B is a union of affine lines.
The χ a -part of the relative singular cohomology H 2 B (X, B, K) and relative de Rham cohomology H 2 dR (X, B/Q) of (X, B) are denoted by H 2 B (X, B, K)(χ a ) and H 2 B (X, B/Q)(χ a ), respectively. By the long exact sequence of relative cohomology, we have an exact sequences: These sequences are compatible with the comparison maps We set Therefore the sequences (1.3.8) together with the comparison maps defines a Yoneda extension class e of mixed K-Hodge structures Under the condition (1.2.7), we have isomorphisms of Hodge structures To prove Theorem 1.1, we use theory of mixed motives and its Hodge realization. By the Lefschetz-Hodge theorem, these isomorphisms arise from algebraic correspondences. Since the pair of varieties (X, B) are defined over Q, the above algebraic correspondence is defined over Q. Let M M (K)/L be the derived category of mixed motives over L with the coefficients in M . Therefore we have the following commutative diagram where the horizontal arrow are induced by algebraic correspondence over Q.
Since the Hodge realization map is given by e M → log(e M ), we have cl(e Hg (X, χ a )) ∈ K log(Q × ) = Im(ρ),

Extension class and relative periods.
The extension class e Hg can be computed by the period integral as follows: Let s B , s dR be liftings of a common bases of H 2 B (X, K)(χ a ) H 2 dR (X, Q)(χ a ) satisfying the conditions (1)-(2) in the last subsection, and b dR be a base of H 1 dR (B/Q)(χ a ). Then we have 2πic(s B ) = s dR + cl(e Hg )b dR , where e Hg = e Hg (X, χ a ). We consider the Betti part of the dual of the exact sequences (1.3.8).
As for the precise argument, see [AOT].
1.4. Period integrals for relative cohomologies and hypergeometric function. We consider a sequence of morphisms: (1.4.10) with the standard orientations and Γ ′′ 1 be a topological cycle on X m defined by Let γ 0 be a one chain defined by Proposition 1.3. Under the above notation, we have pr Since the pairing is given by integrals, we have In the following, we give an explicit formula for (c(pr χ (Γ 1 )), ω) for the classification (1)-(2) of Theorem 1.2. The following proposition give a relation between the period integrals for relative cycles and special values of the hypergeometric function.
Then the integral (1.5.11) is analytic function of α for α ∈ Z and for −1 < α. Thus we have the following theorem.
Theorem 1.5. Let α, β be real numbers such that α, β, α − β ∈ Z. We have the following identity: By the following proposition, the above integral expression gives an explicit formula as an element in Q + Q log(Q × ).
Proposition 1.6. Let α = n m , β = n ′ m ∈ (0, 1) is a rational number. We choose a suitable branch of log and a suitable path [0, 1]. Then we have The second statement follows from the first.

