PDF On adelic Hurwitz zeta measures - Osaka U

On adelic Hurwitz zeta measures

HIROAKI NAKAMURA AND ZDZIS LAW WOJTKOWIAK

Abstract. In this paper we construct a ˆZ-valued measure on ˆZ which interpolates p-adic Hurwitz zeta functions for allp.

Contents

1. Introduction 1

2. The Kummer-Heisenberg measure κ1 3

3. Adelic Hurwitz measure 5

4. Geometrical interpretation of translation of measure 9

5. Consequence of Inversion Formula 11

Appendix A. Cohen’sp-adic Hurwitz zeta function 16

Appendix B. Path conventions 19

References 21

1. Introduction

Let m ≥ 1, 0 < a < m be integers such that a is prime to m, and let p be a rational prime. Set q := 4, q := p according to whether p = 2 or p > 2 respectively, and e :=

|(Z/qZ)^×|. Whenp-m, lethap⁻¹idenote the least positive integer such thathap⁻¹ip≡a mod m. Define the Bernoulli polynomials B_k(T) (k ∈N) by P∞

k=0B_k(T)^w_k!^k = ^we_ew^{T w}−1 and set the Bernoulli numbers Bk:=Bk(0).

In [Sh], Shiratani constructedp-adic Hurwitz zeta functionsζ_p^Sh(s;a, m) (s∈Z^p,s6= 1) characterized by the interpolation property:

(1.1) ζ_p^Sh(1−k;a, m) =

(−^mk−1_k B_k(_m^a), (p|m);

−^mk−1_k B_k(_m^a) +p^k−1mk−1_k B_k(^hap_m⁻¹ⁱ), (p-m)

for all integers k > 1 with k ≡ 0 mod e. In [W3], assuming p - m, the second author introduced ap-adic Hurwitz L-function L^β_p(s;a, m) for β ∈(Z/eZ) which satisfies

(1.2) L^β_p(1−k;a, m) = 1

kB_ka m

−p^k−1 k B_k

hap⁻¹i m

for all integersk > 1 withk ≡β modeusing certainp-adic measures arising in the study of Galois actions on paths onP¹− {0,1,∞}(see also [W4]). The purpose of this paper is to complete the construction to include the casep|mand to lift it over ˆZ= lim←−^N(Z/NZ).

Throughout this paper, we fix an embedding of Q into C. For any subfield F ⊂ C, denote by G_F the absolute Galois group Gal( ¯F /F).

2010Mathematics Subject Classification. 11S40; 11G55, 11F80, 11R23, 14H30.

1

Theorem 1.1. Let m and a be mutually prime integers with m >1, 0< a < m. Then, for every σ ∈G_Q(µm), there exists a certain measure ζˆa,m(σ) in Zˆ[[ˆZ]] such that for every prime p, its image ζˆ_p,a,m(σ) in Zp[[Zp]] has the following integration properties over Z^×p:

Z

Z^×p

b^k−1dζˆ_p,a,m(σ)(b) =

((1−χ_p(σ)^k)·m^k−1 ·_k¹B_k(_m^a) (p|m);

(1−χ_p(σ)^k)·m^k−1

1

kB_k(_m^a)− ^pk−1_k B_k(^hap_m⁻¹ⁱ)

(p-m) for all integers k ≥1, where χp :G_Q →Z^×p denotes the p-adic cyclotomic character, and hap⁻¹i represents the least positive integer such that hap⁻¹ip≡a modm.

Remark 1.2. Note that, in the above theorem, the case m= 1 is excluded. In fact, the case m =a = 1 corresponds to the ˆZ-zeta function treated in [W2]. This separation of treatment is necessary for the appearance of tangential base point −→

10 in the construction of measure, which causes replacements of both Bk(_m^a), Bk(^hap_m⁻¹ⁱ) of RHS by Bk(1).

Remark 1.3. More generally, we construct the measure ˆζ_a,m(σ) ∈Zˆ[[ˆZ]] for m >1 and m - a which satisfies the above integration property for all primes p|m with p - a (cf.

Remark 5.6).

Remark 1.4. Using anyσ ∈G_Q(µm) withχ_p(σ)^e 6= 1, we obtain from ˆζ_p,a,m a set ofp-adic Hurwitz functions{L^[β]p (s, a, m)}β∈(Z/eZ) by the standard integral

L^[β]_p (s;a, m) = 1

1−ω(χ_p(σ))^β[χ_p(σ)]^1−s Z

Z^×p

[b]^1−sb⁻¹ω(b)^βdζˆ_p,a,m(σ)(b)

where ω : Z^×p → µ_e is the Teichm¨uller character, and for every b ∈ Z^×p, [b] ∈ 1 +qZp is defined by b = [b]ω(b). Note that the above integral converges in s ∈ Zp except when it has a pole at s= 1 in the caseβ ≡0 (mod e). It follows from Theorem 1.1 that, for each β ∈Z/eZ, the L-functionL^[β]p (s;a, m) has the interpolation property:

(1.3) L^[β]_p (1−k;a, m) =

(_m^k−1

k B_k(_m^a) (p|m);

m^k−1 k

B_k(_m^a)−p^k−1B_k(^hap_m⁻¹ⁱ)

(p-m)

for all k ≥ 1 with k ≡ β mode. Since Z>0,≡β(mode) is dense in the space β + ^q_pZ^p (=Zp (p > 2), 2Z2 or 1 + 2Z2), the above interpolation property shows that L^[β]p (s, a, m) is determined independently of σ (at least) on that space. In particular when β ≡ 0 (mod e) and p > 2, L^[0]p (s;a, m) = −ζ_p^Sh(s;a, m) for s ∈ Zp− {1}. See also Appendix A for relations of L^[β]p (s, a, m) with Cohen’s Hurwitz zeta functions ζ_p(s, x).

