FINITE AND p-ADIC POLYLOGARITHMS

AMNON BESSER

Abstract. The finite n-th polylogarithm li_{n}(z) ∈ ℤ/p[z] is defined as ∑_{k=1}^{p−1} z^{k}/k^{n}. We state and prove the following theorem. Let Li_{k} : ℂ_{p} → ℂ_{p} be the p-adic polylogarithms defined by Coleman. Then a certain linear combination F_{n} of products of polylogarithms and logarithms, with coefficients which are independent of p, has the property that p^{1−n}DF_{n}(z) reduces modulo p > n + 1 to li_{n−1}(σ(z)) where D is the Cathelineau operator z(1 − z)d/dz and σ is the inverse of the p-power map. A slightly modified version of this theorem was conjectured by Kontsevich. This theorem is used by Elbaz-Vincent and Gangl to deduce functional equations of finite polylogarithms from those of complex polylogarithms.

1. Introduction

The finite logarithm was introduced by Kontsevich (under the name “The 1\frac{1}{2} logarithm”) in [5]. The finite logarithm is the case n = 1 of the n-th polylogarithm li_{n} ∈ ℤ/p[z] defined by li_{n}(z) = ∑_{k=1}^{p−1} z^{k}/k^{n}. In loc. cit. Kontsevich proved that the finite logarithm satisfies a 4-term functional equation, known as the fundamental equation of information theory. The same functional equation is satisfied by the so called infinitesimal dilogarithm −(x log |x| + (1 − x) log |1 − x|). Cathelineau [2] defined general infinitesimal polylogarithms and found that they satisfy interesting functional equations. It was the idea of Elbaz-Vincent and Gangl [4] that these functional equations should be satisfied by finite polylogarithms (The name “finite polylogarithm” is due to them). Inspired by their work, Kontsevich raised the idea that the finite polylogarithm could be a reduction of an infinitesimal version of the p-adic polylogarithm, as defined by Coleman [3]. If such a connection is established, it makes sense to hope that functional equations of the infinitesimal p-adic polylogarithm can be established in a similar way to its complex counterpart and that these then imply by reduction the functional equations of the finite polylogs. A conjectural formula for the precise p-adic polylog whose “derivative”, in the sense to be explained below, reduces to the finite polylog was formulated by Kontsevich and proved by him for small n. The purpose of this short note is to prove such a connection between p-adic polylogarithms and finite polylogarithms.

To state the main result we recall that Coleman defined p-adic polylogarithms, Li_{n} : ℂ_{p} → ℂ_{p}. These functions are locally analytic in the sense that they are given by a convergent power series on each residue disc in ℂ_{p} and satisfy the relations

Li_{0}(z) = \frac{z}{1 − z}

and

d Li_{n}(z) = Li_{n−1}(z)\frac{dz}{z} .

We define the differential operator D by D = z(1 − z)d/dz. Let 𝔽̄_{p} be the algebraic closure of the finite field with p elements and let W = W (𝔽̄_{p}) be the ring of Witt vectors of 𝔽̄_{p}, so W is the ring of integers of the maximal unramified extension of ℚ_{p}. Let σ : 𝔽̄_{p} → 𝔽̄_{p} be the automorphism which is the inverse of the p-power map. Let X = {z ∈ W : |z| = |z − 1| = 1}. Our main result is then: Theorem 1.1. For every n > 1 let

F_{n}(z) = ∑_{k=0}^{n−1} a_{k} log^{k}(z) Li_{n−k}(z) ,

with a_{0} = −n and

a_{k} = \frac{(−1)^{k}}{(k − 1)!} + \frac{(−1)^{k+1}n}{k!} ,

for k > 0. Then the following holds for every p > n+1: One has DF_{n}(X) ⊂ p^{n−1}W and for every z ∈ X one has p^{1−n}DF_{n}(z) ≡ li_{n−1}(σ(z)) (mod p). Furthermore, the choice of the coefficients a_{k} is the unique choice of coefficients in ℚ for which the theorem holds for all p > n + 1.

We remark that for a given p there will be many other choices of coefficients, for example those which are sufficiently congruent to the a_{k}.

The author would like to thank Kontsevich for explaining the conjecture to him and to Gangl for various comments and corrections. He would also like to thank Nekovář who first suggested to him the idea of a connection between finite and p-adic polylogarithms.

2. The proof

The connection between p-adic and finite polylogarithms is made by the following Proposition 2.1. Let Li_{n}^{(p)}(z) := Li_{n}(z) − Li_{n}(z^{p})/p^{n}. Then Li_{n}^{(p)}(X) ⊂ W and the function Li_{n}^{(p)} reduces modulo p to (1 − z^{p})^{−1} li_{n}(z).

