versus ∞ -adic Eulerian

versus ∞ -adic Eulerian

CHIEH-YU CHANG AND YOSHINORI MISHIBA

Abstract. In this paper, we give a simultaneous vanishing principle for thev-adic Carlitz multiple polylogarithms (abbreviated as CMPLs) at algebraic points, where v is a finite place of the rational function field over a finite field. This principle establishes the fact that thev-adic vanishing of CMPLs at algebraic points is equivalent to its∞-adic counterpart being Eulerian. This reveals a nontrivial connection between thev-adic and∞-adic worlds in positive characteristic.

1. Introduction

1.1. Motivation. The study of this paper is motivated by the classical theory and conjec- ture for special zeta values. Letn≥2 be an integer. Then the celebrated formula of Euler for the special values of the Riemann zeta function at even positive integers implies that

ζ(n)/(2π√

−1)ⁿ ∈_Q⇔n is even.

For a prime number p, we consider the Kubota-Leopoldt p-adic zeta function ζp, that interpolates the rational values of the Riemann zeta function at non-positive integers.

When n is even, we know that ζp(n) = 0 (cf. [KL64, Col82, F04]). So this reveals the following interesting phenomena between the archimedean world and its p-adic counter part: for an integern ≥_2,

ζ(n)/(2π√

−1)ⁿ ∈ _Q⇔n is even ⇒ζp(n) =0.

Conjecturally, the reverse direction above is valid. It is known by [S81] that for m ∈ _N, ζp(2m+1)is nonzero when pis regular or(p−1)|2m.

We note that for an integer n ≥ 2, ζ(n) = Lin(1), where Lin is the nth polylogarithm given by

Lin(z) :=

∑

z^m mⁿ.

For a fixed prime number p, we let Cp be the p-adic completion of a fixed algebraic closure of thep-adic numbersQp. The series Lin(z)convergesp-adically on the open unit disc and we denote this function by Lin,p which is called the p-adic nth polylogarithm.

For each branch parameter a ∈ _Cp of the p-adic logarithm, we note that by [F04, Col82] Lin,p can be analytically continued to Cpr{1}, and we denote by Li^a_n,p its analytically

Date: April29,2016.

2010Mathematics Subject Classification. Primary11R58,11J93.

Key words and phrases. Carlitz multiple polylogarithms,t-modules,v-adic vanishing,∞-adic Eulerian.

The first author was partially supported by a Golden-Jade fellowship of the Kenda Foundation and MOST Grant102-2115-M-007-013-MY5.

The second author is supported by JSPS KAKENHI Grant Number15K17525. 1

continued function. Furusho [F04] showed thatζp(n) is the limit of Li^a_n,p when “z →1”

in the sense of [F04], andζp(n) is independent of choices of branch parameters a.

We fix embeddings Q ,→ _C and Q ,→ _Cp. Let u be a nonzero algebraic number for which Lin(u) converges with respect to the archimedean absolute value. Inspired by the conjectural equivalence between the rationality ofζ(n)/(2π√

−1)ⁿ and the vanishing of ζp(n), for a fixed branch parameter a ∈ _Cp of the p-adic logarithm one naturally asks whether there is a criterion ♠_a for the rationality of Lin(u)/(2π√

−1)ⁿ in terms of the vanishing of Li_n,p^a (u):

Lin(u)/(2π√

−1)ⁿ ∈ _Q⇔ ♠_a ⇔Li^a_n,p(u) =0.

For example, forn ≥_{2 and “u} →1”, conjecturally♠_a =_{“n is even”.}

The main purpose of this paper is to give a positive answer of the analogous question above for the Carlitz multiple polylogarithms (abbreviated as CMPLs), which was intro- duced by the first author of the present paper in [C14] generalizing the notion initiated by Anderson and Thakur [AT90].

1.2. Depth one case in function fields. Let A := _Fq[θ] be the polynomial ring in the variableθ over the finite field Fq ofq elements, and k be its quotient field. Denote by∞ the infinity place ofk with an associated absolute value| · |_∞. Let k∞ :=_Fq((1/θ)) be the

∞-adic completion ofk. We fix an algebraic closure k_∞ ofk_∞, and let C∞ be the ∞-adic completion of k_∞. We further fix a fundamental period ˜π ∈ k_∞^× of the Carlitz Fq[t]- module C, where t is an independent variable. Note that C and ˜π play the analogous roles ofGm and 2π√

−1 respectively in the function field setting. A positive integernis called A-even if (q−₁)|n, as q−1 is the cardinality of the unit group A^×.

1.2.1. Special zeta values. Let A+ be the set of monic polynomials in A and consider the Carlitz zeta value atn ∈_N,

ζA(n) :=

∑

a∈A+

1

aⁿ ∈ k^×_∞.

