BOUNDS FOR THE DIMENSIONS OF p-ADIC MULTIPLE L-VALUE SPACES

東京大学大学院数理科学研究科 山下 剛 (Go YAMASHITA) Graduate School of Mathematical Sciences, University of Tokyo

This text is a report of a talk “bounds for the dimensions of p-adic multiple Lvalue spaces” in the symposium “Algebraic Number Theory and Related Topics” (6-10/Dec/2004 at RIMS).

For natural numbers k_{1}, …, k_{d−1} ≥ 1, k_{d} ≥ 2, the following infinite sum

ζ(k_{1}, …, k_{d}) := ∑_{n_{1}<...<n_{d}} \frac{1}{n_{1}^{k_{1}} ⋯ n_{d}^{k_{d}}} (= lim_{ℂ∋z→1}Li_{k_{1},...,k_{d}}(z)) ∈ ℝ

absolutely converges, and is called the multiple zeta value (MZV). Here, Li_{k_{1},...,k_{d}}(z) := ∑_{n_{1}<...<n_{d}} \frac{z^{n_{d}}}{n_{1}^{k_{1}}···n_{d}^{k_{d}}} is the multiple polylogarithm function. The study of MZV’s is

started from Euler. After Zagier made the study of MZV’s revive in the modern times, MZV’s are studied actively by many mathematicians now.

For natural number k_{1}, …, k_{d} ≥ 1 and N-th roots of unity ζ_{1}, …, ζ_{d} satisfying (k_{d}, ζ_{d}) ≠ (1, 1), the multiple L-value (MLV) is defined by the following absolutely converging infinite sum:

L(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d}) := ∑_{n_{1}<...<n_{d}} \frac{ζ_{1}^{−n_{1}}ζ_{2}^{n_{1}−n_{2}} … ζ_{d}^{n_{d−1}−n_{d}}}{n_{1}^{k_{1}} ⋯ n_{d}^{k_{d}}} (= lim_{ℂ∋z→1}Li_{k_{1},...,k_{d};ζ_{1},...,ζ_{d}}(z)) ∈ ℂ.

Here, Li_{k_{1},...,k_{d};ζ_{1},...,ζ_{d}}(z) := ∑_{n_{1}<...<n_{d}} \frac{ζ_{1}^{−n_{1}}ζ_{2}^{n_{1}−n_{2}}...ζ_{d}^{n_{d−1}−n_{d}}z^{n_{d}}}{n_{1}^{k_{1}}···n_{d}^{k_{d}}} is the twisted multiple

polylogarithm function.

Now, we want to consider a p-adic analogue of MZV’s and MLV’s. The infinite sum

∑_{n_{1}<...<n_{d}} \frac{1}{n_{1}^{k_{1}} ⋯ n_{d}^{k_{d}}}

does not converges in the p-adic topology. On the other hand, the multiple polylogarithm function Li_{k_{1},...,k_{d}}(z) has the iterated integral representation:

\frac{dLi_{k_{1},...,k_{d}}(z)}{dz} = \begin{cases}\frac{1}{z}Li_{k_{1},...,k_{d}−1}(z) & if k_{d} > 1, \\ \frac{1}{1−z}Li_{k_{1},...,k_{d−1}}(z) & if k_{d} = 1, and d > 1, \\ \frac{1}{1−z} & if k_{d} = 1, and d = 1. \end{cases}

Considering a p-adic analogue of this iterated integral representation, Furusho defined the p-adic multiple polylogarithm functions Li_{k_{1},...,k_{d}}^{a}(z) by using Coleman’s p-adic iterated integral theory ([C]) (Here, a is a branching parameter. We do not explain it in this article), defined the p-adic multiple zeta values (p-adic MZV’s) to be the limit values of the p-adic multiple polylogarithm functions (cf. [Fu1]):

Date: March/2005.

ζ_{p}(k_{1}, …, k_{d}) := lim_{ℂ_{p}∋z→1}′Li_{k_{1},...,k_{d}}(z) ∈ ℚ_{p},

and studied their properties and relations (cf. [Fu1][Fu2]). Here, ℂ_{p} is the p-adic completion of the algebraic closure of ℚ_{p}． We do not explain the meaning of lim′ in this article.

Example 1.1. (Coleman) For n > 1, we have

ζ_{p}(n) = \frac{p^{n}}{p^{n} − 1}L_{p}(n, ω^{1−n}).

