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Periods of generalized Tate curves

A generalized Tate curve is a universal family of curves with fixed genus and degeneration data which becomes Schottky uniformized Riemann surfaces and Mumford curves by specializing parameters. For each generalized Tate curve, we give explicit formulas of the period isomorphism between its de Rham and Betti cohomology groups, and of the associated objects: Gauss-Manin connection, variation of Hodge structure and monodromy weight filtration. A remarkable fact is that similar formulas hold commonly for families of Riemann surfaces and of Mumford curves. Furthermore, we study the unipotent periods of a generalized Tate curve, and show their limits are described by multiple zeta values.


Introduction
The study of period isomorphisms between the de Rham and Betti cohomology groups of families of algebraic varieties is one of main subjects in algebraic geometry. Their behavior under degeneration of considering varieties is especially interesting, and the associated p-adic version, unipotent version and p-adic unipotent version are also studied. For a universal family of elliptic curves, these objects are described by hypergeometric functions and equations, however, there seems no similar description on curves of genus > 1. The aim of this paper is to give such explicit formulas for a universal family of general curves which are commonly applicable to complex and p-adic analytic cases using the theory of generalized Tate curves. Furthermore, we consider the unipotent version, and show the limits of unipotent periods are expressed by multiple zeta values.
A generalized Tate curve is introduced in [17] as a higher genus version of the Tate elliptic curve which gives a universal deformation of degenerate curves of fixed type. The base ring of a generalized Tate curve is taken to be primitive as far as possible, namely it consists of formal power series in deformation parameters whose coefficients belong to the ring over Z of moduli parameters of the degenerate curves. By specializing these parameters in C (resp. a nonarchimedean complete valuation field), a generalized Tate curve gives families of Schottky uniformized Riemann surfaces (resp. Mumford curves).
In this paper, we give explicit formulas of abelian differentials defined on a generalized Tate curve called universal differentials, and show their variational formulas which imply an important fact that the universal differentials are stable, namely they have only logarithmic poles on the special fiber. By regarding the generalized Tate curve as a family of degenerating Riemann surfaces, we give the explicit description of the period isomorphism between the de Rham and Betti cohomology groups, and of the associated Gauss-Manin connection, variation of Hodge structure and monodromy weight filtration. This description is written by multiple moduli and deformation parameters in the category of formal geometry over Q, and is same to formulas for Mumford curves which is studied by Gerritzen [12] and de Shalit [7,8] using Coleman integration [2,3] in the special case of p-adic base fields.
As a further application of the above result, we study unipotent periods which give isomorphisms between the de Rham and Betti unipotent fundamental groups of curves. By the arithmeticity and stability of universal differentials, we give a Q-structure and Hodge, weight filtrations on the unipotent de Rham bundle on a maximally degenerating family of curves. Furthermore, extending results of Deligne [5], Drinfel'd [9] and Hain [14] on the Knizhnik-Zamolodchikov-Bernard (KZB) equation in the case of genus 0 and 1, we show that the limits of the unipotent periods of a generalized Tate curve are expressed by multiple zeta values. These limits are concerned with the limit Hodge structure of the unipotent fundamental groups, and the p-adic unipotent version using Coleman integration in families of Mumford curves seems to give a higher genus extension of result of Furusho [10,11].

Schottky uniformization
We review the theory of Schottky uniformization of Riemann surfaces. A Schottky group Γ of rank g is a free group with generators γ i ∈ P GL 2 (C) (1 ≤ i ≤ g) which map Jordan curves C i ⊂ P 1 C = C ∪ {∞} to other Jordan curves C −i ⊂ P 1 C by reversing orientation, where C ±1 , ..., C ±g with their interiors are mutually disjoint. Each element γ ∈ Γ−{1} is conjugate to an element of P GL 2 (C) mapping z to β γ z for some β γ ∈ C × with |β γ | < 1 which is called the multiplier of γ. Therefore, we have for some element α γ , α ′ γ of P 1 C called the attractive, repulsive fixed points of γ respectively. Then the discontinuity set Ω Γ ⊂ P 1 C under the action of Γ has a fundamental domain D Γ given by the complement of the union of the interiors of C ±i , and R Γ = Ω Γ /Γ is a (compact) Riemann surface of genus g which is called Schottky uniformized by Γ (cf. [23]). Furthermore, by a result of Koebe, every Riemann surface of genus g can be represented in this manner.

