We study the p-adic counterpart of the GKZ hypergeo- metric system

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THE p-ADIC GELFAND-KAPRANOV-ZELEVINSKY HYPERGEOMETRIC COMPLEX

LEI FU, DAQING WAN AND HAO ZHANG

Abstract. To a torus action on a complex vector space, Gelfand, Kapranov and Zelevinsky introduce a system of differential equations, called the GKZ hypergeometric system. Its solutions are GKZ hypergeometric functions. We study the p-adic counterpart of the GKZ hypergeometric system. In the language of dagger spaces introduced by Grosse-Klönne, thep-adic GKZ hypergeometric complex is a twisted relative de Rham complex of meromorphic differential forms with logarithmic poles for an affinoid toric dagger space over the dagger unit polydisc. It is a complex ofO^†-modules with integrable connections and with Frobenius structures defined on the dagger unit polydisc such that traces of Frobenius on fibers at Techmüller points define the hypergeometric function over the finite field introduced by Gelfand and Graev.

Key words: GKZ hypergeometric system,D^†-modules, twisted de Rham complex, Dwork trace formula.

Mathematics Subject Classification: Primary 14F30; Secondary 11T23, 14G15, 33C70.

Introduction 0.1. The GKZ hypergeometric system. Let

A=







w₁₁ · · · w_1N ... ... wn1 · · · wnN







be an(n×N)-matrix of ranknwith integer entries. Denote the column vectors ofAbyw₁, . . . ,w_N ∈ Zⁿ. It defines an action of then-dimensional torusTⁿ_Z= SpecZ[t^±1₁ , . . . , t^±1_n ]on theN-dimensional affine spaceA^N_Z = SpecZ[x1, . . . , xN]:

Tⁿ_Z×A^N_Z →A^N_Z,

(t1, . . . , tn),(x1, . . . , xN)

7→(t^w₁¹¹· · ·t^w_nⁿ¹x1, . . . , t^w₁^1N· · ·t^w_n^nNxN).

Letγ₁, . . . , γ_n∈C. In [10], Gelfand, Kapranov and Zelevinsky define theA-hypergeometric system to be the system of differential equations

PN

j=1w_ijx_j_∂x^∂f

j +γ_if = 0 (i= 1, . . . , n), Q

λ_j>0

∂

∂xj

^aj

f =Q

λ_j<0

∂

∂xj

−aj

(0.1.1) f,

where for the second system of equations,(λ₁, . . . , λ_N)∈Z^N goes over the family of integral linear relations

N

X

j=1

λjwj = 0

We would like to thank Jiangxue Fang for helpful discussions. The research is supported by the NSFC..

1

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amongw1, . . . ,wN. We call theA-hypergeometric system as theGKZ hypergeometric system. An integral representation of a solution of the GKZ hypergeometric system is given by

f(x₁, . . . , x_N) = Z

Σ

t^γ₁¹· · ·t^γ_nⁿe^P^N^j=1^x^j^t

w1j

1 ···t^wnjn dt₁ t1

· · ·dt_n tn

(0.1.2)

where Σ is a real n-dimensional cycle in Tⁿ. Confer [1, equation (2.6)], [4, section 3] and [8, Corollary 2 in §4.2].

0.2. The GKZ hypergeometric function over finite fields. Let p be a prime number, q a power of p, Fq the finite field with qelements, ψ: Fq →Q

∗ a nontrivial additive character, and χ1, . . . , χn : F^∗q → Q^∗ multiplicative characters. In [7] and [9], Gelfand and Graev define the hypergeometric function over the finite field to be the function defined by the family of twisted exponential sums

Hyp(x₁, . . . , x_N) = X

t₁,...,t_n∈F^∗_q

χ₁(t₁)· · ·χ_n(t_n)ψX^N

j=1

x_jt^w₁^1j· · ·t^w_n^nj , (0.2.1)

where(x1, . . . , xN)varies inA^N(F^q). It is an arithmetic analogue of the expression (0.1.2).

In [5], we introduce the`-adic GKZ hypergeometric sheafHypwhich is a perverse sheaf onA^N_F_q such that for any rational pointx= (x1, . . . , xN)∈A^N(Fq), we have

Hyp(x1, . . . , xN) = (−1)^n+NTr(Frobx,Hyp_x_¯),

whereFrob_xis the geometric Frobenius atx. In this paper, we study thep-adic counterpart of the GKZ hypergeometric system. It is a complex ofO^†-modules with integrable connections and with Frobenius structures defined on the dagger space ([11]) corresponding to the unit polydisc so that traces of Frobenius on fibers at Techmüller points are given byHyp(x₁, . . . , x_N).