In the present paper, we hope to make a small step towards the quest of Coates about existence of zeta functions on ˆZ with values in ˆZ [W2, Introduction].

The mapping ˆζ_a,m in Theorem 1.1 gives a 1-cocycle G_Q(µm) → Zˆ(1)[[ˆZ(−1)]] whose (k−1)st moment integral gives rise to a cohomology class inH¹(G_Q(µm),Zˆ(k)) for k ≥2.

In fact, we will show in Corollary 5.5:

Z

Zp

b^k−1dζˆ_p,a,m(σ)(b) = m^k−1 k B_ka

m

(1−χ_p(σ)^k) (σ∈G_Q(µm), k ≥2)

which implies that the p-adic image of the above cohomology class is torsion with order calculated explicitly by Bernoulli values. It is noteworthy that this cohomology class is

closely related to theξ_m^a-component of theZ(k)-torsor ‘P_m,k+(−1)^kP_m,k’ overµ_m studied by Deligne in [De, Proposition 3.14, Lemma 18.5].

Acknowledgement: This work was partially supported by JSPS KAKENHI Grant Num- ber JP26287006. The authors would like to thank the referee for many valuable sugges- tions including a crucial remark to simplify the proof of Lemma 5.2.

2. The Kummer-Heisenberg measure κ1

2.1. Cyclic coverings. LetF ⊂C be a finite extension of Q with the algebraic closure F ⊂C. For any (normal) algebraic variety V overF and F-rational points x, y ∈V(F), we writeπ^´₁^et(V;y, x) for the set of ´etale paths fromxtoyon the geometric variety V ⊗F, and π^´et₁ (V;x) =π₁^´et(V;x, x) for the ´etale fundamental group with base point x. Denote byπ₁^pro-p(V, x) the maximal pro-p quotient ofπ^´₁^et(V, x), and by π₁^pro-p(V;y, x) the natural push forward ofπ₁^´et(V;y, x) induced from the projection π₁^´et(V, x)π₁^pro-p(V, x).

For each n≥1, write ξ_n := exp(^2πi_n ) so that µ_n:={1, ξ_n, ξ_n², . . . , ξ_nⁿ⁻¹}. Let V_n:=P¹ \ {0, µ_n,∞},

where we understand {0, µ_n,∞} is the abbreviation of {0,∞} ∪µ_n. Regard V_n(C) = C^× \µ_n. Let −→

01_n be the tangential base point on V_n represented by the unit tangent vector and denote for simplicity−→

01. Then, for eachn ≥1, there is a standard cyclic ´etale coverp_n:V_n →V₁ given byz 7→zⁿ which sends−→

01_n to a Galois functor equivalent to−→ 01₁ on V₁. Thus, without ambiguity, we may omit the index of −→

01 on V_n and regard (V_n,−→ 01) as a pointed ´etale cover over (V₁,−→

01). By standard Galois theory, it allows us to identify π^´₁^et(V_n,−→

01) as a subgroup ofπ₁^´et(V₁,−→ 01).

Let x, y be the generators of π^´et₁ (V₁,−→

01) given by the loops based at −→

01 on V₁ =P¹− {0,1,∞} running around 0,1 once anti-clockwise respectively. Then, it is easy to see that, as a subgroup of it, π^´et₁ (Vn,−→

01) is freely generated by xn := xⁿ and yb,n := x^−byx^b (0≤b < n).

2.2. Galois associators and Kummer-Heisenberg measure. Now, let z be an F- point of V₁ =P¹− {0,1,∞}. We have the canonical comparison map

π1(V1(C);z,−→

01)−→π₁^´et(V1;z,−→ 01) from the set of homotopy classes of paths from −→

01 to z on V₁(C) to the ´etale paths from

−

→01 to z on V₁ ⊗F¯. The Galois group G_F acts on the profinite group π^´et₁ (V₁,−→

01) and its torsor of paths π^´et₁ (V₁;z,−→

01).

Let us fix an ´etale pathγ ∈π₁^´et(V1(C);z,−→

01). Forσ ∈GF, define the Galois associator for the path γ by

(2.1) f_γ(σ) :=γ⁻¹·σ(γ)∈π^´₁^et(V₁,−→ 01), where σ(γ) := σ◦γ◦σ⁻¹.

Write π⁰ for the commutator subgroup of a profinite group π. The abelianization of f_γ(σ) is known (cf. [NW1, Proposition 1]) to be expressed as:

(2.2) f_γ(σ)≡x^ρz,γ(σ)y^{ρ1−z,γ(σ)} modπ^´₁^et(V₁,−→ 01)⁰,

with the ˆZ-valued functions

ρ_z,γ, ρ1−z,γ :G_F →Zˆ

the Kummer 1-cocycles associated with the roots of z and 1−z. They are respectively calculated along γ with the above chosen base of the Tate module

(2.3) (ξn)n≥1 ∈Zˆ(1) := lim←−

n

µn.

For the latter ρ1−z,γ, we understand the points −→

01 and 1−z are connected by the unit segment [0,1] on P¹ followed with the reversed path of γ by (∗ 7→1− ∗). We sometimes omit the mention toγ when it is obvious from context.