Degenerate curve
We review a correspondence between certain graphs and degenerate pointed curves, where a (pointed) curve is called degenerate if it is a stable (pointed) curve and the normalization of its irreducible components are all projective (pointed) lines. A graph ∆ = (V, E, T ) means a collection of 3 finite sets V of vertices, E of edges, T of tails and 2 boundary maps such that the geometric realization of ∆ is connected. A graph ∆ is called stable if its each vertex has at least 3 branches. Then for a degenerate pointed curve, its dual graph ∆ = (V, E, T ) is given by the correspondence: V ←→ {irreducible components of the curve}, E ←→ {singular points on the curve}, T ←→ {marked points on the curve} such that an edge (resp. a tail) of ∆ has a vertex as its boundary if the corresponding singular (resp. marked) point belongs to the corresponding component. For a a finite set X, let ♯X denote the number of elements of X. Then under fixing a bijection ν : T ∼ → {1, ..., ♯T }, which we call a numbering of T , a stable graph ∆ = (V, E, T ) becomes the dual graph of a degenerate ♯T -pointed curve of genus rank Z H 1 (∆, Z), where each tail h ∈ T corresponds to the ν(h)th marked point. In particular, a stable graph without tail is the dual graph of a degenerate (unpointed) curve by this correspondence. If ∆ is trivalent, i.e. any vertex of ∆ has just 3 branches, then a degenerate ♯T -pointed curve with dual graph ∆ is maximally degenerate.
An orientation of ∆ = (V, E, T ) means giving an orientation of each e ∈ E. Under an orientation of ∆, denote by the set of oriented edges, and by v h the terminal vertex of h ∈ ±E (resp. the boundary vertex b(h) of h ∈ T ). For each h ∈ ±E, denote by let |h| ∈ E be the edge h without orientation.

Generalized Tate curve
Let ∆ = (V, E, T ) be a stable graph. Fix an orientation of ∆, and take a subset E of ±E ∪ T whose complement E ∞ satisfies the condition that and that v h = v h ′ for any distinct h, h ′ ∈ E ∞ . We attach variables x h for h ∈ E and y e = y −e for e ∈ E. Let A 0 be the Z-algebra generated by x h (h ∈ E), 1/(x e − x −e ) (e, −e ∈ E) and 1/( According to [17,Section 2], we construct the universal Schottky group Γ associated with oriented ∆ and E as follows. For h ∈ ±E, let φ h be the element of P GL 2 (B ∆ ) = GL 2 (B ∆ )/B × ∆ given by where P GL 2 acts on P 1 by linear fractional transformation. We review the following proposition which is shown in [ For any reduced path ρ = h(1) · h(2) · · · h(l) which is the product of oriented It is shown in [17, Section 3] and [18, 1.4] (see also [19,Section 2] when ∆ is trivalent and has no loop) that for any stable graph ∆, there exists a stable pointed curve C ∆ of genus g over A ∆ which satisfies the following: • The closed fiber C ∆ ⊗ A ∆ A 0 of C ∆ obtained by substituting y e = 0 (e ∈ E) becomes the degenerate pointed curve over A 0 with dual graph ∆ which is obtained from the collection of P v := P 1 A 0 (v ∈ V ) by identifying the points x e ∈ P ve and • C ∆ gives rise to a universal deformation of degenerate pointed curves with dual graph ∆. More precisely, if R is a noetherian and normal complete local ring with residue field k, and C is a stable pointed curve over R with nonsingular generic fiber such that the closed fiber C ⊗ R k is a degenerate pointed curve with dual graph ∆, in which all double points are k-rational, then there exists a ring homomorphism and take y e (e ∈ E) as sufficiently small nonzero complex numbers. Then C ∆ becomes a pointed Riemann surface which is Schottky uniformized by the Schottky group Γ over C obtained from Γ ∆ .
We review the construction of C ∆ given in [17,Theorem 3.5]. Let T ∆ be the tree obtained as the universal cover of ∆, and denote by P T ∆ be the formal scheme as the union of P 1 . Then it is shown in [17,Theorem 3.5] that C ∆ is the formal scheme theoretic quotient of P T ∆ by Γ ∆ .