In [Ca35], Carlitz derived an analogue of Euler’s formula on Riemann zeta function at even positive integers. More precisely, we have the following consequence: for a positive integern, we have the equivalence

ζA(n)/ ˜πⁿ ∈ k⇔n is A-even.

Let v be a monic prime of A, and let kv be the completion of k with respect to the normalized v-adic absolute value | · |_v associated to the placev. For a positive integern, in analogy with the special value atnof the Kubota-Leopoldt p-adic zeta functionζp(n) we consider the v-adic Goss zeta function at n denoted by ζA,v(n) ∈ kv (see [Go79]).

Goss [Go79] showed that ζA,v(n) is vanishing for A-even n, and Yu [Yu91] showed the transcendence of ζA,v(n) for A-oddn (ie.,(q−1) _-n), and therefore we have the follow- ing complete story: forn ∈ _N,

(1.2.1) ζ_A(n)/ ˜πⁿ ∈ k⇔ nis A-even⇔ζ_A,v(n) = 0.

1.2.2. Carlitz polylogarithms. Put L₀ := 1 and L_i := (θ−θ^q)· · ·(θ−θ^qi) for i ∈ _{N. Let} log_C be the logarithm of the Carlitz Fq[t]-module Cgiven by the power series

log_C(z) :=

∑

z^qi

L_i (see [Go96, T04]).

In analogy with the classical polylogarithm, Anderson and Thakur [AT90] defined the nth Carlitz polylogarithm for n∈ _N:

Lin(z):=

∑

z^qi Lⁿ_i .

Unlike the simple identity between ζ(n) and the nth polylogarithm at 1 in the classical case, Anderson and Thakur [AT90] proved thatζA(n) _{is a}k-linear combination of Lin at some explicit integral points in A.

We fix an algebraic closure kv of kv, and let Cv be the v-adic completion of kv. Let ¯k be an algebraic closure of k, and fix embeddings ¯k ,→ k_∞ and ¯k ,→ kv. For a positive integern, we denote byC^⊗n thenth tensor power of the Carlitz module Cintroduced by Anderson-Thakur [AT90]. Let u ∈ k^× and put v := (0, . . . , 0,u)^tr ∈ C^⊗n(k). We further put the condition:

♣₁:=“v = (0, . . . , 0,u)^tr is an Fq[t]-torsion inC^⊗n(k^¯)”.

We assume |u|_∞ <|θ|

nq

∞q−1 and note that Lin(u) converges inC∞. Then a consequence of the Anderson-Thakur theory [AT90] is the following equivalence:

(1.2.2) Lin(u)/ ˜πⁿ ∈ _k⇔ ♣_{1 is valid.}

Unlike the analytic continuation of classical p-adic polylogarithms, we shall mention here that the convergence domain of Lin with respect to thev-adic absolute value is the open unit disc{z ∈_Cv;|z|_v <1}. We denote by Lin(z)_vwhen the series Lin(z)converges v-adically. We assume that the given algebraic element u ∈ k has v-adic absolute value less than one, so Lin(u)_v is defined. Then by Yu’s theory [Yu91] one can derive the following equivalence

(1.2.3) Lin(u)_v =0⇔ ♣₁is valid,

and hence one establishes the principle: for u ∈ k^¯× with |u|_∞ <|θ|

nq

∞q−1 and |u|_v <1 we have

Lin(u)/ ˜πⁿ ∈ k⇔ ♣₁ is valid⇔Lin(u)_v =0.

The main result in this paper is to generalize this principle to higher depths.

1.3. Higher depths case. In analogy with the classical multiple polylogarithms, the first author of the present paper introduced the Carlitz multiple polylogarithm (abbreviated as CMPL) for each s= (s₁, . . . ,sr)∈ _N^r _{(see [C}¹⁴_]):

Lis(z₁, . . . ,zr) :=

∑

i₁>···>ir≥0

z^q₁ⁱ¹· · ·z^qr^ir

L^s_i¹

1 · · ·L^s_ir^r ,

whose weight is defined to be wt(s) _:= _∑^r_i=1s_i and whose depth is defined to be r. It is shown in [C14] that each multizeta value ζA(s) initiated by Thakur [T04] is ak-linear

combination of Lis at some integral points in A^r, generalizing the formula of Anderson- Thakur to the higher depth case.