Definition 2.1. Letσ ∈G_F and set

f_γ^[(σ) :=x^{−ρz,γ(σ)}f_γ(σ) (σ∈G_F).

which belongs to the subgroup π₁(V_n,−→

01) ⊂ π₁(V₁,−→

01) by (2.2) for every n ≥ 1. Given 0≤b < n, we define κ⁽ⁿ⁾z,γ(σ)(b)∈Zˆ by the congruence

f_γ^[(σ)≡

n−1

Y

b=0

y_b,n^{κ(n)z,γ(σ)(b)}

modulo π^´₁^et(V_n,−→

01)⁰: the commutator subgroup of π₁^´et(V_n,−→ 01).

Proposition 2.2 (See [NW1] Lemma 1). For each σ∈G_F, the system of functions n

Z/nZ3b 7→κ⁽ⁿ⁾_z,γ(σ)(b)∈Zˆ o

n∈N

running over n≥1 defines a Zˆ-valued measure on Zˆ. We shall denote the above measure by

κ1(γ :−→

0199Kz)(σ) or κ1(z)_γ(σ)

and call it the Kummer-Heisenberg measure associated with the path γ : −→

0199Kz. We view it as an element of the Iwasawa algebra ˆZ[[ˆZ]]. Recall that ˆZ(1) in (2.3) is the Galois module ˆZ acted on by GF by multiplication by the cyclotomic character. Let ˆZ(−1) be its dual.

Proposition 2.3. The function κ1(γ :−→

0199Kz) :G_F →Zˆ(1)[[ˆZ(−1)]]

is a cocycle. Namely it holds that

κ⁽ⁿ⁾_z,γ(στ)(b) =κ⁽ⁿ⁾_z,γ(σ)(b) +χ(σ)·κ⁽ⁿ⁾_z,γ(τ)(χ(σ)⁻¹b) for σ, τ ∈G_F, n ≥1, b∈Z/nZ.

Proof. By the definition of f_γ (2.1), we have f_γ(στ) = f_γ(σ)·σ(f_γ(τ)), hence f_γ^[(στ) ≡ f_γ^[(σ)·σ(f_γ^[(τ)) modulo π₁(V_n)⁰. The assertion follows from this and the observation

σ(y_b,n)≡x^−χ(σ)by^χ(σ)x^χ(σ)b ≡(y_χ(σ)b,n)^χ(σ)

modulo π₁(V_n)⁰.

Remark 2.4. In [NW1, Lemma 1], we introduced a compatible sequence (κ_n)_n in the projective system lim

←−ⁿZˆ[Z/nZ] which forms a measure ˆκ ∈ Zˆ[[ˆZ]]. We call ˆκ (resp. κ¹) the Kummer-Heisenberg measure ine-form (resp. t-form) in the terminology of Appendix B. These two measures are ‘oppositely directed’ mainly because of different choice of path conventions as follows. After identification ˆZ→^∼ Zˆ(1) by 1 7→ (ξn = exp(2πi/n))n, let denote the involution on ˆZ(1) induced by ξ 7→ ξ⁻¹. Then, we have κ¹(σ) = ·κ(σ)ˆ (σ∈GF) as elements of ˆZ[[ˆZ(1)]].

3. Adelic Hurwitz measure

3.1. Paths to roots of unity. Fix m ∈ N>1 and let a be an integer with m - a. Let ι : V₁ → V₁ be the involution given by ι(z) = z⁻¹. For any path γ on V₁(C) from −→

01 to ξ_m^a, we set another path ¯γ from−→

01 to ξ_m^−a by

¯

γ :=ι(γ)·Γ∞

where Γ∞ is a path on V₁(C) from −→

01 to −→

∞1 as in Figure 1.

Figure 1. Γ∞ is a path from −→

01 to −→

∞1

Write _m^a = _a

m

+{_m^a} so that 0 ≤ {_m^a} <1, and define the path Γ_a/m : −→

0199Kξ_m^a to be the composition Γ{a/m}·x^ba/mc, where Γ{a/m} is the path illustrated as in Figure 2.

Figure 2. Γ{a/m} is a path from −→ 01 to ξ^a_m

It is easy to see the following lemma.

Lemma 3.1. Along the above paths Γ_a/m : −→

0199Kξ_m^a and Γ¯_a/m : −→

0199Kξ_m^−a, the associated Kummer 1-cocycles are coboundaries satisfying

ρ_ξm^a_,Γa/m(σ) = a

m(χ(σ)−1), ρ_ξ^−a

m ,Γ¯_a/m(σ) =−a

m(χ(σ)−1) for σ ∈G_Q(µm).

Proof. The first formula is immediate from the definition and the identification ˆZ ∼= Zˆ(1) by 1 7→ (ξn)n≥1. For the second, it suffices to note that the image of ¯Γ_a/m by P¹ − {0,1,∞} ,→ P¹ − {0,∞} is topologically homotopic to the complex conjugate of

Γ_a/m.

Remark 3.2. It is worth noting that Γ_αxⁿ = Γ_α+n for any α ∈ Q and n ∈ Z. This additivity property does not hold for ¯Γα in general. Still, if 0≤α≤1, then it holds that Γ¯α = Γ−α = Γn−αx⁻ⁿ for every n ∈ Z. This last point will play a crucial role later in Lemma 5.7.