Proof. According to [3], the function Li_{n}^{(p)}(z) can be computed as

Li_{n}^{(p)}(z) = ∫_{ℤ_{p}^{×}} x^{−n}dμ_{z}(x) ,

where μ_{z} is the measure on ℤ_{p} defined by

μ_{z}(a + p^{m}ℤ_{p}) = \frac{z^{a}}{1 − z^{p^{m}}} a = 0, 1, …, p^{m} − 1 .

Since for z ∈ X μ_{z} takes integral values, this shows the first statement. Reducing modulo p we may replace the function x ↦ x^{−k} by the function x ↦ a^{−k} if x ≡ a (mod p), which is congruent to it modulo p on ℤ_{p}^{×}. This implies that Li_{n}^{(p)}(z) is congruent modulo p to

∑_{a=1}^{p−1} a^{−n}μ_{z}(a + pℤ_{p}) = ∑_{a=1}^{p−1} a^{−n} \frac{z^{a}}{1 − z^{p}} .

□

Corollary 2.2. Let α ∈ X be a root of unity. Then we have Li_{n}(α) ∈ p^{n}W and p^{−n} Li_{n}(α) ≡ − li_{n}(σ(α))/(1 − α) (mod p).

Proof. Since α ∈ X ⊂ W the order of α is prime to p and we have α^{p^{k}} = α for some k. By using the definition of Li_{n}^{(p)} repeatedly we find

Li_{n}(α) = Li_{n}^{(p)}(α) + p^{−n} Li_{n}(α^{p})
= Li_{n}^{(p)}(α) + p^{−n} Li_{n}^{(p)}(α^{p}) + p^{−2n} Li_{n}(α^{p^{2}})
⋯ = ∑_{i=0}^{k−1} p^{−in} Li_{n}^{(p)}(α^{p^{i}}) + p^{−kn} Li_{n}(α^{p^{k}}) .

Since α^{p^{k}} = α we may move the last term to the left hand side of the equation and obtain

Li_{n}(α) = \frac{1}{1 − p^{−kn}} ∑_{i=0}^{k−1} p^{−in} Li_{n}^{(p)}(α^{p^{i}})

= \frac{p^{n}}{p^{kn} − 1} ∑_{i=0}^{k−1} p^{(k−1−i)n} Li_{n}^{(p)}(α^{p^{i}}) ∈ p^{n}W ,

and dividing by p^{n} and reducing modulo p we obtain using the proposition

− Li_{n}^{(p)}(α^{p^{k−1}}) ≡ (1 − (α^{p^{k−1}})^{p})^{−1} li_{n}(α^{p^{k−1}}) ≡ − li_{n}(σ(α))/(1 − α) (mod p) .

□

Proposition 2.3. Let α be a root of unity in X. Set L̃i_{n}(α) = p^{−n} Li_{n}(α). Then for w ∈ W we have

p^{−n} Li_{n}(α(1 + pw)) ≡ ∑_{k=0}^{n} L̃i_{n−k}(α)\frac{w^{k}}{k!} (mod p) .

Proof. Let g_{n}(w) = p^{−n} Li_{n}(α(1 + pw)). Then one finds

g_{0}(w) = \frac{α(1 + pw)}{1 − α(1 + pw)} = \frac{α}{1 − α} \frac{1 + pw}{1 − \frac{α}{1−α}pw} = L̃i_{0}(α) + ∑_{k=1}^{∞} b_{k}(pw)^{k}

with b_{n} ∈ W. Write

g_{n}(w) = L̃i_{n}(α) + ∑_{k=1}^{∞} d_{k}^{n}w^{k} .

It is easy to verify that

\frac{d}{dw} g_{n}(w) = g_{n−1}(w)\frac{1}{1 + pw} = g_{n−1}(w)(1 + ∑_{k=1}^{∞} c_{k}(pw)^{k}) ,

with c_{k} ∈ W. To find the coefficients d_{k}^{n} modulo p for k < p one can simply reduce the above equations modulo p and one easily finds that

d_{k}^{n} ≡ \begin{cases}\frac{L̃i_{n−k}(α)}{k!} & when k ≤ n \\ 0 & when n < k < p . \end{cases}

It thus remains to show that also for k ≥ p the coefficient d_{k}^{n} is divisible by p. For this it is easier to consider the function f_{n}(u) = Li_{n}(α + u) which satisfies f_{0}(u) ∈ W [[u]], d/duf_{n+1}(u) = g(u)f_{n}(u) with g(u) ∈ W [[u]] and f_{n}(0) ∈ W.