Here, L_{p} is the p-adic L-function of Kubota-Leopoldt, and ω is the Teichmuüller character. In particular, ζ_{p}(2n) = 0 for n ≥ 1. We get the p-adic L-function of Kubota-Leopoldt by p-adically interpolating values at negative integer. Note that this proof of ζ_{p}(2n) = 0 is somewhat indirect, since the above formula is a comparison between the p-adic polylogarithms and the p-adic L-function at positive integer. (Furusho also shows ζ_{p}(2n) = 0 from 2-,3-cycle relations. This comes from the fact that “the angles of the triangle in the 3-cycle relation are 0” in the p-adic world. We also say “π^{2} is 0 in the p-adic world” from the fact ζ_{p}(2n) = 0.)

On the other hand, the values ζ_{p}(2n+1) are difficult. For n ≥ 1, we have the following equivalences: ζ_{p}(2n+1) ≠ 0 ⇔ L_{p}(2n+1, ω^{−2n}) ≠ 0 ⇔ H^{2}(ℤ[1/p], ℚ_{p}/ℤ_{p}(−n)) = 0 (higher Leopoldt conjecture). This holds in the case where p is a regular prime or the case where p − 1 devides n. However, it is not known whether this holds or not in general. □

Analogously, we can define twisted p-adic multiple polylogarithms Li_{k_{1},...,k_{d};ζ_{1},...,ζ_{d}}^{a}(z) and p-adic multiple L-values

L_{p}(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d}) := lim_{ℂ_{p}∋z→1}′Li_{k_{1},...,k_{d};ζ_{1},...,ζ_{d}}^{a}(z) ∈ ℚ_{p}(μ_{N})

for p ∤ N (cf. [Y]). For w > 0, we define Z_{w}^{p}[N] ⊂ ℚ_{p} to be the following:

Z_{w}^{p}[N] := ⟨L_{p}(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d})| \begin{matrix}d ≥ 1, k_{i} ≥ 1, ζ_{i} ∈ μ_{N} for i = 1, …, d, \\ k_{1} + ⋯ + k_{d} = w, (k_{d}, ζ_{d}) ≠ (1, 1) \end{matrix} ⟩_{ℚ} ,

(the ℚ-vector space generated by L_{p}(k_{1}, …, k_{d}; ζ_{1}, …, ζ_{d})’s), and Z_{0}^{p}[N] := ℚ. Put Z_{•}^{p}[N] := ⊕_{w}Z_{w}^{p}[N] (formal direct sum), Z_{w}^{p} := Z_{w}^{p}[1], and Z_{•}^{p} := Z_{•}^{p}[1]. We call Z_{w}^{p} (resp. Z_{w}^{p}[N]) the space of p-adic multiple zeta values of weight w (resp. the space of p-adic multiple L-values of weight w).

It is known that there are many relations between p-adic MZV’s and p-adic MLV’s as the usual MZV’s and MLV’s (cf. [Fu1][Fu2][Y]). For the relations of p-adic MZV’s and p-adic MLV’s, we have some conjectures, which are analogous to the complex case.

Conjecture 1.2. (Furusho) All linear relations between p-adic MZV’s are derived from 2-,3-,5-cycle relations.

We do not explain 2-,3-,5-cycle relations in this article.

Conjecture 1.3. (isobar conjecture, Furusho) All linear relations between p-adic MZV’s are linear combinations of relations between p-adic MZV’s of the same weights. In particular, the formal direct sum Z_{•}^{p} := ⊕_{w}Z_{w}^{p} has the natural embedding into ℚ_{p}.

For the p-adic MLV’s, we also conjecture that all linear relations between p-adic MZV’s are linear combinations of relations between p-adic MZV’s of the same weights ([Y]).

The main result of the talk at RIMS (6/Dec/2004) concerns the dimensions of the space of p-adic MLV’s. First, we review the complex case. Zagier conjectures the dimensions of the space of MZV’s as follows:

Conjecture 1.4. (dimension conjecture, Zagier) We define a sequence {D_{n}}_{n} to be D_{0} = 1, D_{1} = 0, D_{2} = 1, D_{n+3} = D_{n+1} + D_{n} (n ≥ 0). (The generating function is ∑_{n=0}^{∞} D_{n}t^{n} = 1/(1 − t^{2} − t^{3}).) Then, we have dim_{ℚ} Z_{w} = D_{w} for w ≥ 0. Here, using MZV’s and MLV’s, we define Z_{w}, Z_{w}[N] by the same way of Z_{w}^{p}, Z_{w}^{p}[N].