0.3. The p-adic GKZ hypergeometric complex. For any v = (v₁, . . . , v_N)∈ Z^N≥0 andw = (w1, . . . , wn)∈Zⁿ, write

x^v=x^v₁¹· · ·x^v_N^N, t^w=t^w₁¹· · ·t^w_nⁿ, |v|=v1+· · ·+vN. LetK be a finite extension ofQp containing an elementπsatisfying

π^p−1+p= 0.

Denote by | · | the p-adic norm on K defined by |a| = p^−ord^p^(a). For each real number r > 0, consider the algebras

K{r⁻¹x} = { X

v∈Z^N≥0

avx^v: av∈K, |av|r^|v|are bounded}, Khr⁻¹xi = { X

v∈Z^N≥0

avx^v: av∈K, lim

|v|→∞|av|r^|v|= 0}.

They are BanachK-algebras with respect to the norm

k X

v∈Z^N≥0

a_vx^vk_r= sup|a_v|r^|v|.

We haveKhr⁻¹xi ⊂K{r⁻¹x}.Elements inKhr⁻¹xiare exactly those power series converging in the closed polydisc{(x1, . . . , xN) : xi∈Qp, |xi| ≤r}.Moreover, for anyr < r⁰, we have

K{r⁰⁻¹x} ⊂Khr⁻¹xi ⊂K{r⁻¹x}.

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Let

Khxi^†= [

r>1

K{r⁻¹x}= [

r>1

Khr⁻¹xi.

K{x}^† is the ring ofover-convergent power series, that is, series converging in closed polydiscs of radii>1.

Let ∆ be the convex hull of{0,w1, . . . ,wN} in Rⁿ, and let δ be the convex polyhedral cone generated by{w1, . . . ,w_N}. For anyw∈δ, define

d(w) = inf{a >0 : w∈a∆}.

We have

d(aw) =ad(w), d(w+w⁰)≤d(w) +d(w⁰)

whenevera≥0andw,w⁰ ∈δ. There exists an integerd >0such that we have d(w)∈ ¹_dZfor all w∈Zⁿ. For any real numbersr >0ands≥1, define

L(r, s) = { X

w∈Zⁿ∩δ

a_w(x)t^w: a_w(x)∈K{r⁻¹x}, ka_w(x)k_rs^d(w) are bounded}

= { X

v∈Z^N≥0,w∈Zⁿ∩δ

avwx^vt^w: avw∈K, |avw|r^|v|s^d(w)are bounded},

L^† = [

r>1, s>1

L(r, s).

Note thatL(r, s)andL^† are rings. Let F(x,t) =

N

X

j=1

x_jt^w₁^1j· · ·t^w_n^nj,

Consider thetwisted de Rham complex C^·(L^†)defined as follows: We set C^k(L^†) ={ X

1≤i₁<···<i_k≤n

fi₁...i_k

dti₁

t_i₁ ∧ · · · ∧dti_k

t_i_k : fi₁...i_k∈L^†} ∼=L^†(ⁿk) with differentiald:C^k(L^†)→C^k+1(L^†)given by

d(ω) =

t^γ₁¹· · ·t^γ_nⁿexp(πF(x,t))−1

◦dt◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(x,t)) (ω)

= dtω+

n

X

i=1

γi+π

N

X

j=1

wijxjt^w^jdt_i ti

∧ω

for anyω ∈C^k(L^†), where dt is the exterior derivative with respect to the t variable. For each j∈ {1, . . . , N}, define∇ ∂

∂xj :C^·(L^†)→C^·(L^†)by

∇ ∂

∂xj(ω) =

◦ ∂

∂xj

◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(x,t))

= ∂ω

∂xj

+πt^w^jω.

Since _∂x^∂

j commutes withd_t, ∇ ∂

∂xj commutes with d: C^k(L^†)→C^k+1(L^†). We have integrable connections

∇:C^·(L^†)→C^·(L^†)⊗_Khxi†Ω¹_Khxi†

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defined by

∇(ω) =

N

X

j=1

∇ ∂

∂xj(ω)⊗dxj, whereΩ¹_Khxi† is the freeK{x}^†-module with basisdx1, . . . ,dxN.

Consider the lifting of the Frobenius correspondence in the variabletdefined by Φ(f(x,t)) =f(x,t^q).

One verifies directly that Φ(L(r, s)) ⊂ L(r,√^q

s) and hence Φ(L^†) ⊂ L^†. It induces maps Φ : C^k(L^†)→C^k(L^†)on differential forms commuting withd_t:

Φ X

1≤i1<···<i_k≤n

fi₁...i_k(x,t)dti₁

ti₁

∧ · · · ∧ dti_k

ti_k

= X

1≤i1<···<i_k≤n

q^kfi₁...i_k(x,t^q)dti₁

ti₁

∧ · · · ∧dti_k

ti_k

Suppose furthermore that γ1, . . . , γn ∈ _1−q¹ Z and (γ1, . . . , γn) ∈ δ. Consider the maps F : C^k(L^†)→C^k(L^†)defined by

F =

◦Φ◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(x^q,t)) (0.3.1)

=

t^γ₁¹^(q−1)· · ·t^γ_nⁿ^(q−1)exp πF(x^q,t^q)−πF(x,t)

◦Φ.