3.2. Translation of a measure. Let p_n : V_n → V₁ be the cyclic cover of degree n considered in§2.1. For an ´etale path γ :−→

0199Kz onV₁, we shall write γ_n = (γ)_n

:−→

01_n99Kz^1/n

to denote the lift of γ to V_n for all n ≥ 1. Let us fix σ ∈ G_F. Note that the end point z^1/n may or may not be fixed by the σ. By (2.2), we have

(3.1) f_γ^[(σ) = (γ·x^ρz,γ(σ))⁻¹σ(γ)∈π^´₁^et(V₁,−→ 01)⁰. But since V_n is an abelian cover of V₁, π^´et₁(V₁,−→

01)⁰ is contained in π₁(V_n,−→

01). Therefore (3.1) implies that the lifts of γ ·x^ρz,γ(σ) and of σ(γ) departing at −→

01n on Vn end at the same point σ(z^1/n) = ξn^ρz,γ(σ)z^1/n. Since the lift (x^ρz,γ(σ))_n of x^ρz,γ(σ) from −→

01_n ends at ξn^ρz,γ(σ)

−

→01_n, the subsequent pathγ_n,σ should lift γ so as to start from that pointξn^ρz,γ(σ)

−

→01_n with ending at the point σ(z^1/n) on V_n:

−

→01_n ^{xρz,γ(σ)//}

(σ(γ))n

66ξ^ρz,γ(σ)n

−

→01_n ^{γn,σ //}σ(z^1/n). (3.2)

In summary, writing (σ(γ))_n for the lift of σ(γ) from −→

01_n onV_n, we may express f_γ^[(σ) as the composition of those three paths

f_γ^[(σ) = (x^ρz,γ(σ))⁻¹_n ·γ_n,σ⁻¹ ·(σ(γ))_n onV_n.

Below, we shall see magnification of the base space ˆZon a cosets+rZˆ (s, r∈Z,r ≥1) under the measure κ1(z)_γ(σ) can be interpreted as a twisted lifting of the reference path γ :−→

0199Kz toVr followed with ‘s-rotated’ embedding by Vr ,→V1. Set an ‘s-modified’ pathγh−si :−→

0199Kz, for the given path γ :−→

0199Kz on V₁, by

(3.3) γ_h−si:=γ·x^−s.

It follows easily that

(3.4) ρz,γ_h−si(σ) =ρz,γ(σ)−s(χ(σ)−1) (σ∈GF).

Suppose that ξ_r, z^1/r ∈F. Write

γ_r = (γ)_r :−→

01_r99Kz^1/r, γ_h−si,r = (γ_h−si)_r :−→

01_r99Kξ_r^−sz^1/r

for the lifts of the paths γ and γ_h−si by p_r:V_r →V₁ respectively, and γ_r∗ =_r(γ_r) :−→

0199Kz^1/r, γ_h−si,r∗ =_r(γ_h−si,r) :−→

0199Kξ_r^−sz^1/r

for the images of paths γ_r, γh−si,r on V_r by the immersion _r : (V_r,−→

01_r) ,→ (V₁,−→ 01) respectively. It follows that

ρ_z1/r,γ_h−si,r∗(σ) =ρ_z1/r,(γ)r∗(σ)−s

r(χ(σ)−1) (3.5)

= 1

rρ_z,γ(σ)− s

r(χ(σ)−1) for every σ ∈G_F.

Lemma 3.3. Notations being as above, with assumptions ξ_r, z^1/r ∈F and σ ∈G_F. (i) For every n ≥1, it holds that

κ^(nr)_z,γ (σ)(vr+sχ(σ)) =κ⁽ⁿ⁾

ξ^−sr z^1/r,γh−si,r∗(σ)(v) (v = 0, . . . , n−1), where vr+sχ(σ) in LHS is regarded ∈(Z/nrZ).

(ii) For any continuous function ϕ on Zˆ, we have Z

sχ(σ)+rZˆ

ϕ(b)dκ1(γ :−→

0199Kz)(σ)(b) = Z

Zˆ

ϕ(rv+sχ(σ))dκ1(γh−si,r∗ :−→

0199Kξ_r^−sz^1/r)(σ)(v).

Proof. In this proof, forn ≥1, we denote π(n) :=π^´₁^et(V_n,−→

01) and write

(3.6) $_nr :π(nr)π(n)

for the surjection induced from the open immersion (Vnr,−→

01nr) ,→ (Vn,−→

01n). Note that, among the standard generators x^nr, yb,nr (b = 0, . . . , nr−1) of π(nr), onlyx^nr and yvr,nr

(v = 0, . . . , n−1) survive via $_nr to be xⁿ, y_v,n (v = 0, . . . n−1) in π(n).

Noting thatx^−uyx^u =y_u,nr ≡y_u+nrk,nr modπ(nr)⁰ foru, k ∈Z, we see from Definition 2.1 that

x^sχ(σ)·f_γ^[(σ)·x^−sχ(σ) ≡

nr−1

Y

u=0

y−sχ(σ)+u,nrκ^(nr)z,γ (σ)(u) mod π(nr)⁰ (3.7)

≡

nr−1

Y

v=0

yv,nrκ^(nr)z,γ (σ)(v+sχ(σ))

mod π(nr)⁰ which should map via $nr to the product over those v multiples ofr:

(3.8) $_nr x^sχ(σ)·f_γ^[(σ)·x^−sχ(σ)

≡

n−1

Y

v=0

y_v,n^κ(nr)z,γ (σ)(rv+sχ(σ)) mod π(n)⁰

as π(n)⁰ ⊃ $_nr(π(nr)⁰). We shall interpret the LHS of the above expression (3.7) by applying the composition diagram (3.2) to the pathγ_h−si :−→

0199Kz (3.3) on V₁ and its lift

(γ_h−si)_r =γ_h−si,r onV_r:

V_nr ^{ /}

V_n

[γh−si,r :−→

0199Kξ_r^−sz^1/r] ^ V_r ^{ /}

V₁ ^{? _}[γh−si,r∗]

[γh−si :−→

0199Kz] ^ V₁ .

