Multiple ζ-Values, Galois Groups, and Geometry of Modular Varieties

Abstract

We discuss two arithmetical problems, at first glance unrelated:

• 1) The properties of the multiple ζ-values

  $$\zeta ({n_{1, \ldots,}}{n_m}): = \sum\limits_{0 < {k_1}{ < _2} \cdots < {k_m}} {\frac{1}{{k_1^{{n_1}}k_2^{{n_2}} \cdots k_m^{{n_m}}}}} {n_m} >1$$

  (1)

  and their generalizations, multiple polylogarithms at N-th roots of unity.

• 2) The action of the absolute Galois group on the pro-l completion

  $$\pi _1^{(l)}({X_N}): = \pi _1^{(l)}({\mathbb{P}^1}\backslash \{ 0{,_{\mu N}},\infty \},\upsilon )$$

  of the fundamental group of X _N := ℙ¹\{0, ∞ and all N-th roots of unity}.

These problems are the Hodge and l-adic sites of the following one:

• 3) Study the Lie algebra of the image of motivic Galois group acting on the motivic fundamental group of ℙ ¹\{0, µN, ∞}.

We will discuss a surprising connection between these problems and geometry of the modular varieties

$${Y_1}(m:N): = {\Gamma _1}(m;N)\backslash G{L_m}(\mathbb{R})/{O_m}.\mathbb{R}_ + ^*$$

where Γ₁ (m; N) is the subgroup of GL _m (ℤ) stabilizing (0, …, 0,1) mod N.

In particular using this relationship we get precise results about the Lie algebra of the image of the absolute Galois group in Aut π ^(l)₁ (X _N ), and sharp estimates on the dimensions of the ℚ-vector spaces generated by the multiple polylogarithms at N-th roots of unity, depth m and weight w := n ₁ + … +n _m .

The simplest case of the problem 3) is related to the classical theory of cyclotomic units. Thus the subject of this lecture is higher cyclotomy.