(0.3.2)

Even thought^γ₁¹· · ·t^γ_nⁿexp(πF(x,t))does not lie inL^†and multiplication by it does not define an endomorphism onC^·(L^†), the next Lemma 0.4 (i) shows thatt^γ₁¹^(q−1)· · ·t^γnⁿ^(q−1)exp

πF(x^q,t^q)− πF(x,t)

lie inL^†, and hence the expression (0.3.2) shows thatF defines endomorphism on each C^k(L^†).

Lemma 0.4.

(i)t^γ₁¹^(q−1)· · ·t^γnⁿ^(q−1)exp πF(x^q,t^q)−πF(x,t)

andt^γ₁¹^(1−q)· · ·t^γnⁿ^(1−q)exp πF(x,t)−πF(x^q,t^q) lie inL(r, r⁻¹p^p−1^pq )for any 0< r≤p^p−1^pq .

(ii) Let C^(1)·(L^†)be the twisted de Rham complex so that C^(1),j(L^†) =C^j(L^†)for each k, and d⁽¹⁾:C^(1),k→C^(1),k+1 is given by

d⁽¹⁾ =

t^γ₁¹· · ·t^γ_nⁿexp(πF(x^q,t))−1

◦dt◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(x^q,t))

= dt+

n

X

i=1

γi+π

N

X

j=1

wijx^q_jt^w^jdti

ti

. Let∇⁽¹⁾ be the connection onC^(1)·(L^†)defined by

∇⁽¹⁾∂

∂xj

=

t^γ₁¹· · ·t^γ_nⁿexp(πF(x^q,t))−1

◦ ∂

∂xj

◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(x^q,t))

= ∂

∂xj

+qπx^q−1_j t^w^j.

ThenF defines a horizontal morphism of complexes of Khxi^†-modules with connections F : (C^(1)·(L^†),∇⁽¹⁾)→(C^·(L^†),∇).

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(iii) LetE(0,1)^N be the closed unit polydisc with the dagger structure sheaf ([11]) associated to the algebraKhxi^†, and letFrbe the lifting

Fr :E(0,1)^N →E(0,1)^N, (x1, . . . , xN)→(x^q₁, . . . , x^q_N) of the geometric Frobenius correspondence. We have an isomorphism

Fr^∗(C^·(L^†),∇)∼= (C^(1)·(L^†),∇⁽¹⁾).

Proof. (i) Writeexp(πz−πz^q) = 1 +P∞

i=1cizⁱ. We have|ci| ≤p⁻^p−1^pq ⁱby [15, Theorem 4.1]. Write exp(πz^q−πz) = 1−(

∞

X

i=1

c_izⁱ) + (

∞

X

i=1

c_izⁱ)²− · · ·

= 1 +

∞

X

i=1

c⁰_izⁱ.

Then we also have the estimate|c⁰_i| ≤p⁻^p−1^pq ⁱ. For the monomial xjt^w^j, we have

exp π(xjt^w^j)^q−πxjt^w^j

=

∞

X

i=0

c⁰_ixⁱ_jt^iw^j, kc⁰_ixⁱ_jk_r≤p⁻^p−1^pq ⁱrⁱ=

r⁻¹p^p−1^pq −i

≤

r⁻¹p^p−1^pq −d(iwj)

Here for the last inequality, we use the fact that d(iw_j) ≤ i and the assumption that r ≤ p^p−1^pq . So we have exp π(x_jt^w^j)^q −πx_jt^w^j

∈ L(r, r⁻¹p^p−1^pq ). Since r⁻¹p^p−1^pq ≥ 1, the space L(r, r⁻¹p^p−1^pq ) is a ring. So t^γ₁¹^(q−1)· · ·t^γnⁿ^(q−1)exp πF(x^q,t^q)−πF(x,t)

lies in L(r, r⁻¹p^p−1^pq ).

Similarlyt^γ₁¹^(1−q)· · ·t^γnⁿ^(1−q)exp πF(x,t)−πF(x^q,t^q)

lies inL(r, r⁻¹p^p−1^pq ).

(ii) Using the fact that Φ◦dt= dt◦ΦandΦ◦ _∂x^∂

j =_∂x^∂

j ◦Φ, one checks thatF◦d⁽¹⁾=d◦F andF◦ ∇⁽¹⁾=∇ ◦F.