Lemma 2.4. In the situation above we have v_{p}(a_{k}) ≥ −v_{p}(k!), where v_{p} is the p-adic valuation and a_{k} is the k-th coefficient in the power series expansion with respect to u of any of the functions f_{n}.

Proof. Let a_{k}(h) be the k-th coefficient of h for any power series h. We have

v_{p}(a_{k}(f_{n}g)) ≥ min_{l≤k}[v_{p}(a_{l}(f_{n}))] .

This implies that

v_{p}(a_{k}(f_{n+1})) ≥ min_{l<k}[v_{p}(a_{l}(f_{n}))] − v_{p}(k) .

The lemma is clearly true for n = 0. Suppose it is true for n. Then

v_{p}(a_{k}(f_{n+1})) ≥ min_{l<k}[−v_{p}(l!)] − v_{p}(k) ≥ −v_{p}((k − 1)!) − v_{p}(k) = −v_{p}(k!) .

□

Since g_{n}(w) = p^{−n}f_{n}(αpw), to finish the proof we have to check that for every k ≥ p we have k − n − v_{p}(k!) > 0. The well known estimate v_{p}(k!) ≤ k/(p − 1) and the assumption n ≤ p−2 imply that it is sufficient to require k(1−1/(p−1)) > p−2, and this is satisfied for k ≥ p. □

Proof of Theorem 1.1. Using the fact that D is a derivation and that

D log^{k}(z) = z(1 − z)\frac{d}{dz} log^{k}(z) = z(1 − z)k log^{k−1}(z)\frac{1}{z} = (1 − z)k log^{k−1}(z)

and

D Li_{k}(z) = z(1 − z)\frac{d}{dz} Li_{k}(z) = z(1 − z) Li_{k−1}(z)\frac{1}{z} = (1 − z) Li_{k−1}(z)

we see that

DF_{n}(z) = D ∑_{k=0}^{n−1} a_{k} log^{k}(z) Li_{n−k}(z)

= (1 − z) ∑_{k=0}^{n−1} a_{k}(k log^{k−1}(z) Li_{n−k}(z) + log^{k}(z) Li_{n−k−1}(z))

= (1 − z) ∑_{k=0}^{n−1} log^{k}(z) Li_{n−k−1}(z)(a_{k} + (k + 1)a_{k+1}) .

Here we understand that a_{n} = 0. Every z ∈ X can be written as α(1 + pw) with α a root of unity in X and w ∈ W. We have

p^{−1} log(α(1 + pw)) = p^{−1} log(1 + pw) ≡ w (mod p) .

If we assume that a_{k} ∈ ℤ[1/(n + 1)!] we now find from the last computation and from proposition 2.3,

p^{1−n}DF_{n}(α(1 + pw))

≡ (1 − α) ∑_{k=0}^{n−1} w^{k}(a_{k} + (k + 1)a_{k+1}) ∑_{m=0}^{n−k−1} \frac{1}{m!}L̃i_{n−k−1−m}(α)w^{m}

≡ (1 − α) ∑_{l=0}^{n−1} w^{l} ∑_{k=0}^{l}(a_{k} + (k + 1)a_{k+1})\frac{1}{(l − k)!}L̃i_{n−l−1}(α) .

It follows that to make the reduction of p^{1−n}DF_{n} independent of w for all p > n+1 it is necessary and sufficient that for l = 1, …, n − 1 we have

(2.1)  ∑_{k=0}^{l}(a_{k} + (k + 1)a_{k+1})\frac{1}{(l − k)!} = 0 .

If this is satisfied then the reduction of p^{1−n}DF_{n}(α(1 + pw)) is (a_{0} + a_{1})(1 − α)L̃i_{n−1}(α) ≡ −(a_{0} + a_{1}) li_{n−1}(σ(α)) so we should also require a_{0} + a_{1} = −1.

Let A(t) = ∑_{k=0}^{n−1} a_{k}t^{k}. Then the relations (2.1) can be written as e^{t}(A(t) + dA(t)/dt) ≡ a (mod t^{n}) where a is a constant. This implies that A(t)+dA(t)/dt ≡ ae^{−t} (mod t^{n}) and after solving the resulting differential equation that the n − 2 first equations in (2.1) are equivalent to A(t) ≡ (at + b)e^{−t} (mod t^{n−1}) for some other constant b. We have −1 = a_{0} + a_{1} = b + (a − b) = a. Now, in (b − t)e^{−t} the coefficient of t^{n} is

b\frac{(−1)^{n}}{n!} − \frac{(−1)^{n−1}}{(n − 1)!} = \frac{(−1)^{n−1}}{(n − 1)!} (1 − b/n) .