Universal differential
We define universal differentials on a generalized Tate curve. Let ∆ be a stable graph, and let Γ ∆ = Im (π 1 (∆, v b ) → P GL 2 (B ∆ )) be the universal Schottky group as above. Then it is shown in [17, Lemma 1.3] that each γ ∈ Γ ∆ − {1} has its attractive (resp. repulsive) fixed points α (resp. α ′ ) in P 1 B ∆ and its multiplier β ∈ e∈E A ∆ · y e which satisfy Fix a set {γ 1 , ..., γ g } of generators of Γ ∆ , and for each γ i , denote by α i (resp. α −i ) its attractive (resp. repulsive) fixed points, and by β i its multiplier. Then under the assumption that there is no element of ±E ∩ E ∞ with terminal vertex v b , for each 1 ≤ i ≤ g, we define the associated universal differential of the first kind as Assume that Then for each t ∈ T with v t = v b and m > 1, we define the associated universal differential of the second kind as whose cardinality is 0 or 1, and take a maximal subtree T ∆ of ∆, and for each t ∈ T , take the unique path we define the associated universal differential of the the third kind as , which has only (simple) poles at the points p t 1 (resp. p t 2 ) corresponding to t 1 (resp t 2 ) with residue 1 (resp. −1).
(4) Take x h (h ∈ ±E ∪ T ) and y e (e ∈ E) be complex numbers as in 2.3. Then ω i , ω t,m , ω t 1 ,t 2 are abelian differentials on the Riemann surface R Γ , where Γ is the Schottky group obtained from Γ ∆ .

Stability of universal differentials
For a vertex v ∈ V , denote by C v the corresponding irreducible component of Proof. For the proof of (1), we may assume that γ i = φ h i (1) · · · φ h i (l i ) . Let γ be an element of Γ ∆ . Then by Proposition 2.1, putting y e = 0 (e ∈ E), belongs to γ γ i , and becomes 0 otherwise. Therefore, the assertion follows from the definition of ω i .
The assertion (2) immediately follows from Proposition 2.1, and (3) can be shown in the same way as above.
Remark. Let the notation be as above, and for a family of Schottky uniformized Riemann surfaces obtained from C ∆ as in 2.3, let c e be a cycle corresponding to e ∈ E − E ′ which is oriented by the right-hand rule for γ j . Then the restriction of ω j to an irreducible component of C ∆ ′ is characterized analytically by that its integral along c e is 2π √ −1 if e = |h j,k | for some integer 1 ≤ k ≤ l(j), and is 0 otherwise. Therefore, Theorem 3.2 can be extended for this family degenerating as y e → 0 (e ∈ E ′ ). Since Schottky uniformized Riemann surfaces make a nonempty open subset in the moduli space of Riemann surfaces, by the theorem of identity in the function theory of several variables, the extended version of Theorem 3.2 also holds for families of general Riemann surfaces. This modification gives a more explicit formula than [15, Corollary 4.6].
A stable differential on a stable curve is defined as (regular or meromorphic) section of the dualizing sheaf on this curve ( [6]). By Theorems 3.1 and 3.2, we have: (1) For each 1 ≤ i ≤ g, ω i is a regular stable differential on C ∆ .
(2) For each t ∈ T with v t = v b and m > 1, ω t,m is a meromorphic stable differential on C ∆ which has only pole (of order m) at the point p t corresponding to t.