(iii) Consider the K-algebra homomorphism

Khy1, . . . , yNi^† →Khx1, . . . , xNi^†, yj7→x^q_j. This makesKhxi^† a finiteKhyi^†-algebra. We have a canonical isomorphism

L˜^†⊗_Khyi^†Khxi^{† ∼}→⁼ L^†,

where L˜^† is defined in the same way as L^† except that we change the variables from xj to yj. The connection∇ onL˜^† defines a connection on L˜^†⊗_Khyi^†Khxi^† via the Leibniz rule. Via the above isomorphism, it defines the connectionFr^∗∇ on L^†. Let’s verify that it coincides with the connection∇⁽¹⁾ onL^†. Any element in L^† can be written as a finite sum of elements of the form

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f(x)g(y,t)withf(x)∈K[x]andg(y,t)∈L˜^†. By the Leibniz rule, we have (Fr^∗∇) ∂

∂xj(f(x)g(y,t)) = ∂

∂xj

(f(x))g(y,t) +f(x)

∇(g(y,t)), ∂

∂xj

= ∂

∂xj

(f(x))g(y,t) +f(x) X

m

∇ ∂

∂ym(g(y,t))dym, ∂

∂xj

= ∂

∂xj

(f(x))g(y,t) +f(x)∇ ∂

∂yj(g(y,t))qx^q−1_j

= ∂

∂xj

(f(x))g(y,t) +f(x) ∂

∂yj

(g(y,t)) +πt^w^jg(y,t) qx^q−1_j

= ∂

∂xj

(f(x))g(y,t) +f(x) ∂

∂xj

(g(y,t)) +qπx^q−1_j t^w^jf(x)g(y,t)

= ∂

∂xj

f(x)g(y,t)

+qπx^q−1_j t^w^jf(x)g(y,t)

= ∇⁽¹⁾(f(x)g(y,t)).

This proves our assertion. Similarly, one verifies that the connectionFr^∗∇ on Fr^∗C^j( ˜L^†) can be

identified with the connection∇⁽¹⁾ onC^·(L^†).

Definition 0.5. Supposeγ1, . . . , γn ∈ _1−q¹ Zand(γ1, . . . , γn)∈δ. Thep-adic GKZ hypergeometric complexis defined to be the tuple(C^·(L^†),∇, F)consisting of the complexC^·(L^†)ofKhxi^†-module modules with the connection∇and the horizontal morphismF : Fr^∗(C^·(L^†),∇)→(C^·(L^†),∇).

0.6. The GKZ hypergeometric D^†-module. Let D^†= [

r>1, s>1

{ X

v∈Z^N_≥0

fv(x) ∂^v

π^|v| : fv(x)∈K{r⁻¹x}, kfv(x)krs^|v| are bounded}, where for anyv= (v₁, . . . , v_N)∈Z^N≥0, we set∂^v= ^∂^v^{1 +}^···+^vN

∂x^v₁¹···∂x^vN_N .D^† is a ring of differential operators possibly of infinite orders. ThisD^† is also used in [14]. LetD^†

P^N,Q(∞)be the sheaf of differential operators of finite level and of infinite order on the formal projective space P^N over the integer ring ofK with over-convergent poles along the∞divisor. For the definition of this sheaf, see [3].

By [13], we have Γ(P^N,D^†

P^N,Q(∞)) = [

r>1, s>1

{ X

v∈Z^N_≥0

fv(x)∂^v v!

: fv(x)∈K{r⁻¹x}, kfv(x)krs^|v|are bounded}, wherev! =v1!· · ·vN!. In section 1, we prove the following proposition.

Proposition 0.7. We haveD^†= Γ(P^N,D^†

P^N,Q(∞)).

In particular, by the result in [13], D^† is a coherent ring. Let _∂x^∂

j ∈ D^† act via∇ ∂

∂xj. ThenL^† is a leftD^†-module, and the twisted de Rham complex C^·(L^†)is a complex ofD^†-modules. The cohomology groupsH^k(C^·(L^†))are also leftD^†-modules.

Let

C(A) = {k1w1+· · ·+kNwN : ki∈Z≥0},

L^†0 = [

r>1, s>1

{ X

w∈C(A)

a_w(x)t^w: a_w(x)∈K{r⁻¹x}, kaw(x)krs^d(w)are bounded}.

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C(A)is a submonoid ofZⁿ∩δ, andL^†0 is both a subring and aD-submodule ofL^†. Let C^k(L^†0) ={ X

1≤i1<···<ik≤n

f_i₁_...i_kdt_i₁ ti1

∧ · · · ∧dt_i_k ti_k

: f_i₁_...i_k ∈L^†0} ∼=L^†0(ⁿj). Note thatd:C^k(L^†)→C^k+1(L^†)(resp. ∇ ∂

∂xj) maps C^k(L^†0) to C^k+1(L^†0) (resp. C^k(L^†0)). So C^·(L^†0)is a subcomplex ofD^†-modules ofC^·(L^†). Let

Fi,γ =ti

∂

∂ti

+γi+π

N

X

j=1

wijxjt^w^j.