It is easy to see that for the equation (2.1) with l = n − 1 to be satisfied, this coefficient must be 0 and hence b = n. This gives the choice of the coefficients in the theorem and shows that they are the unique choice in ℤ[1/(n + 1)!]. Now if we have coefficients in ℚ satisfying the theorem, then we may clear denominators not dividing (n + 1)! and using only independence of w we obtain that these must be a rational multiple of our a_{k}. But since the reduction of p^{1−n}DF_{n} is non-trivial the multiplier must be 1. □

Remark 2.5. We have the equation F_{n}(z) = −nL_{n}(z) − L_{n−1}(z) log(z), where

L_{n}(z) = ∑_{m=0}^{n−1} \frac{(−1)^{m}}{m!} Li_{n−m}(z) log^{m}(z)

is the function defined in [1]. By loc. cit. the function F_{n} satisfies F_{n}(z)+(−1)^{n}F_{n}(1/z) = 0. Differentiating this relation one gets the relation zDF_{n}(1/z)+(−1)^{n}DF_{n}(z) = 0. Reducing modulo p we find z li_{n−1}(1/z)+(−1)^{n} li_{n−1}(z) = 0. This relation is easily verified directly.

3. Another proof of the main result

In this section we sketch another proof of the main result. This proof has two interesting features: First of all, it proves directly the formula for F_{n} in terms of the functions L_{n}(z) defined at the end of the last section. This formula is of course simpler than the original formula. The other feature is that the key ingredient in the proof is a formula, discovered by Rob de Jeu and the author, which seems to be of some further importance. This formula shows up in the computation of syntomic regulators. These two features suggest that the proof to be described below may in some way be more “correct” than the first one, although it is if anything slightly more complicated.

The formula alluded to above is the content of the following Proposition 3.1. We define a sequence of functions f_{k}(z, S) inductively as follows:

f_{0}(z, S) = \frac{S}{1 − S} , f_{k+1}(z, S) = ∫_{z}^{S} f_{k}(z, t)d log t .

Then, when z, S ∈ W and z ≡ S (mod p) the following formula holds:

(3.1) − ∑_{k=0}^{n}(−1)^{k}k! · \binom{n}{k}f_{k+1}(z, S) log^{n−k}(S) = (−1)^{n}n! (L_{n+1}(S) − L_{n+1}(z)) .

We would like to remark on the potential importance of this formula. The construction of p-adic polylogarithms by Coleman is an inductive procedure. At each step the degree n polylogarithm Li_{n} is constructed as a locally analytic function satisfying the differential equation d Li_{n}(z) = Li_{n−1}(z)d log z. This determines Li_{n} up to a locally constant function and a Frobenius condition replaces this by a globally constant function ambiguity. As remarked by Kontsevich the distribution relation removes the ambiguity completely. The formula above allows for a different approach: The functions f_{n} have no ambiguity in their definitions and are in fact given by converging power series in S and z − S. Once the f_{n} are given, the formula determines L_{n} up to a locally constant function and the distribution relation determines it completely. The functions Li_{n} can be determined from the L_{n}.

For our purposes, the formula is also useful because it allows us to relate the values of L_{n} at two congruent points. Lemma 3.2. Suppose z ∈ X and w ∈ W. Then

p^{−n}f_{n}(z, z(1 + pw)) ≡ \frac{z}{1 − z} \frac{w^{n}}{n!} (mod p) .

The proof is a direct computation similar to the proof of proposition 2.3. Suppose now that z is a root of unity. Then log(z(1 + pw)) ≡ pw (mod p^{2}). Thus, using (3.1) we immediately obtain

p^{−n−1}(−1)^{n}n!(L_{n+1}(z(1 + pw)) − L_{n+1}(z)) ≡ c_{n+1} \frac{z}{1 − z} w^{n+1} (mod p)

with

c_{n+1} = ∑_{k=0}^{n}(−1)^{k} \frac{k!}{(k + 1)!}\binom{n}{k} = \frac{1}{n + 1} ∑_{k=0}^{n}(−1)^{k}\binom{n + 1}{k + 1} = \frac{1}{n + 1} .

so

p^{−n}(L_{n}(z(1 + pw)) − L_{n}(z)) ≡ −(−1)^{n} \frac{1}{n!} \frac{z}{1 − z} w^{n} (mod p) .