Period map
Let ∆ = (V, E, T ) be a stable graph, and assume that there are a vertex Then there is an element of T with terminal vertex v b which we denote by t 0 . Then Proof. To prove this theorem, we can regard C • ∆ as a family of open Riemann surfaces obtained from C ∆ by removing sections associated with p t (t ∈ T ). For small (counterclockwise oriented) closed paths where δ i,j denotes the Kronecker delta. Furthermore, for small closed paths a t (t ∈ T ) in C • ∆ around p t , Let b i (1 ≤ i ≤ g) be a closed path in C • ∆ corresponding to a path from p ∞ to γ i (p ∞ ). Then by Proposition 2.1, b i ω t 0 ,m (m > 1) belongs to A ∆ ⊗Q whose constant term is By Proposition 2.1, the difference between 2 distinct elements of {α ±i (1 ≤ i ≤ g)} belongs to A × ∆ times a certain product of y e (e ∈ E). Therefore, the assertion follows from that the Vandermonde determinant det (α i − x t 0 ) j 1≤i,j≤g is an invertible element of B ∆ .

Gauss-Manin connectiom and variation of Hodge structure
Let ∇ denote the Gauss-Manin connection for C ∆ ⊗ A ∆ (B ∆ ⊗Q) regarded as a family of curves over B ∆ ⊗Q. (1) There exist elements where P ij ∈ B ∆ denote the universal periods [17, Section 3] defined as exp γ i ω j .
(2) The both sets {η j | 1 ≤ j ≤ g} and {ω t 0 ,m | 2 ≤ m ≤ g + 1} give bases of the Hodge component H 0, Proof. By Theorem 4.1 and its proof, there are (B ∆ ⊗Q)-linear sums η i of ω t 0 ,m such that a i η j = 0 and b i η j = δ i,j . Then the assertion (1) follows. Therefore, the Griffiths transversality implies that {η j } generates a sub (B ∆ ⊗Q)-module of H 0,1 with rank g which is also generated by {ω t 0 ,m }. Then the assertion (2) follows. Proof. The universal differential ω i becomes a differential of the first kind on C whose residue (cf. [22]) for each edge in E is constant, and η j becomes a differential of the second kind on C whose Coleman integration (cf. [2,3]) along γ i is δ i,j . Then the assertion follows from Theorems 2.8 and 2.9 of [7].

Monodromy weight filtration
Let C ∆ be the generalized Tate curves associated with a stable graph ∆ = (V, E, T ), the monodromy weight filtration and operator with respect to the closed fiber of C ∆ obtained by y e (e ∈ E ′ ).

Theorem 4.4.
(1) For each e ∈ E, let Res e be the linear map on H defined by Res e (ω t 0 ,m ) = 0 and by (2) Let ∆ ′ be the graph obtained from ∆ by shrinking each edge in E − E ′ to one point. Then {Res e } e∈E ′ gives rise to an isomorphism from H/F 1 E ′ H onto the group H 1 ∆ ′ , B ∆ ⊗Q of singular cohomology classes. Furthermore, the line integral in C ∆ corresponding to cycles in ∆ ′ gives an isomorphism and N E ′ gives the identity map on H 1 ∆ ′ , B ∆ ⊗Q under the above isomorphisms.
Proof. The assertion (1) follows from that Res e gives the residue map for the family of Riemann surfaces associated with C ∆ . Then by Theorem 4.2, F 0 E H = F 1 E H has a basis {ω t 0 ,m }, and by Theorem 4.1, the line integrals from