Fixing any s = (s₁, . . . ,sr) ∈ _N^r _and u = (u₁, . . . ,ur) ∈ (k^×)^r, one has an associated t-module G := Gs,u defined over k and an associated algebraic point v := v_s,u ∈ G(k) given in [CPY14] (for avoiding the heavy notation, we drops and u without confusion when it is clear from the contents). Suppose that Li(s<sub>`</sub>,...,sr)(u`, . . . ,ur) _{converges ∞-} adically for each`=1, . . . ,r. It is shown in [CPY14] that

♣_r :=“vis an Fq[t]-torsion point in G(k)” is valid if and only if

Li(s₁,...,s_r)(u1, . . . ,ur)/ ˜π^s1+···+sr, Li(s₂,...,s_r)(u2, . . . ,ur)/ ˜π^s2+···+sr, . . . , Lisr(ur)/ ˜π^sr

are simultaneously ink. Note that the value Lis(u)is calledEulerianif the ratio Lis(u)/ ˜π^wt(s) lies ink(cf. [T04, CPY14]).

Our main result, stated as Theorem4.1.2, is to prove thev-adic counterpart fitting into the reflection principle. Suppose that |ui|_v <1 for all i. We show that ♣_r is valid if and only if

Li(sr,...,s₁)(ur, . . . ,u1)_v=Li(sr,...,s₂)(ur, . . . ,u2)_v=· · · =Lisr(ur)_v =0,

where the subscript v above denotes by the v-adic convergence. Note that when we restrictr =1, then we recover the result in (1.2.3).

We shall mention that for thes and u given above, if Li(s₁,...,s_r)(u1, . . . ,ur) is Eulerian, then the following values

Li_(s2,...,sr)(u2, . . . ,ur), . . . , Lisr(ur)

are automatically Eulerian. This fact is proven in [CPY14] using the ABP-criterion [ABP04], which is a strong tool in the transcendence theory of ∞-adic case. However, we do not know whether its analog is true in the v-adic case, which needs additional work on developing somev-adic transcendence theory.

We mention that the similar criterion for Thakur’s multizeta values (abbreviated MZVs) to beEulerianis given in [CPY14]. One then naturally asks whether one has the criterion for its v-adic counterpart as v-adic MZVs were introduced in [T04]. Such a criterion is related to how to express the givenv-adic MZV as some coordinate logarithm of certain t-module, which is not clear to the authors at this moment how to solve this issue. In the depth one case, the far reaching theory of [AT90] do relate both ζA(n) andζA,v(n)to the logarithm ofC^⊗n, but for higher depth MZVs it is an open question although certain

∞-adic MZVs of higher depths are done in [C15] byt-motivic methods.

The first step of proving the main result above is to write down thet-moduleGand the algebraic pointv explicitly. We mention that in [CPY14], G and v are only theoretically constructed without being written down explicitly. We further use some techniques in [AT90] to compute the coefficient matrices of the logarithm of G explicitly, and then show that the Carlitz multiple star polylogarithms (abbreviated as CMSPLs) defined in (2.2.1) occur as the coordinate logarithms of G. Then we establish an identity of the CMSPLs (stated as Lemma 4.1.3) in terms of linear combination of products of CMPLs and CMSPLs. These properties together with Yu’s theory [Yu91] enable us to derive the desired results.

2. Preliminaries 2.1. Notation. We adopt the following notation.

Fq = the finite field withq elements, forq a power of a prime number p.

θ, t = independent variables.

A = _Fq[θ], the polynomial ring in the variableθ over Fq. v = a monic irreducible polynomial in A.

k = _Fq(θ), the fraction field of A.

kv = the completion ofkwith respect to the place v.

kv = a fixed algebraic closure of kv. k = the algebraic closure of kinkv.

Cv = the completion ofkvwith respect to the canonical extension of v.

| · |_v = a fixed absolute value onCv so that|v|_v =1/q^degv. Ga = the additive group scheme over A.

2.2. CMPLs and CMSPLs. We recall the Carlitz multiple polylogarithms [C14] that are generalization of the polylogarithms initiated in [AT90]. Put L₀ := 1 and L_i := (θ−θ^q)· · ·(θ−θ^qi) for i ∈ N. Given anys := (s1, . . . ,sr) ∈ _N^r, the associated Carlitz multiple polylogarithm (abbreviated as CMPL) and the Carlitz multiple star polyloga- rithm (abbreviated as CMSPL, compared with the terminology in [FKMT15]) are defined by the series

Lis(z₁, . . . ,zr):=

∑

i₁>···>ir≥0

z^q₁ⁱ¹ · · ·z^q_r^ir L^s_i¹

1 · · ·L^s_ir^r and

(2.2.1) Li^?_s(z₁, . . . ,zr) _:=

∑

i₁≥···≥ir≥0

z^q₁ⁱ¹ · · ·z^q_r^ir L^s_i¹

1 · · ·L^s_i^r

r

.