We first derive:

x^sχ(σ)·f_γ^[(σ)·x^−sχ(σ) =x^sχ(σ)·x^{−ρz,γ(σ)}γ⁻¹σ(γ)·x^−sχ(σ) (3.9)

=xs(χ(σ)−1)−ρz,γ(σ)·(γx^−s)⁻¹σ(γx^−s)

= (γh−si·x^{ρz,γh−si(σ)})⁻¹·σ(γh−si).

By (3.4), the former factor of path composition reads onV_r (γh−si)_r,σ· x^{ρz,γh−si(σ)}

r = (γx^−s)_r,σ· x^ρz,γ(σ)−s(χ(σ)−1)

r

where (γh−si)_r,σ stands for a suitable lift of γh−si on V_r, which arrives at the same end point on V_r as the latter σ-transformed factor

(σ(γh−si))_r :−→

01_r99Kσ(ξ_r^−sz^1/r).

It turns out that (γ_h−si)_r,σ starts at ξ^ρz,γ(σ)−s(χ(σ)−1)

r ·−→

01_r which is equal to −→

01_r by our assumption ξ_r, z^1/r ∈F. Thus we conclude

(3.10) (γh−si)_r,σ =γh−si,r = (γ ·x^−s)_r .

By virtue of this and (3.5), applying to (3.9) the surjection $_r :π(r)π(1) determined byx^r 7→x, y₀ 7→y and y₁, . . . , yr−1 7→1 as the case n= 1 of (3.6), we obtain

$_r

(x^{−ρz,γh−si(σ)})_r·(γh−si)⁻¹_r,σ·σ(γh−si,r)

=$_r

(x^r)^{−ρz1/r ,γr(σ)+sr(χ(σ)−1)}γ_h−si,r⁻¹ ·σ(γh−si,r)

=x^{−ρz1/r ,γh−si,r∗(σ)}·γh−si,r∗−1·σ(γh−si,r∗)

=f_γ^[

h−si,r∗(σ)

≡

n−1

Y

v=0

y_v,n

κ⁽ⁿ⁾

ξ−s

r z1/r ,γh−si,r∗

(σ)(v)

modπ(n)⁰.

This, combined with (3.8) and (3.9) and the compatibility$_nr =$_r|_π(nr), proves (i). The

assertion (ii) is just a formal consequence of (i).

Suppose we are given a measure µ ∈ Zˆ[[ˆZ]]. Let m, a ∈ Z be integers as in §3.1 and pick ν ∈Zˆ^×. Consider the coset Q_aν,m := ^aν_m + ˆZof ˆZ in Qf = ˆZ⊗Q. Then, obviously,

R_aν,m:=m·Q_aν,m =aν+mZˆ ⊂Zˆ.

We define the measure [m, aν]∗(µ) on R_aν,m by assigning to each open subset U ⊂R_aν,m the valueµ(U⁰), whereU⁰ is the inverse image ofU by the affine mapt 7→mt+aν (t ∈Zˆ).

Note that Lemma 3.3 (ii) reads:

(3.11) κ1(γ :−→

0199Kz)(σ)|_sχ(σ)+mZˆ = [m, sχ(σ)]∗

κ1(γh−si,m∗ :−→

0199Kξ_m^−sz^1/m)(σ) for σ ∈G_F, m, s∈ Z,m ≥1, where ∗|_sχ(σ)+mZˆ in LHS designates the restricted measure onR_sχ(σ),m =sχ(σ) +mZˆ ⊂Zˆ.

Letιdenote the action of the complex conjugation on ˆZ(1)[[ˆZ(−1)]], that is, the action of −1∈Zˆ^×. It is straightforward to see

(3.12) ι◦[m, aν]∗ = [m,−aν]∗◦ι.

Now we are ready to introduce the fundamental object of our study. Let m > 1 and a∈Z as above, and let Γ_a/m ∈π(V₁(C);ξ^a_m,−→

01) be the path introduced in§3.1.

Definition 3.4 (ˆZ-Hurwitz and adelic Hurwitz measure). For each σ ∈G_Q(µm) we define the ˆZ-Hurwitz measureζ_a/m(σ)∈Zˆ[[ˆZ]] and the adelic Hurwitz measure ˆζ_a,m(σ)∈ Zˆ[[ˆZ]] by the formulas

ζ_a/m(σ) :=κ1

−

→01

Γ¯a/m

−99Kξ_m^−a

(σ) +ι

κ1

−

→01

Γa/m

−99Kξ^a_m (σ)

; ζˆ_a,m(σ) := [m, aχ(σ)]∗ζ_a/m(σ).

4. Geometrical interpretation of translation of measure

In this section we address the fact that translation of Kummer-Heisenberg measure by [m, aχ]∗ corresponds to path composition with the loop x^α. We work however only with p-adic measures.

4.1. p-adic Galois polylogarithms. Let γ be an ´etale path on V₁ from −→

01 to an F- rational (possibly tangential) point z. Let QphhX, Yii be the non-commutative power series ring in two variables X, Y, and write E : π₁^pro-p(V1,−→

01) ,→ Q^phhX, Yii for the embedding that sends the standard generators x, y to exp(X), exp(Y). We define I_Y to be the ideal of QphhX, Yii generated by monomials containing Y twice or more.

Forσ ∈G_F, set

Λ_γ(σ) :=E(f_γ(σ));

Λ_γ(σ) :=E(f_γ^[(σ)) = exp(−ρ_z,γ(σ)X)·Λ_γ(σ).