Coleman integration in families of Mumford curves
We consider the problem of constructing Coleman integrals in families raised by Besser [1, 1.6] in the case of Mumford curves. Let K be a p-adic field, namely a local field of characteristic 0 whose residue field is of characteristic p > 0. Denote by | · | a multiplicative valuation on K, and by S ∆ the K-analytic subspace of K E ×K E consists of (x h , y e ) h∈E,e∈E satisfying Then as is seen in 2.3, we have a family C ∆ /S ∆ of Mumford curves over K from the generalized Tate curve C ∆ by specializing parameters. Therefore, the universal differentials ω i and ω t,m in 3.1 give differentials on C ∆ /S ∆ of the first and second kind respectively, and hence there exist their Coleman integrals given by [3] under fixing a logarithm homomorphism log : K × → K.
The Coleman integrals of ω i and ω t,m (determined unique up to constants) on C ∆ /S ∆ are given by where Γ denotes Schottky groups over K associated with Γ ∆ , and z 0 is a point on P 1 K outside the limit set of Γ.
Proof. We use the description given by de Shalit [8] of Coleman integrals on semistable curves. Let C K be a Mumford curve over K as a member of C ∆ . Then it is shown in [8, 0.4 and 1.1] that there exists a cover C K of C K as a K-analytic space with action of Γ such that C K /Γ ∼ = C K , and the Coleman integrals on C K are functions on C K obtained by anti-differentiating. Therefore, the assertion follows from the explicit formulas of ω i and ω t,m . Corollary 4.6. Let C K be a Mumford curve over a p-adic field K obtained from C ∆ by the base change corresponding to a ring homomorphism ϕ from A ∆ to the valuation ring of K, and put E ′ = {e ∈ E | ϕ(y e ) = 0}. Then the monodromy weight filtration and monodromy operator given in Theorem 4.4 become the p-adic monodromy weight filtration and monodromy operator for C K (cf. [4]).

Unipotent fundamental group and period
Assume that n is an integer > 1, and let C • be an algebraic curve over a subfield K of C which is obtained from a proper smooth curve C of genus g by removing n points. Since C • is not complete, H 1 dR (C • ) has a basis B C • consisting of 2g + n − 1 meromorphic differentials on C/K of the first or second kind which may have poles outside C • . For w 1 , ..., w m ∈ B C • , we define a D-module D(w 1 , ..., w m ) on C • given by the trivial bundle K m+1 × C • with connection form d − m i=1 e i,i+1 w i , where e i,j is the square matrix of degree m + 1 whose (k, l)-entry is δ i,k · δ j,l . We consider the tannakian subcategory of D-modules on C • generated by D(w 1 , ..., w m ) (w i ∈ B C • ). Since these underlying bundles are trivial, for each K-rational point x on C • , one can define the fiber functor on this category by taking the (trivial) fibers over x. Denote by π dR 1 (C • ; x) the tannakian fundamental group of this category which becomes a profinite algebraic group over K, and by A dR (C • ; x) the enveloping algebra of the Lie algebra Lie π dR 1 (C • ; x) . Let (C • ) an be a Riemann surface associated with C • ⊗ K C, and for K-rational points x, y on C • , denote by π 1 ((C • ) an ; x, y) the set of homotopy classes of paths from x to y in (C • ) an . When x = y, π 1 ((C • ) an ; x, y) becomes the fundamental group π 1 ((C • ) an ; x) of (C • ) an with base point x. We consider the tannakian category of unipotent local systems on (C • ) an with fiber functor obtained from taking the fiber over x. Then it is shown in [5] that its tannakian fundamental group π Be 1 ((C • ) an ; x) is a pro-algebraic group over Q, and the associated enveloping algebra where I denotes the augmentation ideal of the group ring Q [π 1 ((C • ) an ; x)]. Since π 1 ((C • ) an ; x) is a free group of rank 2g + n − 1, A Be ((C • ) an ; x) becomes the ring of noncommutative formal power series over Q in 2g + n − 1 variables.
Proof. We may show the assertion in the case when x = y. For each D-module D(w 1 , ..., w m ), there exists the associate local system on (C • ) an described by the iterated integrals w ′ 1 · · · w ′ r for r ≤ m and w ′ i ∈ {w 1 , ..., w m }. By this association, we have a group homomorphism which gives a ring homomorphism Let B m ((C • ) an ) be the C-vector space of iterated integrals spanned by and H 0 (B m ((C • ) an ); x) be the space consisting of elements of B m ((C • ) an ) whose restriction to {loops in (C • ) an based at x} is homotopy functional. Then it is shown in [13, (5.3)] that H 0 (B m ((C • ) an ); x) is the dual space of C [π 1 ((C • ) an ; x)]/ I m+1 , and that by taking leading terms of iterated integrals, we have an exact sequence Denote by (I m /I m+1 ) ∨ the dual space of I m /I m+1 . Then there exists the associated injective linear map which is the inverse of the map induced from the above homomorphism ϕ. This implies that ϕ is an isomorphism. The remaining assertions follow from that the tannakian fundamental group π ♯ 1 (♯ = dR, Be) and its Lie algebra are given as the subsets of A ♯ which consist of grouplike elements and primitive elements in A ♯ respectively.
Following [13], we define the Hodge (resp. weight) filtrations F • (resp. W • ) on π dR 1 (C • ; x, y) as follows. First, F p is spanned by D-modules D(w 1 , ..., w r ) for at least p elements of B C • which are holomorphic on C or having only simple poles at D = C−C • . Second, W l is spanned by D(w 1 , ..., w r ) for at most l elements of B C • which have only simple poles at D. Take a K-basis {w 1 , ..., w 2g+n−1 } of H 1 dR (C • ) consisting of meromorphic differentials on C, and a point x on D such that w i (1 ≤ i ≤ 2g + n − 1) are regular or having simple poles at x. Then for a point y on C • , Theorem 5.1 also holds by replacing x with a tangential point on C • starting at x, where the associated iterated integrals are y x w 1 · · · w m (w 1 : regular at x).
Let x, y are points or tangential points on C • . Then by Theorem 5.1, for a topological path t xy and the de Rham path d xy from x to y, d −1 xy t xy gives an element of π dR 1 (C • ; x) ⊗ K C. If we fix a K-basis {w i } of H 1 dR (C • ) and denote by W i the symbols corresponding to w i , then we have an injection from π dR 1 (C • ; x) into the ring C W i of noncommutative formal power series of W i which sends the dual of w i to e W i . The unipotent periods are coefficients of d −1 xy t xy ∈ π dR 1 (C • ; x) ⊗ K C ֒→ C W i which are given by iterated integrals where if x (resp. y) are tangential points, then w 1 (resp. w m ) are taken to be regular at x (resp. y).

