Periods for the Fundamental Group

Pierre Deligne

Arizona Winter School 2002

Course description

Let X be a non singular complex algebraic variety. By Grothendieck, its complex cohomology can be described as (a) singular cohomology, with complex coefficients; (b) hypercohomology of the algebraic de Rham complex Ω_{X}^{∗}; for X affine: the cohomology of the complex of algebraic differential forms on X. The description (a) gives a rational structure: use rational coefficients. If X is defined over k^{⊂}ℂ, the description (b) gives a k-structure: use forms defined over k. The period matrix is the change of basis matrix, from a rational basis for the ℚ-structure (a), to a k-basis for the k-structgure (b). Basic example: k = ℚ, X the multiplicative group 𝔾_{m}. Here, singular H_{1} (dual to H^{1}) is generated by a loop around 0, while de Rham H^{1} is generated by \frac{dz}{z}. The period matrix is one by one; it is 2πi.

Fix a base point o. Algebraic geometry has few tools to understand π_{1}(X, o) itself. If we make π_{1} abelian, we obtain H_{1}, which has a de Rham description, periods, …. The story is almost as good for the group algebra ℚ[π_{1}(X, o)], divided by a power of the augmentation ideal I, for instance because this quotient has a description as some relative homology group in X^{N}. While periods in cohomology have mainly been considered for projective X, ℚ[π_{1}]/I^{N+1} is interesting for X as simple as ℙ^{1} − {0, 1, ∞}. For any X, I/I^{2} is H_{1}, hence I^{N}/I^{N+1} is a quotient of ⊗^{N} H. For X = ℙ^{1} − {0, 1, ∞}, the interest lies in the extensions.

The course will explain how for X = ℙ^{1}−{0, 1, ∞}, the periods of ℚ[π_{1}]/I^{N+1} are encoded in the multi-zeta values: the values of

ζ(s_{1}, …, s_{r}) = ∑_{n_{1}>...>n_{r}>0} \frac{1}{n_{1}^{s_{1}} … n_{r}^{s_{r}}}

for s_{i} integers ≥ 1. This for a suitable (tangential) base point.

If we consider X defined over ℚ, with “good reduction” mod p, the de Rham analog of ℚ[π_{1}]/I^{N+1}, tensored with ℚ_{p}, depends only on the reduction mod p of X, and is acted upon by its Frobenius endomorphism. For X = ℙ^{1} − {0, 1, ∞}, this de Rham analog is the quotient of the algebra of non commutative formal power series ℚ ≪ e_{0}, e_{1} ≫ by the part of degree ≥ N + 1. For the same base point as previously, the Frobenius action on ℚ_{p} ≪ e_{0}, e_{1} ≫ is of the form

e_{1} → p e_{0}
e_{1} → g^{−1}.p e_{1}.g,

where in g the coefficient of 1 is 1 and that of e_{1}^{n} (n > 0) is 0.

Define ζ^{(p)}(s_{1}, …, s_{r}) to be the coefficient in g of e_{0}^{s_{1}−1}e_{1}e_{0}^{s_{2}−1}e_{1} … e_{0}^{s_{r}−1}e_{1}. We will explain why the ζ^{(p)}(s_{1}, …, s_{r}) ∈ ℚ_{p} should satisfy the same polynomial identities (with rational coefficients) as the ζ(s_{1}, …, s_{r}) ∈ ℝ, plus the analog of “2πi = 0” (vanishing of the ζ^{(p)}(2n)).