We denote by Lis(z1, . . . ,zr)_v and Li^?_s(z1, . . . ,zr)_v while working on the v-adic conver- gence. Both series converge when|z₁|_v <1 and|z2|_v, . . . ,|zr|_v≤1. Indeed, since v²does not divide θ−_θ^qi_{, ifzi}’s satisfy the above condition, then we have

z^q₁ⁱ¹ · · ·z^qr^ir

L^s_i¹

1 · · ·L^s_i^r

r

v

≤ |v|⁻⁽_v ^{s1i1+···+srir)}|z1|^q_vⁱ¹· · · |zr|^q_v^ir ≤ |v|⁻_v ^wt(s)i1|z1|^q_vⁱ¹ →0 (i1→_∞).

2.3. t-modules. In this section, we review the theory of t-modules introduced by An- derson [A86]. For an A-algebra R, we denote by τ the Frobenius qth power operator τ := (x 7→x^q) : R → R. For convenience, we extend the action of τ on the matrices with entries in R by componentwise action. We denote by Cv[τ] the non-commutative polynomial generated byτ overCv subject to the relation

τα=_α^q_{τ forα} ∈ _Cv.

For each ϕ ∈ _Matd(_Cv[_τ])_{, we write ϕ} = _∑^∞_i=0αiτⁱ _{with each αi} ∈ _Matd(_Cv) _{and αi} = ₀ fori 0, and further define∂ϕ:=_α0.

Let t be a new variable and d be a positive integer. By a d-dimensional t-module, we mean a pair G= (_G^d_a,ρ), where ρ is anFq-linear ring homomorphism

ρ: Fq[t] →Mat_d(_Cv[_τ])

so that∂ρt−_θI_d is a nilpotent matrix. Here we denote byρa the image ofa∈ _Fq[t] byρ.

For a subring A⊆R ⊆_Cv, we say that thet-module is defined over Rif all the coefficient matrices of ρt are in Mat_d(R). In this situation,G^d_a(R) = R^dhas an Fq[t]-module via the mapρ.

Let K be either ¯k or Cv and let G = (_G^d_a,ρ) be a d-dimensional t-module defined over K. Then we have the exponential function of G which is an Fq-linear d-variable power series of the form exp_G = I_d+_∑^∞_i=1αiτⁱ with αi ∈ Mat_d(K), satisfying the following identity:

(2.3.1) exp_G◦∂ρa =ρa◦exp_G for all a ∈_Fq[t].

The logarithm ofGdenoted by log_G, is defined to be the formal inverse of exp_G that has the property:

(2.3.2) log_G◦ρa =∂ρa◦log_G for all a∈ _Fq[t].

The logarithm log_G will be the primary interest for our study as it could provide rich source of transcendental values (see [Yu91, Yu97]).

3. Computation on the logarithms

In this section, our goal is to give an explicit construction of an appropriate t-module G over ¯k associated to an index s ∈ _N^r and an algebraic point u ∈ (k^¯×)^r, and show that the CMSPLs in question occur as coordinate logarithms of Gat an explicit algebraic point of G.

3.1. Construction of the t-module G. In what follows, we fixs = (s₁, . . . ,sr) ∈ _N^r and u = (u₁, . . . ,ur) ∈ (k^×)^r. For 1≤` ≤r, we set d` :=s`+· · ·+sr and d :=d₁+· · ·+dr. Let B be ad×d-matrix of the form







B[11] · · · B[1r] ... ... B[r1] · · · B[rr]







where B[`<sub>m</sub>] <sub>is a d`</sub>×_dm-matrix for each ` <sub>and</sub> m. In this paper, B[`_m] is called the (`_,m)-th block matrix of B.

For 1≤`≤m ≤r, we set

N` :=

















0 1 0 · · · 0 0 1 . .. ...

. .. ... 0 . .. 1 0

















∈ Mat_d`(k),

N :=







 N1

N₂ . ..

Nr









∈ _Matd(k)_,

E[`m] :=











0 · · · ₀ ... . .. ... 0 . .. ...

1 0 · · · 0











∈ Matd<sub>`</sub>×d<sub>m</sub>(k) (if` =m),

E[`m] :=











0 · · · 0

... . .. ...

0 . .. ...

(−₁)<sup>m−`</sup><sub>∏</sub><sup>m</sup><sub>e=`</sub>⁻¹ue 0 · · · ₀











∈ Mat<sub>d`</sub>×dm(k) (if` <m),

E :=











E[11] E[12] · · · E[1r] E[22] . .. ...

. .. E[r−1,r] E[rr]











∈ Mat_d(k).