Limits of unipotent periods
Let the notation be as in 4.1, and assume that ∆ is trivalent, namely C ∆ is maximally degenerate. Then one can take hence A 0 = Z, and each points p x h on C 0 = C ∆ ⊗ A ∆ A 0 has two Z-rational tangential points over Z. Let be as in Theorems 4.1 and 4.2.
For h 1 , h 2 ∈ ±E ∪ T such that p h i = p t 0 , take a tangential point p h i over Z starting at p h i for each i = 1, 2. Then a unipotent period for C ∆ from p h 1 to p h 2 is defined as an iterated integral where w k ∈ {ω i , η j , ω t } such that w 1 | Cv (resp. w m | Cv ) have no simple pole at p h 1 (resp. p h 2 ). Since {ω i , η j , ω t } is a (B ⊗Q)-basis of H 1 dR C ∆ ⊗ A ∆ B ∆ ⊗Q , the (B ⊗Q)module generated by D-modules D(w 1 , ..., w m ) (w k ∈ {ω i , η j , ω t }) gives rise to a natural Q-structure of the vector bundle associated with π dR 1 on a deformation of C 0 . Furthermore, F p and W l are defined as its subbundles such that F p is generated by D(w 1 , ..., w m ) for at least p elements of {ω i , ω t }, and W l is generated by D(w 1 , ..., w m ) for at most l elements of {ω t }.
Theorem 5.2. The above unipotent period is expressed as a Laurent power series in y e (e ∈ E) whose minimal terms are generated over Q by π √ −1 and multiple zeta values.
Proof. We may show the assertion in the case that v h 1 = v h 2 , and hence we may assume that p h 2 is the half rotation of p h 1 or that ( p h 1 , p h 2 ) is identified with the interval (0, 1). By the universal expression of ω i , η j , ω t (Theorems 3.1 and 4.2), and the stability of ω i , ω t (Theorem 3.2), in the former case, the minimal terms are positive powers of π √ −1, and in the latter case, the minimal terms are Q-linear sums of multiple zeta values by the following lemma. Then the iterated integral γ w 1 · · · w m