Algebraic cycles and one forms with constant residues
2.1. Algebraic cycles on Fermat surfaces. As for the case (1)-(2) in Theorem 1.2, algebraic cycles generating the Hodge classes are given as follows (see [AS]). We define varieties Z Proposition 2.1. We have the following inclusions i,k,± ) we have i,k,± ) Thus we have the required inclusions.
By the above proposition, we have Then the varieties Z is stable under the action of G pm = µ 3 pm given as (1.2.5). We set x = u m , y = v m , z = w m and for an element of α ∈ 1 pm Z and use a notation x α = u mα , etc.
We define characters φ p of G pm by setting φ p = χ a , where Here the Kummer character χ a is defined by (1.2.6) The following proposition is proved in [AS].
Proposition 2.2 ( [AS]). The φ p parts of the cohomology classes of the components in Z p generates H 2 (X pm )(φ p ). In other words, the map H 2 Z p (X pm )(φ p ) → H 2 (X pm )(φ p ) is surjective. As a consequence, the map is the zero map by the localization exact sequence.
2.2. Algebraic cycles and differential forms. Let p = 1, 2, 3. We express the de Rham cohomology H 2 (X pm )(φ p ) by that of an open set X 0 p defined by X 0 pm = {(u, v, w) ∈ X pm | uvw = 0} By branch condition, we have Let Ω = Ω p ∈ Γ(X 0 pm , Ω 2 )(φ p ) be a closed form whose cohomology class generates H 2 (X pm )(φ p ). We set U p = X 0 pm − Z p and the inclusion U p → X 0 pm is denoted by j. We consider the following diagram: By Proposition 2.2, there exists an element ψ ∈ Γ(X 0 pm , Ω 1 X pm (log(Z p ))(φ p ) such that dψ = j * Ω.
By the above diagram, res(ψ) is a locally constant function. Vice versa, if res(ψ) of a one form ψ is a constant function on each component, then Ω = dψ is a closed form and an image under the map is not zero, it becomes a base. In the next subsection, we construct ψ satisfying the above equation.
2.3. One forms ψ with constant residues along Z p . For p = 1, 2, 3, 4, we consider varieties and consider a sequence of morphisms: We define one form ψ 0 on X pm by and rational functions f by Since f | Z p is a constant function, the residue of a differential form ψ = f α ψ 0 is constant on each component of Z p . Using relations dψ 0 = 0 and By setting x p = ξ, y p = η, we have the following multi valued differential form on A 2 : Then the integral Γ i Ω converges, if α and β are real numbers and satisfy the following conditions. 3.1. Blowing up and Stokes' formula. For a real number α, we set (−1) α = e(α/2). Let Γ i and be (i = 1, 2, 3) topological chains defined in (1.6.12). We define a topological cycle Γ ′′ 1 , Γ ′′ 2 , Γ ′′ 3 on X pm for p = 1, 2, 3, 4 by Here π ′ •π is a morphism defined in (2.3.1). The image π(Γ ′′ i ) of Γ ′′ i under the map π defined in (2.3.1) is denoted by Γ ′ i . Then we have the following diagram: In the last section, we have a relation dψ = j * Ω on X p . Since the pole of the differential form ψ p passes through the singular points , we can not apply Stokes' formula for Γ ′ i in X p . Thus we consider the blowing up b : X p → X p of the variety X p with the center B ′ ∩ Z ′ . Let E 2 and E 3 be the exceptional divisors over the points p 2 and p 3 , respectively.
The chain Γ i is called the proper transform of Γ ′ i for short. From now on we assume that α and β satisfy the convergent condition Table 1. Via the birational map b, we have an equality Let B i and R be the proper transform of B i (i = 2, 3, x, y) and R = {x p + y p = 1} ∪ {x = 0} ∪ {y = 0}, respectively. Then we have Since d ψ p = Ω on X p and the pole of the rational differential form ψ p does not intersect with the proper transforms Γ i for i = 2, 3, we have 3.2. Computation of restrictions for Γ 1 . The restriction of the rational one form ψ to B i (i = x, y, 2, 3) and E j (j = 1, 2) are denoted by ψ B i = ψ| B i and ψ E j = ψ| E j , respeictively. We have ψ | x p +y p =1 = 0 on X p for p = 1, 2, 3. The restrictions ψ B 2 , ψ B 3 are computed as follows.
Theorem 3.1. Suppose that α, β be real number satisfying the condition of Table 1. Then we have the following equalities.
4.2.1. Hypergeometric identities for X 2m . By Stokes's theorem on X 2m , we have the following equality: By analytic continumation, we have the following theoreum.
Theorem 4.1. Let 2α ∈ Z. Then we have the following equalities.

4.2.2.
Hypergeometric identities for X 3m , X 4m . Using Stokes' theorem, the integral is computed using the boundary integral on Γ 2 . By Proposition 1.4, we have the following theorem.

Moreover the first line is equal to the third line if 4α ∈ Z.
Proof.
(2) is similar by applying Stokes' formula to X 4m .

5.
Hypergeometric identities for Γ 3 5.1. Restriction of ψ to the boundary for Γ 3 . In this section, we give a hypergeometric identities arising from the integration on Γ 3 for p = 3, 4.
5.2. Hypergeometric identities for Γ 3 . We similarly compute the integral on Γ 3 and get the following theorem.