It follows from the definition of the twisted de Rham complex that the homomorphism L^†0→Cⁿ(L^†0), f 7→fdt1

t₁ ∧ · · · ∧dtn

t_n induces an isomorphism

L^†0/

n

X

i=1

Fi,γL^†0∼=Hⁿ(C^·(L^†0)).

Let’s give an explicit presentation of theD^†-moduleHⁿ(C^·(L^†0)). Let Λ = {λ= (λ₁, . . . , λ_N)∈Z^N :

N

X

j=1

λ_jw_j= 0},

^λ = Y

λj>0

1 π

∂

∂xj

^λj

− Y

λj<0

1 π

∂

∂xj

^−λj

(λ∈Λ)

Ei,γ =

N

X

j=1

wijxj

∂

∂xj

+γi(i= 1, . . . , n), Consider the map

ϕ:D^†→L^†0, X

v∈Z^N≥0

fv(x)∂^v

π^|v| 7→( X

v∈Z^N≥0

fv(x)∂^v

π^|v|)·1 = X

v∈Z^N≥0

fv(x)t^v¹^w^N^+···+v^N^w^N.

It is a homomorphism ofD^†-modules. In §1, we prove the following theorems.

Theorem 0.8. ϕinduces isomorphisms D^†/X

λ∈Λ

D^†^λ →^∼⁼ L^†0,

D^†/(

n

X

i=1

D^†E_i,γ+X

λ∈Λ

D^†^λ) →^∼⁼ L^†0/

n

X

i=1

F_i,γL^†0∼=Hⁿ(C^·(L^†0)).

Moreover, there exist finitely manyµ⁽¹⁾, . . . , µ^(m)∈Λsuch that

m

X

i=1

D^†µ⁽ⁱ⁾ =X

λ∈Λ

D^†λ.

Theorem 0.9. C^·(L^†)andC^·(L^†0)are complexes of coherentD^†-modules.

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Definition 0.10. TheGKZ hypergeometric D^†-module is defined to be the leftD^†-module D^†/(

n

X

i=1

D^†E_i,γ+X

λ∈Λ

D^†λ)∼=Hⁿ(C^·(L^†0)).

The GKZ hypergeometric D^†-module is the p-adic analogue of the (complex) hypergeometric D-module ([1]) associated to the GKZ hypergeometric system of differential equations (0.1.1).

0.11. Fibers of the GKZ hypergeometric complex. Leta = (a₁, . . . , a_N)be a point in the closed unit polydiscE(0,1), where a_i∈K⁰ for some finite extensionK⁰ of K. Let’s specialize at x= a, that is, apply the functor -⊗_Khxi† K⁰, where K⁰ is regarded as a Khxi^†-algebra via the homomorphism

Khxi^† →K⁰, x_i7→a_i. Let

L^†₀ = [

s>1

{ X

w∈Zⁿ∩δ

a_wt^w: a_w∈K⁰, |aw|s^d(w) are bounded}.

In section 1, we prove the following.

Lemma 0.12. L^† is flat over Khxi^† and

L^†⊗_Khxi†K⁰∼=L^†₀.

Consider the twisted de Rham complexC^·(L^†₀)defined as follows: We set C^k(L^†₀) ={ X

1≤i1<···<ik≤n

fi₁...i_k

dt_i₁ ti₁

∧ · · · ∧dt_i_k ti_k

: fi₁...i_k∈L^†₀} ∼=L^†(ⁿk)

0

with differentiald:C^k(L^†₀)→C^k+1(L^†₀)given by

d(ω) =

t^γ₁¹· · ·t^γ_nⁿexp(πF(a,t))−1

◦dt◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(a,t)) (ω)

= dtω+

n

X

i=1

γi+π

N

X

j=1

wijajt^w^jdti

ti

∧ω

for anyω∈C^k(L^†₀). By Lemma 0.12, we have the following corollary.

Corollary 0.13. In the derived category of complexes of Khxi^†-modules, we have C^·(L^†)⊗^L_Khxi†K⁰∼=C^·(L^†₀).

The specialization of Φatais the lifting of the Frobenius correspondence defined by Φ_a:L^†₀→L^†₀, f(t)7→f(t^q).

It induces the mapsΦa:C^k(L^†)→C^k(L^†)on differential forms commuting withdt:

Φ_a X

1≤i1<···<ik≤n

f_i₁_...i_k(t)dt_i₁ ti₁

∧ · · · ∧dt_i_k ti_k

= X

1≤i1<···<ik≤n

q^kf_i₁_...i_j(t^q)dt_i₁ ti₁

∧ · · · ∧dt_i_k ti_k

The specialization ofF :C^·(L^†)→C^·(L^†)at ais given by

Fa =

◦Φa◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(a^q,t))

=

t^γ₁¹^(q−1)· · ·t^γ_nⁿ^(q−1)exp πF(a^q,t^q)−πF(a,t)

◦Φ_a.