We also define

Em :=





0 0 0

0 E[mm] 0

0 0 0



∈Mat_d(k)

to be the d×d-matrix such that the (m,m)-th block matrix is E[mm] and the others are zero matrices.

We define thet-module G =Gs,u := (_G^d_a,ρ)by

(3.1.1) ρt =θId+N+Eτ ∈Matd(k[τ]). Note that G depends only onu₁, . . . ,ur−1.

Example3.1.2. When r=1, we have

ρt =C_t^⊗s1 :=











 θ 1

θ 1 . .. ...

θ 1

τ θ













∈ Mat_d(k[_τ])

which is called the s₁th tensor power of the Carlitz module.

When r=2, we have

ρt =



















C^⊗(_t ^s1+s2)

0 · · · 0 ... ...

−u₁τ · · · ₀ 0 C_t^⊗s2



















∈_Matd(k[_τ])_.

When r=3, we have

ρt =

































C_t^{⊗(s1+s2+s3)}

0 · · · 0 ... ...

−u1τ · · · 0

0 · · · 0 ... ... u1u2τ · · · 0 0 C_t^⊗(s2+s3)

0 · · · 0 ... ...

−u₂τ · · · ₀

0 0 C^⊗_t ^s3

































∈ Mat_d(k[τ]).

3.2. Logarithm of G. We denote by

log_G =

∑

i≥0

Piτⁱ the logarithm of thet-module G, where we put

P₀:= I_d, P_i :=





P_i[11] · · · P_i[1r] ... ... P_i[r1] · · · P_i[rr]



∈_Matd(k)_, P_i[`m] ∈<sub>Matd`×dm</sub>(k)_.

Proposition 3.2.1. We have Pi[`m] = 0 for ` > m. For ` ≤ m, if we denote by yi[`m] := Pi[`m]<sub>d`dm</sub>, which is the most lower right corner of Pi[`m], then we have

yi[`m] = ¹

L^d_i^m (if`=m), (3.2.2)

and

yi[`<sub>m</sub>] = (−<sub>1</sub>)<sup>m−`</sup>

∑

0≤i_`≤···≤im−1<i

u^q<sub>`</sub><sup>i`</sup>· · ·u^q_m^im−1₋₁ L^s_i^`

` · · ·L<sup>s</sup><sub>im−1</sub><sup>m−1</sup>L<sup>d</sup><sub>i</sub><sup>m</sup> (<sub>if</sub> ` <_m)_. (3.2.3)

Proof. For any matrix M = (M_ij) with M_ij ∈ _Cv, we set M^(s) := (M_ij^qs) for s ∈ _{Z. From} the functional equation

log_G◦ρt =∂ρt ◦log_G,

by comparing the coefficient matrices we have the following identity P_i+1(_θ^qi+1I_d+N) +P_iE⁽ⁱ⁾ = (θI_d+N)P_i+1 fori ∈ _Z≥0.

Given two square matrices of the same size X,Y, we denote by ad(X)⁰(Y) := Y and ad(X)^j+1(Y) :=X(ad(X)^j(Y))−(ad(X)^j(Y))X. From the identity above, we obtain

P_i+1−^ad(N)¹(P_i+1)

θ^qi+1−_θ =− ^PiE

(i)

θ^qi+1 −_θ and hence for eachi ∈_N≥0,

Pi+1 =−

2d₁−2 j

∑

ad(N)^j(P_iE⁽ⁱ⁾) (_θ^qi+1−_θ)^j+1 (3.2.4)

because ad(N)^2d1−1(Pi+1) =0 from the fact N^d1 =0. Therefore we have (3.2.5) P_i[`m] =0 for` > m

by the induction oni.

Letting

Y =













∗ ∗ ∗

∗ ∗ ∗

y

∗ ∗ ∗













be a d×d-matrix such that the most lower right corner of the(`,m)-th block matrix isy, we note that E<sup>tr</sup><sub>`</sub>YEm is of the form

E^tr_`YEm =













0 0 0

y

0 0 0

0 0 0













(the d×d-matrix such that the most upper left corner of the (`,m)-th block matrix is y and the others are zero matrices).