Definition 4.1. Define p-adic Galois polylogarithms `ik(z)γ,Lik(z)γ : GF → Q^p by the congruence expansion

log Λ_γ(σ)≡ρ_z,γ(σ) +

∞

X

k=1

(−1)^k−1`i_k(z)_γ(σ) (adX)^k−1(Y), log Λ_γ(σ)≡

∞

X

k=1

(−1)^k−1Li_k(z)_γ(σ) (adX)^k−1(Y) modulo the ideal I_Y.

Proposition 4.2. The family of functions {ρ_z,γ, `i_k(z)_γ, Li_k(z)_γ :G_F →Q^p}_k≥1 satisfy Lik(z)γ =

k

X

i=1

ρ^k−i_z,γ

(k+ 1−i)!`ii(z)γ, (i)

`i_k(z)_γ =

k−1

X

s=0

B_s

s!ρ^s_z,γLi_k−s(z)_γ (ii)

for k = 1,2, . . ..

In fact, this proposition is a formal consequence of the following lemma:

Lemma 4.3. Let K be a field of characteristic zero, and suppose that two sequences {b_i}i≥0, {u_i}i≥0 in K satisfy the congruence

e^−u0Xe^{u0X+P∞k=0uk+1(adX)k(Y)} ≡e^{b0X+P∞k=0bk+1(adX)k(Y)} modI_Y as non-commutative power series in X, Y. Then, b0 = 0 and, for k= 1,2, . . .,

b_k =

k

X

i=1

(−u₀)^k−i

(k+ 1−i)!u_i, u_k =

k−1

X

s=0

B_s

s!(−u₀)^sbk−s, where B₀, B₁, . . . are Bernoulli numbers defined by P∞

s=0 Bs

s!T^s= _eT^T−1. Proof. We use the classical Campbell-Hausdorff formula

log(e^αe^β)≡β+

∞

X

n=0

B_n

n!(adβ)ⁿ(α) mod deg(α)≥2.

Set −α = P∞

i=0bi+1(adX)ⁱ(Y) and −β = u0X so that congruences mod deg(α) ≥ 2 derive those mod I_Y. It follows that log(e^u0Xe^{b1Y+b2[X,Y]+...}) is congruent to u₀X + P∞

k=0

Pk s=0

Bs

s!(−u₀)^sbk+1−s

(adX)^k(Y) mod I_Y. This is equivalent to the equality (4.1)

∞

X

k=0

u_k+1T^k =

−u₀T e^−u0T −1

^∞ X

k=0

b_k+1T^k.

The assertion follows from this immediately.

4.2. Extension for Qp-paths. Letπ_Qp(−→

01) be the pro-algebraic hull of the image of the above embedding E : π₁^pro-p(V₁,−→

01),→QphhX, Yii, and extend it to the inclusion of path torsors π^pro-p₁ (V₁;z,−→

01) ,→ π_Qp(z,−→

01) naturally. The elements of π_Qp(−→

01), π_Qp(z,−→ 01) will be simply called Qp-paths, and the action of the Galois group G_F on the pro-p paths extends to that on the Qp-paths in the obvious manner.

For each Q^p-path γ : −→

0199Kz and σ ∈ G_F, we may define the Galois associator f_γ(σ) := γ⁻¹ ·σ(γ) ∈ π_Qp(−→

01) extending (2.1). Then, define ρ_z,γ, li_k(z, γ) : G_F → Qp

(k = 1,2, . . .) as the coefficients in log(f_γ(σ)) so as to extend the congruence in Definition 4.1 mod I_Y, and then, define Li_k(z)_γ : G_F → Qp (k = 1,2, . . .) as the coefficients of log(exp(−ρ_z,γ(σ)X)·f_γ(σ)) again as the extension of Definition 4.1. Then, it is simple to see that the identities in Proposition 4.2 hold true for Q^p-paths γ : −→

0199Kz in the same forms.

4.3. Relation with κ^1,p. We now arrive at the stage to connect the `-adic polyloga- rithms Li_kand the Kummer-Heisenberg measureκ1. In [NW1], we showed that, for pro-p paths γ :−→

0199Kz, the function Li_k(z)_γ multiplied by (k−1)! can be written by a certain polylogarithmic character ˜χ_k(z)_γ :G_F →Zp defined by Galois transformations of certain sequence of numbers of forms Qpⁿ−1

s=0 (1−ξ^sz^1/pn)^sk−1/pn (ξ ∈ µ_pⁿ, n ≥ 1). This enabled us to express Li_k(z)_γ(σ) (σ ∈ G_F) by the moment integral _(k−1)!¹ R

Zpb^k−1dκ1,p(σ)(b) over the p-adic measure κ1,p(σ) which is by definition the image of the Kummer-Heisenberg measureκ1(σ) (§2.2) by the projection ˆZ[[ˆZ]]→Zp[[Zp]].

A generalization of this phenomenon has been investigated in [NW3] for some more generalQp-paths of the formγx^a/m. We summarize the result as follows:

Proposition 4.4 ([NW3]§7). Let γ :−→

0199Kz be a pro-p path. Then, for any α∈Qp, we have

Li_k(z)_γx^α(σ) = 1 (k−1)!

Z

Zp

(b+αχ(σ))^k−1dκ1,p(−→

0199K^γ z)(b) (σ ∈G_F).

Proof. We just translate [NW3,§7] frome-form to t-form in the terminology of Appendix B. In e-form, it reads (with δ:=γ, α:=−_n^s, ˆκ_p :=κ_z,γ in [NW3, §7]):

˜

χ^x_k^αδ(σ) = Z

Zp

(b−αχ(σ))^k−1dˆκ_p(σ)(b).