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By Lemma 0.4 (i),t^γ₁¹^(q−1)· · ·t^γnⁿ^(q−1)exp πF(a^q,t^q)−πF(a,t)

lie inL^†₀, and henceF_a defines an endomorphism on eachC^k(L^†₀).

From now on, we assume thata is a Techmüller point, that is,a^q_j =a_j (j= 1, . . . , N). Thena is a fixed point ofFr. In this case Fa:C^·(L^†₀)→C^·(L^†₀)commutes withd:C^j(L^†₀)→C^j+1(L^†₀) and hence is a chain map.

Consider the operator Ψ_a:L^†₀→L^†₀defined by Ψa(X

w

cwt^w) =X

w

cqwt^w. We extend it to differential forms by

Ψ_a X

1≤i1<···<ik≤n

f_i₁_...i_k(t)dti₁

t_i₁ ∧ · · · ∧dti_k

t_i_k

= X

1≤i1<···<ik≤n

q^−kΨ_a(f_i₁_...i_j(t))dti₁

t_i₁ ∧ · · · ∧dti_k

t_i_k . It commutes withdt. LetGa:C^·(L^†₀)→C^·(L^†₀)be the map defined by

Ga =

◦Ψa◦

t^γ₁¹· · ·t^γ_nⁿexp(πF(a,t))

= Ψa◦

t^γ₁¹^(1−q)· · ·t^γ_nⁿ^(1−q)exp πF(a,t)−πF(a^a,t^q) . Here by Lemma 0.4 (i), t^γ₁¹^(1−q)· · ·t^γnⁿ^(1−q)exp πF(a,t)−πF(a,t^q)

lies in L^†₀ and hence G_a defines an operator onC^·(L^†₀). Then G_a commutes withd:C^k(L^†₀)→C^k+1(L^†₀). We thus get a chain mapG_a:C^·(L^†₀)→C^·(L^†₀).

Lemma 0.14. We have G_a◦F_a= id andF_a◦G_a is homotopic toid. In particular, F_a andG_a induce isomorphisms onH^·(C^·(L^†₀)).

In section 3, we show that each Ga : C^k(L^†₀) → C^k(L^†₀) is a nuclear operator and hence the homomorphism on eachH^k(C^·(L^†₀))induced byGais also nuclear. We can talk about their traces and characteristic power series. ButFa does not have this property. Let

Tr G_a, C^·(L^†₀)

=

n

X

k=0

(−1)^kTr G_a, C^k(L^†₀)

=

n

X

k=0

(−1)^kTr Ga, H^k(C^·(L^†₀))

=

n

X

k=0

(−1)^kTr F_a⁻¹, H^k(C^·(L^†₀)) ,

det I−T Ga, C^·(L^†₀)

=

n

Y

k=0

det I−T Ga, C^k(L^†₀)(−1)^k

=

n

Y

k=0

det I−T Ga, H^k(C^·(L^†₀))⁽⁻¹⁾^k

=

n

Y

k=0

det I−T F_a⁻¹, H^k(C^·(L^†₀))(−1)^k

.

Letχ:F^∗q →Qpbe the Techmüller character which maps eachuinF^∗q to its Techmüller lifting.

By [15, Theorems 4.1 and 4.3], the formal power seriesθ(z) = exp(πz−πz^p)converges in a disc

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of radius>1, and its valueθ(1)atz= 1is a primitivep-th root of unity inK. Letψ:Fq→K^∗ be the additive character defined by

ψ(¯a) =θ(1)^Tr^{Fq /Fp}^(¯^a) for any¯a∈Fq. Let¯a_j∈Fq be the residue classa_j modp, let

Sm(F(¯a,t))

= X

¯

u1,...,¯un∈F^∗_qm

χ1(Norm_Fm

q/Fq(¯u1))· · ·χn(Norm_Fm

q/Fq(¯un))ψ

Tr_F_qm_/_F_qX^N

j=1

¯

aju¯^w₁^1j· · ·u¯^w_n^nj be the twisted exponential sums for the multiplicative characters χi = χ^(1−q)γⁱ, the nontrivial additive characterψ:Fq →C^∗p, and the polynomial F(¯a,t), and let

L(F(¯a,t), T) = expX^∞

m=1

Sm(F(¯a,t))T^m m

be theL-function for the twisted exponential sums. The following theorem is well-known in Dwork’s theory. Its proof is given in section 2 for completeness.