Theorem 1.5. (Goncharov, Terasoma, Deligne-Goncharov [G1][T][DG]) For w ≥ 0, we have dim_{ℚ} Z_{w} ≤ D_{w}.

This theorem says that there are enormous relations between MZV’s. The opposite inequality seems to be a trancedental number theoric problem, and that we cannot prove it by the present algebraic geometical methods. For MLV’s, we have the following.

Theorem 1.6. (Deligne-Goncharov[DG]) For N = 2 (resp. N > 2), we define a sequence {D_{n}[N]}_{n} by a generating function ∑_{n=0}^{∞} D_{n}[2]t^{n} = 1/(1 − t − t^{2}) (resp. ∑_{n=0}^{∞} D_{n}[N]t^{n} = 1/(1 − (\frac{φ(N)}{2} + ν)t + (ν − 1)t^{2})). Here, φ is the Euler function, and ν is the number of prime numbers dividing N. Then, we have dim_{ℚ} Z_{w}[N] ≤ D_{w}[N] for w ≥ 0 and N ≥ 1.

Remark . For N > 4, it is known that the equality does not hold in general (Goncharov[G2]). The gap is related to the space of cusp forms for Γ_{1}(N) of weight 2 when N is a prime number (loc. cit.).

Now, we return to the p-adic case. The following is the p-adic analogoue of Zagier’s conjecture.

Conjecture 1.7. (dimension conjecture, Furusho-Y.) We define a sequence {d_{n}}_{n} to be d_{0} = 1, d_{1} = 0, d_{2} = 0, d_{n+3} = d_{n+1} + d_{n} (n ≥ 0). (The generating function is ∑_{n=0}^{∞} d_{n}t^{n} = (1 − t^{2})/(1 − t^{2} − t^{3}).) Then, we have dim_{ℚ} Z_{w}^{p} = d_{w} for w ≥ 0.

The main result is the following:

Theorem 1.8. (Y.[Y]) For N = 2 (resp. N > 2), we define a sequence {d_{n}[N]}_{n} by a generating function ∑_{n=0}^{∞} d_{n}[2]t^{n} = (1 − t^{2})/(1 − t − t^{2}) (resp. ∑_{n=0}^{∞} d_{n}[N]t^{n} = (1 − t)/(1 − (\frac{φ(N)}{2} + ν)t + (ν − 1)t^{2})). Here, φ is the Euler function, and ν is the number of prime numbers dividing N. Then, we have dim_{ℚ} Z_{w}^{p}[N] ≤ d_{w}[N] for w ≥ 0 and N ≥ 1.

This theorem also says that there are enormous relations between p-adic MLV’s. The opposite inequality seems to be a p-adic trancedental number theoric problem, and that we cannot prove it by the present algebraic geometical methods.

Remark . It is not known that dim_{ℚ} Z_{w}^{p}[N] does not depend on p. It seems to be a difficult problem (cf. the higher Leopoldt conjecture in Example 1.1).

Remark . For N > 4, it is known that the equality does not hold in general by the same reason. The gap is related to the space of cusp forms for Γ_{1}(N) of weight 2 when N is a prime number.

These sequences have K-theoric meanings, and we prove the upper bounds by relating the K-theory. For example, we have

\frac{1}{1 − t^{2} − t^{3}} = \frac{1}{1 − t^{2}} \frac{1}{1 − \frac{t^{3}}{1−t^{2}}} = \frac{1}{1 − t^{2}} \frac{1}{1 − (t^{3} + t^{5} + t^{7} + ⋯)}.