Let γx^α be the t-path reciprocally corresponding to the e-path x^αδ. In RHS, we regard the measure ˆκ_p as thep-adic image of ˆκ(δ) of Remark 2.4 which can be switched into the e-form κ1,p(γ) to obtain

Z

Z^p

(b−αχ(σ))^k−1dˆκp(σ)(b) = Z

Z^p

(−b−αχ(σ))^k−1dκ^1,p(σ)(b).

At the same time, we may convert the LHS to t-form by (B.11) and (B.13) as

˜

χ^x_k^αδ(σ) =−(k−1)!Li_k(z)_xαδ:−→ 01 z(σ)

= (−1)^k−1(k−1)! Lik(z)_γxα:−→

0199Kz(σ).

The formula of the proposition follows from combination of these identities.

5. Consequence of Inversion Formula

5.1. Pro-p inversion formula. We start this section with the main technical result.

Let a, m be integers with m > 1, m - a, and fix the m-th root of unity z :=ξ_m^a ∈ µ_m and set F = Q(z). Pick any path γ : −→

0199Kz in π^pro-p₁ (P¹ − {0,1,∞}, z,−→

01) and let

¯ γ :−→

0199Kz⁻¹ be the associated path defined in§3.1.

By the assumption z ∈ µ_m, using the p-adic cyclotomic character χ_p : G_F → Z^×p, we may suppose that the Kummer 1-cocycle ρ_z,γ :G_F →Zp (written justρ_z for simplicity) is of a 1-coboundary form

(5.1) ρz,γ(σ) =ρz(σ) = α(χp(σ)−1) (σ ∈GF)

with a unique constant α∈ _m^a +Zp. Since we do not assume p-m, the constant α∈Qp

may generally have denominator, while ρ_z(σ)∈Z^p.

Theorem 5.1. Notations being as above, we have

Li_k(ξ_m^−a)_γx¯ ^α(σ) + (−1)^kLi_k(ξ_m^a)_γx^−α(σ) = 1

k!B_k(α)(1−χ_p(σ)^k).

for σ ∈G_F and k ≥1.

This result generalizes [W3, Theorem 10.2], where only the case p-m was considered.

Here, we shall present a proof using the inversion formula forp-adic Galois polylogarithms [NW2]. Forσ ∈G_F, consider the`-adic polylogarithmic characters (for`=p) ˜χ_k(z)_γ(σ),

˜

χ_k(¹_z)_γ¯(σ) along those pro-ppathsγ and ¯γ. In [NW2, 6.3], we showed an inversion formula for γ and ¯γ in the following form^∗

(5.2) χ˜_k(z)_γ(σ) + (−1)^kχ˜_k(¹_z)_¯γ(σ) =−1

k{B_k(−ρ_z(σ))−B_k·χ_p(σ)^k} (σ∈G_F), where B_k(T) is the Bernoulli polynomial defined by P∞

k=0B_k(T)^w_k!^k = ^we_ew^{T w}−1 and B_k = B_k(0). Apply to (5.2) the translation formula

(5.3) Li_k(z)_γ(σ) = (−1)^k−1

(k−1)! χ˜_k(z)_γ(σ) (σ∈G_F, k≥1) for which we refer the reader to (B.11), (B.13) and (B.14), and obtain (5.4) Li_k(¹_z)_¯γ(σ) + (−1)^kLi_k(z)_γ(σ) = 1

k!(B_k(−ρ_z(σ))−B_k·χ_p(σ)^k) (σ ∈G_F).

Observe that this formula already gives a special case of Theorem 5.1 where α = 0 and ρ_z(σ) = 0. What we shall do from now is to deform this formula into a form involved with the Qp-paths γx^−α and ¯γx^α. In fact, it follows from Proposition 4.4, we generally have

Li_k(z)_γx^α(σ) =

k−1

X

i=0

Lik−i(z)_γ(σ)(αχ_p(σ))ⁱ i! , hence the LHS of Theorem 5.1 can be written as:

Li_k(¹_z)_γx¯ ^α(σ) + (−1)^kLi_k(z)_γx^−α(σ) (5.5)

=

k−1

X

i=0

(αχ_p(σ))ⁱ i!

Li_k−i(¹_z)_γ¯(σ) + (−1)^k−iLi_k−i(z)_γ(σ)

.

To complete the proof of Theorem 5.1, by comparing (5.4) and (5.5), we are now reduced to the following core lemma:

Lemma 5.2. Letk be a positive integer, and set J_s :=−¹_s{B_s(−ρ_z(σ))−B_s·χ_p(σ)^s} for s= 1, . . . , k and σ ∈G_F. Then, we have

1

kB_k(α) χ_p(σ)^k−1

=

k−1

X

i=0

k−1 i

αⁱχ_p(σ)ⁱJk−i(σ).

∗See (B.14). The path ¯γ from −→

01 to z⁻¹ ∈ µm in t-form here reciprocally corresponds to the path h0,1i[1^∞₀ ]h1,∞i ·f₂(γ) ine-form with the notation of [NW2,§6.3].

Proof. For simplicity, we omitσ in this proof. To simplify the RHS of the lemma, we use the Bernoulli addition formula

(5.6) B_k(y+x) =

k

X

s=0

k s

B_s(y)x^k−s.

Applying (5.6) with x = αχp, y = −ρz so that x+y = α by (5.1) (resp. with x = α, y= 0 so that x+y=α), we obtain:















B_k(α) =

k−1

X

i=0

k i

B_k−i(−ρ_z)(αχ_p)ⁱ+ (αχ_p)^k, B_k(α) =

k−1

X

i=0

k i

αⁱB_k−i+α^k.