Theorem 0.15. Suppose γ₁, . . . , γ_n ∈ _1−q¹ Z, γ = (γ₁, . . . , γ_n) ∈δ, and suppose K⁰ contains all (q−1)-th root of unity. Let a= (a1, . . . , an) be a Techmüller point, that is, a^q_j =aj. Then each Ga:C^k(L^†₀)→C^k(L^†₀)is nuclear. Moreover, we have

Sm(F(¯a,t)) = Tr (qⁿGa)^m, C^·(L^†₀)

=

n

X

k=0

(−1)^kTr (qⁿF_a⁻¹)^m, H^k(C^·(L^†₀)) , L(F(¯a,t), T) = det I−qⁿT G_a, C^·(L^†₀)−1

=

n

Y

k=0

det I−qⁿT F_a⁻¹, H^k(C^·(L^†₀))(−1)^k+1

,

In [2], Adolphson shows thatL(F(¯a,t), T)depends analytically on the parameters aandγ.

0.16. The GKZ hypergeometric F-crystal. It follows from the definition of the twisted de Rham complex that the homomorphism

L^† →Cⁿ(L^†), f 7→fdt1

t1

∧ · · · ∧dtn

tn

induces an isomorphism

L^†/

n

X

i=1

Fi,γL^† ∼=Hⁿ(C^·(L^†)).

∇defines a connection onHⁿ(C^·(L^†)), and F defines a horizontal morphism F : Fr^∗(Hⁿ(C^·(L^†)),∇)→(Hⁿ(C^·(L^†)),∇).

LetU be the affinoid subdomain of the closed unit polydisc E(0,1)^N parametrizing those points a= (a1, . . . , aN)so thatF(¯a,t) =PN

j=1¯ajt^w^j is non-degenerate in the sense that for any faceτ of∆not containing the origin, the system of equations

∂

∂t1

Fτ(¯a,t) =· · ·= ∂

∂tn

Fτ(¯a,t) = 0

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has no solution in(F^∗p)ⁿ, whereFτ(¯a,t) =P

w_j∈τa¯jt^w^j. When restricted to U, we have H^k(C^·(L^†)) = 0

for k 6= n, and Hⁿ(C^·(L^†)) defines a vector bundle on U of rank n!vol(∆). Denote this vector bundle byHyp.

Definition 0.17. We define theGKZ hypergeometric crystal to be(Hyp,∇, F).

Leta= (a₁, . . . , a_N)be a point inU with coordinates inK⁰, and letHyp(a)be the fiber ofHyp at a. By Corollary 0.13, the fact C^k(L^†) = 0 for k > n, and the fact that − ⊗_Khxi^† K⁰ is right exact, we have

Hⁿ(C^·(L^†))⊗_Khxi^†K⁰∼=Hⁿ(C^·(L^†₀)).

So we have

Hyp(a)∼=L^†₀/

n

X

i=1

F_i,γ,aL^†₀, whereFi,γ,a=ti ∂

∂t_i+γi+πPN

j=1wijajt^w^j.Ifais a Techmüller point, then we have Sm(F(¯a,t)) = (−1)ⁿTr (qⁿF_a⁻¹)^m,Hyp(a)

, L(F(¯a,t), T) = det I−qⁿT F_a⁻¹,Hyp(a)(−1)ⁿ⁻¹

.

Leta= (a1, . . . , aN)andb= (b1, . . . , bN)be points inU with coordinates inK⁰, and let T_a,b: Hyp(a)→^∼⁼ Hyp(b)

be the parallel transport forHyp. It is well-defined if|bi−ai|<1 for alli. It can be described as follows: For any formal power seriesf(t)∈Qp[[Zⁿ∩δ]], we have

∇ ∂

∂xj

exp(−πF(x,t))f(t)

= exp(−πF(x,t))◦ ∂

∂x_j ◦exp(πF(x,t))

exp(−πF(x,t))f(t)

= 0.

Soexp(−πF(x,t))f(t)is horizontal with respect to∇. But it is only a formal horizontal section since it may not lie in L^†. Formally, Ta,b maps exp(−πF(a,t))f(t) to exp(−πF(b,t))f(t). So Ta,b: Hyp(a)→^∼⁼ Hyp(b)can be identified with the isomorphism

Ta,b:L^†₀/

n

X

i=1

Fi,γ,aL^†₀→L^†₀/

n

X

i=1

Fi,γ,bL^†₀, g(t)7→exp πF(a,t)−πF(b,t) g(t).

This is well-defined if|bi−ai|<1for allisince we then haveexp πF(a,t)−πF(b,t)

∈L^†₀. SinceF : Fr^∗(Hyp,∇)→(Hyp,∇)is a horizontal morphism, we have a commutative diagram

Hyp(a^q) ^T^{aq ,x}→^q Hyp(x^q)

Fa↓ ↓Fx

Hyp(a) ^T→^a,x Hyp(x).