The term 1/(1−t^{2}) corresponds to π^{2} in the weight 2, and t^{3} +t^{5} +t^{7} +⋯ corresponds to

rankK_{2n−1}(ℤ) = \begin{cases}0 & for n: even or n = 1, \\ 1 & for n: odd and n ≠ 1. \end{cases}

In the p-adic case, the generating function (1 − t^{2})/(1 − t^{2} − t^{3}) loses the factor 1/(1 − t^{2}). It corresponds to the fact that “π^{2} = 0 in the p-adic world”. The difference between the complex case and p-adic case of the generating functions is 1/(1 − t), not 1/(1 − t^{2}) for N > 2. It corresponds to the fact that in the complex case, we have − log(1 − ζ) + log(1 − ζ^{−1}) = − log(−ζ) = (rational number) · π in the weight 1, and that it vanishes in the p-adic case, since “π = 0 in the p-adic world”.

The ingredients of the main theorem is Deligne-Goncharov’s category of mixed Tate motives over ℤ[μ_{N}, \{\frac{1}{1−ζ_{w}} \}_{w|N}] ([DG]), Deligne-Goncharov’s motivic pro-unipotent fundamental groupoids of 𝕌_{N} := ℙ^{1} − {0, ∞} ∪ μ_{N} ([DG]), and Tannakian interpretations ([Fu2]) using Besser’s Frobenius invariant path ([B]).

We briefly explain the proof of the main theorem. We construct an element φ_{p} deeply related to the p-adic MLV’s in the ℚ_{p}(μ_{N})-valued point of a pro-unipotent group U_{ω} deeply related to the K-theory. (Roughly speaking, φ_{p} is an element representing “the difference between de Rham and rigid”.) By φ_{p} ∈ U_{ω}(ℚ_{p}(μ_{N})), φ_{p} satisfies the defining equations of U_{ω}. (The author does not know the concrete defining equations). The scheme U_{ω} is “small enough” by the relation to K-theory. Thus, we have enormous relations between p-adic MLV’s from the fact that φ_{p} satisfies the defining equations. From this, we get the upper bounds. This proof is the p-adic analogue of Deligne-Goncharov’s proof in the complex case. They construct an element a_{σ}^{0} deeply related to MLV’s in the ℂ-valued point of a pro-unipotent group U_{ω} (the same one in the above) deeply related to the K-theory. (Roughly speaking, a_{σ}^{0} is an element representing “ the diference between de Rham and Betti”.) They prove the upper bounds of the space of MLV’s from this a_{σ}^{0} ∈ U_{ω}(ℂ).

For this element a_{σ}^{0}, we have the following conjecture:

Conjecture 1.9. (Grothendieck, [DG]) The element a_{σ}^{0} ∈ U_{ω}(ℂ) is ℚ-Zariski dense.

In the case where N = 1, this conjecture (+α) induces Zagier’s dimension conjecture and the isobar conjecture (cf. isobar conjecture 1.3 in the p-adic case) that all linear relations between MZV’s are linear combinations of relations between MZV’s of the same weights. In the p-adic case, we have the following conjecture:

Conjecture 1.10. (Y., [Y]) The element φ_{p} ∈ U_{ω}(ℚ(μ_{N})) is ℚ-Zariski dense.

In the case where N = 1, this conjectrue (+α) induces the dimension conjecture 1.7 in the p-adic case and the isobar conjecture 1.3 in the p-adic case.

References

[B] Besser, A. Coleman integration using the Tannakian formalism. Math. Ann. 322 (2002), no. 1, 19–48.

[C] Coleman, R. F. Dilogarithms, regulators and p-adic L-functions. Invent. Math. 69 (1982), no. 2, 171–208.

[DG] Deligne, P., Goncharov, B. Groupes fondamentaux motiviques Tate mixte. preprint NT/0302267 [Fu1] Furusho, H. p-adic multiple zeta values. I. p-adic multiple polylogarithms and the p-dic KZ

equation. Invent. Math. 155(2004) , no. 2, 253–286.

[Fu2] Furusho, H. p-adic multiple zeta values. II. Tannakian interpretations. preprint.

[G1] Goncharov, B. Multiple ζ-Values, Galois Groupes, and Geometry of Modular Varieties. preprint AG/0005069

[G2] Goncharov, B. The dihedral Lie algebras and Galois symmetries of π_{1}(ℙ^{1} −{0, ∞}∪μ_{N}). Duke. Math. J. 110, (2001), 397–487.

[T] Terasoma, T. Mixed Tate motives and multiple zeta values. Invent. Math. 149(2002) 339–369.

[Y] Yamashita, G. bounds for the dimensions of p-adic multiple L-value spaces. preprint.

  E-mail address: gokun@ms.u-tokyo.ac.jp