Each of the above identities respectively simplifies the former and latter term of the following computation of the RHS of the lemma. In fact, noting _k−i^{1 k−1}_i

= ¹_k ^k_i , one computes:

RHS =

"

−

k−1

X

i=0

(αχ_p)ⁱ k

k i

Bk−i(−ρz)

# +

"

χ^k_p k

k−1

X

i=0

k i

αⁱBk−i

#

=

−1

kB_k(α) + 1 kα^kχ^k_p

+

"

χ^k_p

k (B_k(α)−α^k)

#

which coincides with the LHS of the aimed identity.

Thus, our proof of Theorem 5.1 is completed.

Remark 5.3. We mention that the proof in [W3] also carries over in the case p|m; One needs only consider rational paths in π₁^pro-p(V₁;ξ_m^a,−→

01) ˆ⊗Q and in π^pro-p₁ (V₁,−→

01) ˆ⊗Q. The embedding of the latter π₁⊗ˆQ into QphhX, Yii extends to that of the former pro-p path space into QphhX, Yii.

Remark 5.4. If p - m, then for any α ∈ Zp there is γ ∈ π₁^pro-p(V₁;ξ_m^a,−→

01) such that ρ_z,γ =α(χ_p−1). Hence we have

(k−1)!

χ_p(σ)^k−1 Lik(ξ_m^−a)¯γx^α(σ) + (−1)^kLik(ξ_m^a)_γx^−α(σ)

=−B_k(α) k

as long asχp(σ)^k 6= 1. A key observation here is the following: Takingα= 0 we get values of the Riemann zeta function at negative integers (cf. [W2]), while taking α = _m^a ∈ Q^× we get values of Hurwitz zeta function ζ(s,_m^a) at negative integers. If we chooseγ from topological paths Γ_a/m ∈ π(V₁(C);ξ_m^a,−→

01) (§3.1), then we get −¹_kB_k(_m^a) for every choice of rational prime p.

5.2. Moment integrals of p-adic Hurwitz measure. First we shall rewrite the for- mula in Theorem 5.1 in terms of measures κ^1,p(−→

0199K^γ ξ_m^a) andκ^1,p(−→

0199K^¯γ ξ_m^−a) after multi- plied by m^k−1. Set α:= _m^a ∈Q. By Proposition 4.4 we find that, for σ∈G_F,

m^k−1Li_k(ξ_m^−a)_¯γx^α(σ) = m^k−1 (k−1)!

Z

Zp

(v +αχ_p(σ))^k−1d

κ1,p(−→

0199K^¯γ ξ_m^−a)(σ) (v) (5.7)

= 1

(k−1)!

Z

aχp(σ)+mZp

b^k−1d [m, aχ_p(σ)]∗κ1,p(ξ_m^−a)_γ¯(σ) (b)

= 1

(k−1)!

Z

Zp

b^k−1d [m, aχ_p(σ)]_∗κ^1,p(ξ_m^−a)_¯γ(σ) (b),

where the last equality follows as the measure [m, aχ_p(σ)]∗κ1,p(ξ_m^−a)_¯γ is supported on aχp(σ) +mZ^p ⊂Z^p. In the same way, we get that

m^k−1Li_k(ξ_m^a)_γx^−α(σ) = m^k−1 (k−1)!

Z

Zp

(v−αχ_p(σ))^k−1dκ1,p(−→

0199K^γ ξ_m^a)(σ)(v) (5.8)

=−(−1)^k−1 m^k−1 (k−1)!

Z

Zp

(v+αχ_p(σ))^k−1d

ι·κ1,p(−→

0199Kξ_m^a)_γ(σ) (v)

= (−1)^k (k−1)!

Z

Zp

b^k−1d

[m, aχ_p(σ)]∗(ι·κ1,p(ξ_m^a)_γ(σ)) (b).

Now, we enter the situation of Theorem 1.1 and §3, that is, a, m ∈ Z (m > 1) are integers with m-a, and set γ := Γ_a/m, α:=a/m.

Corollary 5.5. For the adelic Hurwitz measure ζˆ_a,m = [m, aχ_p(σ)]∗ζ_a/m ∈ Zˆ[[ˆZ]], the p-adic image ζˆ_p,a.m(σ)∈Zp[[Zp]] satisfies

Z

Zp

b^k−1dζˆ_p,a,m(σ)(b) = m^k−1 k B_ka

m

(1−χ_p(σ)^k) (σ∈G_Q(µm), k ≥2).

Proof. Combining the above calculations (5.7) and (5.8), we obtain from Theorem 5.1:

m^k−1

k B_k(α)(1−χ_p(σ)^k) = (k−1)!m^k−1 Li_k(ξ^−a)_γx¯ ^α(σ) + (−1)^kLi_k(ξ^a)_γx¯ ^−α(σ)

= Z

Zp

b^k−1d[m, aχ_p(σ)]∗

κ1,p(−→

0199K^¯γ ξ_m^−a)(σ) +ι·κ1,p(−→

0199K^γ ξ_m^a)(σ) (b)

= Z

Zp

b^k−1d[m, aχ_p(σ)]∗ζ_p,a/m(σ)(b),

where ζ_p,a/m(σ) is the image of ζ_a/m(σ) by the projection ˆZ[[ˆZ]] → Zp[[Zp]]. This con-

cludes the proof of the corollary.

5.3. Proof of Theorem 1.1. Note first that the support of the measure ˆζ_p,a,m(σ) is aχ_p(σ) +mZp. When p | m and p - a, it is included in Z^×p so that the above Corollary proves the case.

Remark 5.6. It is worth noting that we do not need to assume 0 < a < m for the construction of the measure ˆζ_a,m and the integration property in the above case of p|m.

This leads to Remark 1.3 of Introduction.