Since E^tr_` N = 0 and ad(N)^j(P_iE⁽ⁱ⁾) can be expressed as ad(N)^j(P_iE⁽ⁱ⁾) = NB+ (−1)^jP_iE⁽ⁱ⁾N^j for someB ∈Mat_d(k), we have

E^tr_`P_i+1Em = −

2d₁−2 j

∑

E^tr_` ad(N)^j(PiE⁽ⁱ⁾)Em

(θ^qi+1−θ)^j+1 (3.2.6)

=

2d₁−2 j

∑

E^tr_`P_iE⁽ⁱ⁾N^jEm

(θ−θ^qi+1)^j+1

= ^E

tr

`P_iE⁽ⁱ⁾N^dm−1Em

(θ−θ^qi+1)^{dm .}

Note that the last equality above comes from the facts that E⁽ⁱ⁾N^jEm = _{0 if} j 6= dm−_1, and

E⁽ⁱ⁾N^dm−1Em =

d₁+· · ·+d_m−1

z }| {

dm

z}|{

d_m+1+· · ·+dr

z }| {



















E[1m]^{(i) }



























d₁+· · ·+dm

...

0

0 0

if j =dm−1. Note that we also have

E^tr_`P_i =

d₁th

↓

(d₁+d₂)th

↓

(d)th

↓













































0

0

∗ yi[`1] ∗ yi[`2] · · · ∗ yi[`r] ←( d1+· · ·+d`−1+1)th

0

0

By comparing with the most upper left corner of the (`,m)-th block matrix (which is the (d<sub>1</sub>+· · ·+d`−1+1,d₁+· · ·+dm−1+1)-th entry) of the both sides of the equation (3.2.6), we have from (3.2.5) that

L^d_i+^m₁y_i+1[`m] = L<sup>d</sup><sub>i</sub><sup>m</sup>y<sub>i</sub>[`m] (_if `=m) (3.2.7)

and that

L^d_i+^m₁y_i+1[`m] = L_i^dm

m−1 n

∑

y_i[`n](−1)^m−n

m−1 e

∏

u^q_eⁱ+L_i^dmy_i[`m] (if` < m) (3.2.8)

(note thatLi+1 = (θ−θ^qi+1)Li). By definition, we havey0[`m] =1 if`=mandy0[`m] =0 if ` < m. We fix` and show the equalities (3.2.2) and (3.2.3) hold by the induction on m (≥`).

When m = `, we can show this easily by the recurrence relation (3.2.7). Let m > ` and assume that the equality (3.2.3) is true for y_i[`n] <sub>with</sub> ` ≤n <m and i ≥_{0. By the} recurrence relation (3.2.8), we have

L^d_i+^m₁y_i+1[`m] = L^d_i^m

m−1 n=`+

∑

(−1)^n−`

∑

0≤i_`≤···≤in−1<_i

u^q<sub>`</sub><sup>i`</sup>· · ·u^q_nⁱⁿ⁻¹₋₁ L^s_i^`

` · · ·L^s_iⁿ⁻¹

n−1L^d_iⁿ(−1)^m−n

m−1 e

∏

u^qeⁱ

+L^d_i^m 1

L^d_i^{`</sup>(−1)<sup>m−`}

m−1 e

∏

u^q_eⁱ+L^d_i^my_i[`m]

= (−1)^m−`

m−1

n=`+

∑

∑

0≤i_`≤···≤i_n−1<i

u^q<sub>`</sub><sup>i`</sup>· · ·u^q_nⁱⁿ⁻¹₋₁ u^qnⁱ· · ·u^q_mⁱ₋₁ L^s_i^`

` · · ·L^s_iⁿ⁻¹

n−1L^s_iⁿ· · ·L^s_i^m−1 + (−1)^m−`u

qⁱ

` · · ·_u^q_mⁱ₋₁

L^s_i<sup>`</sup>· · ·L<sup>s</sup><sub>i</sub><sup>m−1</sup> +L<sup>d</sup><sub>i</sub><sup>m</sup>yi[`m]

= (−1)^m−`

∑

0≤i_`≤···≤i_m−1 im−1=i

u^q<sub>`</sub><sup>i`</sup>· · ·_u^q_m^im−1₋₁ L^s_i^`

` · · ·L^s_i^m−1

m−1

+L_i^dmyi[`m].

Sincey₀[`m] = 0, we have

L^d_i^myi[`m] = (−1)<sup>m−`</sup>

i−1

i

∑

∑

0≤i_`≤···≤i_m−1 im−1=i⁰

u^q<sub>`</sub><sup>i`</sup> · · ·_u^q_m^im−1₋₁ L^s_i^`

` · · ·L^s_i^m−1

m−1

= (−1)^m−`

∑

0≤i_`≤···≤im−1<i

u^q<sub>`</sub><sup>i`</sup> · · ·u^q_m^im−1₋₁ L^s_i^`

` · · ·L^s_i^m−1

m−1

and hence the equalities (3.2.2) and (3.2.3) are true for eachm≥`by the induction.