Let{e₁(x), . . . , e_M(x)} be a local basis forHyp overU. Write (qⁿF_x⁻¹)

e₁(x), . . . , e_M(x)

= (e₁(x^q), . . . , e_M(x^q))Q(x), T_a,x(e₁(a), . . . , e_M(a)) = (e₁(x), . . . , e_M(x))P(x)

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whereP(x)andQ(x)are matrices of power series. Then we have Q(x) =P(x^q)Q(a)P(x)⁻¹ and hence

(−1)ⁿSm(F(¯x,t)) = Tr (P(x^q)Q(a)P(x)⁻¹)^m , (0.17.1)

L(F(¯x,t), T)⁽⁻¹⁾ⁿ⁺¹ = det I−T P(x^q)Q(a)P(x)⁻¹ (0.17.2)

wheneverx^q−1_j = 1anda^q−1_j = 1. Write

∇ ∂

∂xj

e1(x), . . . , eM(x)

= (e1(x), . . . , eM(x))Aj(x).

As∇ ∂

∂xj(Ta,x(ek(a))) = 0for allk,P(x)satisfies the system of differential equations

∂

∂xj

(P(x)) +Aj(x)P(x) = 0.

(0.17.3)

Equations (0.17.1)-(0.17.3) give formulas for calculating the exponential sums and theL-function using a solution of a system of differential equations.

1. D^†-modules Lemma 1.1. Letm be a positive integer and let

m=a0+a1p+a2p²+· · · be itsp-expansion, where0≤ai≤p−1 for alli. Define

σ(m) =a₀+a₁+a₂+· · · . (i) We have

ordp

π^m m!

= σ(m) p−1. (ii) For any real number >0, there existsδ >0 such that

σ(m)≤m+δ.

Proof. (i) We have

ordp(m!) = hm p i

+hm p² i

+· · ·

= (a1+a2p+· · ·) + (a2+a3p+· · ·) +· · ·

= a1+a2(1 +p) +a3(1 +p+p²) +· · ·

= a1(p−1)

p−1 +a2(p²−1)

p−1 +a3(p³−1) p−1 +· · ·

= m−σ(m) p−1 . So we have

ordp

π^m m!

= m

p−1 −m−σ(m)

p−1 =σ(m) p−1. (ii) ChooseM sufficiently large so that for any x≥M, we have

(p−1)(x+ 1)≤p^x.

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Let

m=a₀+a₁p+· · ·+a_lp^l

be the expansion of m, where 0≤a_i ≤p−1 and a_l6= 0. Ifm ≥p^M, then we have l ≥M and hence(p−1)(l+ 1)≤p^l.So we have

σ(m) =a0+a1+· · ·+al≤(p−1)(l+ 1)≤p^l≤m

for anym≥p^M. Takeδ= max(σ(1), . . . , σ(p^M)). Then we haveσ(m)≤m+δfor allm.

1.2. Proof of Proposition 0.7. Set B^†= [

r>1, s>1

{ X

v∈Z^N_≥0

fv(x)∂^v v!

: fv(x)∈K{r⁻¹x}, kfv(x)krs^|v|are bounded}.

Let’s proveB^† =D^†. Given P

v∈Z^N≥0f_v(x)^∂_v^v

! inB^†, choose real numbersr >1, s >1 andC >0 such that

kfv(x)krs^|v|≤C.

We have

X

v∈Z^N≥0

f_v(x)∂^v v!

= X

v∈Z^N≥0

f_v(x)π^|v|

v!

∂^v π^|v|. By Lemma 1.1 (i), we haveord_p

π^|v|

v!

≥0.Hence

fv(x)π^|v|

v!

_rs^|v|≤ kfv(x)krs^|v|≤C.

SoP

v∈Z^N≥0fv(x)^∂_v^v

! lies inD^†. Conversely, given P

v∈Z^N≥0fv(x)_π^∂_|v|^v in D^†, choose real numbersr > 1, s >1 and C >0 such that

kfv(x)krs^|v|≤C.

We have

X

v∈Z^N≥0

f_v(x)∂^v

π^|v| = X

v∈Z^N≥0

f_v(x) v!

π^|v|

∂^v v!

. Choose >0 so that

s > p^p−1 , and chooseδas in Lemma 1.1 (ii). We have

ordp

π^|v|

v!

≤|v|+δn p−1 . Lets⁰ =sp⁻^p−1 >1 and letC⁰=Cp^p−1^δn. We have

f_v(x) v!

π^|v|

_rs^0|v| ≤ kfv(x)krp^|v|+δn^p−1 s^0|v|

= kfv(x)krs^|v|p^p−1^δn

≤ C⁰. SoP

v∈Z^N≥0f_v(x)^∂_v^v

! lies inB^†.