We can define the derivation D on functions of two variables as D = S(1−S)∂/∂S+ z(1 − z)∂/∂z. One easily obtains the following Lemma 3.3. If z, S ∈ X and z ≡ S (mod p) then p^{k}|Df_{k}(z, S).

Differentiating the key formula we find, with S = z(1 + pw),

p^{−n}(−1)^{n}n!(DL_{n+1}(S) − DL_{n+1}(z))

= p^{−n} ∑_{k=0}^{n}(−1)^{k}k!\binom{n}{k} [Df_{k+1}(z, S) log^{n−k}(S) + (n − k)f_{k+1}(z, S) log^{n−k−1}(S)] (1−S)

≡ ∑_{k=0}^{n}(−1)^{k}k!\binom{n}{k}(n − k)f_{k+1}(z, S) log^{n−k−1}(S)(1 − z) ≡ d_{n+1}zw^{n} (mod p) ,

where

d_{n+1} = ∑_{k=0}^{n}(−1)^{k} \frac{k!}{(k + 1)!}\binom{n}{k}(n − k) = − ∑_{l=1}^{n}(−1)^{l}\binom{n}{l} = 1 .

Let z be a root of unity. Since log(z) = 0 we have

DL_{n}(z) = z(1 − z)[ Li_{n−1}(z)\frac{1}{z}

+ ∑_{m=1}^{n−1} \frac{(−1)^{m}}{m!} (Li_{n−m−1}(z)\frac{1}{z} log^{m}(z) + m Li_{n−m}(z) log^{m−1}(z)\frac{1}{z}) ]

= (1 − z)(Li_{n−1}(z) − Li_{n−1}(z)) = 0 .

Thus we find

p^{−n}DL_{n}(S) ≡ −(−1)^{n} \frac{n}{n!}zw^{n−1} (mod p) .

Suppose now that we let F (z) = ∑_{m=1}^{n} e_{m}L_{m}(z) log^{n−m}(z). Substituting first of all a root of unity in X we find DF (z) = e_{n−1}(1 − z)L_{n−1}(z) so we should have e_{n−1} = −1. Then

p^{1−n}DF (S) = ∑_{m=1}^{n} e_{m}(DL_{m}(S) log^{n−m}(S)

+ (n − m)L_{m}(S) log^{n−m−1}(S)(1 − S))p^{1−n}

≡ ∑_{m=1}^{n} e_{m}( − (−1)^{m} \frac{m}{m!}zw^{m−1} · w^{n−m}

+ (n − m)(p^{−m}L_{m}(z) − (−1)^{m} \frac{1}{m!} \frac{z}{1 − z} w^{m})w^{n−m−1}(1 − z))

= −zw^{n−1} ∑_{m=1}^{n}((−1)^{m}e_{m}(\frac{m}{m!} + (n − m)\frac{1}{m!}))

+ ∑_{m=1}^{n} e_{m}(n − m)p^{−m}L_{m}(z)w^{n−m−1}(1 − z) (mod p) .

We wish to choose the coefficients e_{m} in such a way that this expression is independent of w. Since in the second sum we know by corollary 2.2 that p^{−m}L_{m}(z) is not congruent to 0 modulo p (as a function of z) we see that the only non zero coefficients can be e_{n} and e_{n−1}. By fixing e_{n−1} = −1 we get the equation

(−1)^{n}e_{n} \frac{n}{n!} + (−1)^{n} (\frac{n − 1}{(n − 1)!} + \frac{1}{(n − 1)!}) = 0

from which we can recover e_{n} = −n.

References

[1] A. Besser and R. de Jeu. The syntomic regulator for K–theory of fields. Work in progress, 2001.

[2] J.L. Cathelineau. Remarques sur les différentielles des polylogarithmes uniformes. Ann. Inst. Fourier (Grenoble), 46(5):1327–1347, 1996.

[3] R. Coleman. Dilogarithms, regulators, and p-adic L-functions. Invent. math., 69:171–208, 1982.

[4] P. Elbaz-Vincent and H. Gangl. On poly(ana)logs I. Max Planck Institut für Mathematik preprint MPI-2000-62, To appear in Comp. Math., 2000.

[5] M. Kontsevich. The 1 \frac{1}{2} -logarithm. Unpublished note, 1995. Reproduced as an appendix to [4].

8 AMNON BESSER

Department of Mathematics, Ben-Gurion University of the Negev, P.O.B. 653, Be’er-Sheva 84105, Israel