3.3. v-adic convergence of CMPLs. Next, we consider when the sum log_Gx converges in LieG(_Cv) = _C^d_v for x ∈ G(_Cv) = _C^d_v. For any matrix M = (M_ij) with M_ij ∈ _Cv, we denote |M|_v := max{|M_ij|_v}. Assume that |um|_v ≤ 1 for each 1 ≤ m < r. Then log_Gx converges for eachx ∈ G(Cv)with|x|_v <1. Indeed, it is clear that|N|_v =1 and|E⁽ⁱ⁾|_v ≤ 1. Sincev²does not divide θ^qi+1−θ for eachi ≥0, we have 1≥ |θ^qi+1 −θ|_v ≥ |v|_v. Thus by the equation (3.2.4), we obtain

|P_i+1|_v≤ |P_i|_v|θ^q

i+1−θ|⁻⁽_v ^2d1−2+1) ≤ |P_i|_v|v|⁻⁽_v ^2d1−1). SinceP0= Id, we have an upper bound

|Pi|_v ≤ |v|⁻_v^i(2d1−1) for eachi ≥0. Therefore if|x|_v <1, we have

|P_ix⁽ⁱ⁾|_v ≤ |P_i|_v|x|^q_vⁱ ≤ |v|⁻_v^i(2d1−1)|x|^q_vⁱ →0 (i→_∞), and hence the sum converges.

Let

(3.3.1) v=v_s,u :=



















































0 

























































 d1

... 0

(−1)^r−1u₁· · ·ur

0 







 d2

... 0

(−1)^r−2u₂· · ·ur

... ...

0 







 dr

... 0 ur

∈ G(k).

It is clear that |v|_v < 1 when |um|_v ≤ 1 for each 1≤ m < r and |ur|_v <1. Thus in this case, the sum log_Gvconverges v-adically.

Remark3.3.2. The authors do not know the precisev-adic convergence domain of log_G.

Theorem3.3.3. Let u1, . . . ,ur ∈ k^× with|um|_v ≤1for each 1≤ m<r and|ur|_v <1. Let G andvbe as above. Then we have

log_Gv=



















































∗ ^

























































 d1

...

∗ (−1)^r−1Li^?_(s

r,...,s₁)(ur, . . . ,u₁)_v

∗ ^







 d₂ ...

∗ (−1)^r−2Li^?_(sr,...,s

2)(ur, . . . ,u2)_v

... ...

∗ ^







 dr

...

∗ Li^?_sr(ur)_v

∈ LieG(_Cv).

Proof. Let 1 ≤ ` ≤ r. By Proposition 3.2.1 and the definition of v, the last coordinate of the`-th block (=the(d₁+· · ·+d`)-th component) of the vector log_Gvis

i

∑

∑

yi[`m](−1)^r−mu^qmⁱ· · ·ur^qi

=

∑

i≥0

1

L^d_i^{`</sup>(−1)<sup>r−`}u^q_`ⁱ· · ·u^qrⁱ

+

∑

(−₁)^m−`

∑

0≤i_`≤···≤im−1<i

u^q<sub>`</sub><sup>i`</sup>· · ·u^q_m^im−1₋₁ L^s_i^`

` · · ·L^s_im−1^m−1L^d_i^m(−₁)^r−m_u^q_mⁱ · · ·_u^q_rⁱ





= (−1)^r−`

∑

i≥0





u^q<sub>`</sub><sup>i</sup>· · ·u<sup>q</sup><sub>r</sub><sup>i</sup> L<sup>s</sup><sub>i</sub><sup>`</sup>· · ·L^s_i^r +

∑

m=`+₁

∑

0≤i_`≤···≤im−1<i

u^q<sub>`</sub><sup>i`</sup>· · ·u^q_m^im−1₋₁ u^q_mⁱ· · ·u^q_rⁱ L^s_i^`

` · · ·L^s_i^m−1

m−1L^s_i^m· · ·L^s_i^r





= (−₁)^r−`

∑

0≤i_`≤···≤ir

u^q<sub>`</sub><sup>i`</sup>· · ·u^qr^ir

L^s_i^`

` · · ·L^s_i^r

r

= (−₁)^r−`_Li^?_(s

r,...,s_{`</sub>)(<sub>ur, . . . ,u`})_.

Remark 3.3.4. The t-module G and algebraic point v defined above are exactly iden- tified with Ext¹_F (1,M⁰) and M respectively in [CPY14, Thm. 4.3.2] through [CPY14, Thms. 5.2.1, 5.2.3].

Remark 3.3.5. If we replace u₁, . . . ,ur by r independent variables t₁, . . . ,tr, then the t- module G is defined over A[t₁, . . . ,tr] and the formula of log_Gvin Theorem 3.3.3is still valid as power series in the variables t₁, . . . ,tr (when ignoring convergence). Multiple several variable zeta functions are studied in [ADR16], but the authors do not know whether the formula mentioned above has connection with those in [ADR16] or has application tov-adic MZVs.

4. v-adic vanishing principle forCMPLs

In what follows, we fix a monic primev of Aand an r-tuples = (s1, . . . ,sr)∈ _N^r. We further fix u= (u₁, . . . ,ur) ∈ (k)^r so that|u₁|_v, . . . ,|ur−1|_v ≤1 and|ur|_v <1. Therefore

Li^?_(s

r,...,s<sub>`</sub>)(ur, . . . ,u`)_v converges v-adically for`=1, . . . ,r.

4.1. The vanishing principle. We continue the notation and hypothesis above. We let G = (_G^d_a,ρ)be thet-module defined in (3.1.1). Definev =v_s,u ∈ G(k)to be the algebraic point given in (3.3.1). Although we have the functional identity for the logarithm of a t-module 2.3.2, we can not evaluate at arbitrary points as the logarithm in question is not an entire function. The following lemma provides the situation fitting into our need for later study on log_G.

Lemma 4.1.1. Let H = (_Gⁿ_a,σ) be a t-module defined over the ring of integers of Cv and x ∈ H(_Cv) be a point such that |x|_v < 1. Then for each a ∈ _Fq[t], we have log_H(_σa(x)) =

∂σa(log_H(x)).

Proof. Let X be a new variable and consider a map log_H,X: (_XCv[[X]])ⁿ →(_XCv[[X]])ⁿ by log_H,Xg :=

∑

Qig⁽ⁱ⁾,

where Q_i ∈ Matn(k) are the coefficient matrices of log_H and g⁽ⁱ⁾ is obtained by raising each component of g to the qⁱth power. Then we have log_H,X(σa(g)) = ∂σa(log_H,X(g)) for each a ∈ _Fq[t] and g ∈ (_XCv[[X]])ⁿ. Let Cv{X} be the ring consisting of all power series which converge on |X|_v ≤1, and we callCv{X} the Tate algebra over Cv.

Letm_v be the maximal ideal of the ring of integers ofCv. For eachx∈ mⁿ_v andi ≥1, it is clear that log_H,X(Xⁱx) ∈ _Cv{X}ⁿ and its values at X =_{1 is logH}x. Since log_H,X, log_H and the evaluation map X 7→1 are additive, we can extend these operations from{Xⁱx} to(Xm_v[X])ⁿ, and hence we have the commutative diagram

(Xmv[X])^{n //}

log_H,X

mⁿ_v

log_H

Cv{X}^{n //} _Cⁿ_v =LieH

where the horizontal maps are the evaluating map g(X) 7→g(1).

Let x ∈ H(_Cv) be a point such that |x|_v < 1. Note that log_H(σa(x)) converges since

|_σa(x)|_v ≤ |x|_v <1. From the fact thatσa(Xx)∈ (Xmv[X])ⁿ, we have log_H(σa(x)) =log_H(σa(Xx)|_X=1) =log_H,X(σa(Xx))|_X=1

=_∂σa(log_H,X(Xx))|_X=1 =_∂σa(log_H(x)).

Theorem 4.1.2. Let notation and hypotheses be given as above and put v(t) := v|_θ=t ∈ _Fq[t]. Then the following are equivalent.

(i) The following v-adic values are simultaneously vanishing:

Li^?_(sr,...,s

1)(ur, . . . ,u₁)_v=_Li^?_(s

r,...,s2)(ur, . . . ,u₂)_v=_{. . .} =_Li^?_sr(ur)_v =_0.

(ii) vis a v(t)-power torsion point in G(k). (iii) vis anFq[t]-torsion point in G(k).

Furthermore, we assume that|u_i|_v <1for all i. ThusLi<sub>(s`,...,sr)</sub>(u`, . . . ,ur)_vconverges v-adically for`=_{1, . . . ,}r. In this situation, the conditions above are also equivalent to the following one.

(iv) The following v-adic values are simultaneously vanishing:

Li_(s1,...,sr)(u₁, . . . ,ur)_v=Li_(s2,...,sr)(u₂, . . . ,ur)_v=. . . =Lisr(ur)_v =0.

Proof. (i)⇒(ii) To simplify the notation, we denote by [−] the Fq[t]-actions on any t- modules. For 1 ≤i ≤ r, let G_i be the t-module whose t-action is defined by the square matrix of size d1+· · ·+di cut from the upper left square of the matrix (3.1.1). We also set G0 :=0. Thus we have the exact sequence

0 ^// G_i−1 //G_i ^{πi //}C^{⊗di //}0 for each 1≤i ≤r. Note that Gr =G.