[Paper Review] Multiple zeta-values, Galois groups, and geometry of modular varieties

A. B. Goncharov|ArXiv.org|May 8, 2000

Advanced Mathematical Identities4 citations

TL;DR

This paper establishes a deep connection between multiple zeta values (MZVs), Galois group actions on fundamental groups of modular curves, and the geometry of modular varieties. Using motivic Galois theory and polylogarithmic currents on curves, it proves sharp bounds on the dimension of MZV spaces and conjectures a Lie-theoretic structure for their algebraic relations, linking number theory, arithmetic geometry, and quantum field theory via motivic periods.

ABSTRACT

We discuss the following two problems: 1) The properties of the multiple zeta-values and their generalizations, multiple polylogarithms at N-th roots of unity; 2) The action of the absolute Galois group on the pro-l-completion of the fundamental group of the projective line without zero, infinity, and all N-th roots of unity; and a surprising connection of these problems with the geometry and topology of modular varieties for GL_m.

Motivation & Objective

• To understand the algebraic and arithmetic structure of multiple zeta values (MZVs) and their generalizations at roots of unity.
• To describe the action of the absolute Galois group on the pro-l fundamental group of the punctured projective line minus roots of unity.
• To relate these arithmetic problems to the geometry of modular varieties $Y_1(m;N)$ via motivic Galois representations.
• To establish sharp dimension estimates for the Q-vector spaces spanned by MZVs of given weight and depth.
• To formulate a motivic and l-adic version of Zagier's conjecture on special values of L-functions via polylogarithmic periods.

Proposed method

• Uses the theory of framed mixed Tate motives over Spec(Z) to model MZVs as periods of motivic objects.
• Applies Voevodsky's triangulated category of motives and Levine's framework for mixed Tate motives to construct a motivic realization of MZVs.
• Introduces higher polylogarithmic currents $G_n(x,y)$ on algebraic curves using Green's functions and symmetric powers of cohomology.
• Constructs generating series for multiple polylogarithms via skew-symmetrized integrals over symmetric products of curves.
• Relates the special values of $L(S^n H^1(X), n)$ to the polylogarithmic periods $G_n(x,y)$, especially when $x,y$ are torsion points on modular curves.
• Uses Feynman-type path integrals with matrix-valued fields to model the asymptotic behavior of multiple Green functions, linking to quantum field theory.

Experimental results

Research questions

• RQ1 What is the precise structure of the Q-vector space generated by multiple zeta values of given weight and depth?
• RQ2 How does the absolute Galois group act on the pro-l fundamental group of $\mathbb{P}^1 \setminus \{0, \mu_N, \infty\}$?
• RQ3 What is the Lie algebra of the image of the motivic Galois group acting on the motivic fundamental group of $X_N$?
• RQ4 Can the special values of $L$-functions associated to symmetric powers of elliptic curves be expressed via polylogarithmic periods?
• RQ5 What is the role of modular varieties $Y_1(m;N)$ in unifying the arithmetic of MZVs and Galois actions?

Key findings

• The dimension of the Q-vector space of multiple zeta values of weight $k$ is bounded above by the dimension of the graded dual of the universal enveloping algebra of a free Lie algebra generated by elements of odd degree $\geq 3$.
• For weight 12, the space of primitive MZVs has dimension 2, spanned by $[e_5,e_7]$ and $[e_3,e_9]$ in the Lie algebra $\mathcal{F}(3,5,\ldots)$.
• The space of decomposable MZVs of weight 12 is generated by 10 specific monomials involving $\pi^2$, $\zeta(3)$, $\zeta(5)$, $\zeta(3,5)$, etc., consistent with the conjectural structure.
• When $x$ and $y$ are cusps on a modular curve, the polylogarithm $G_n(x,y)$ becomes the regulator of a motivic Ext class, and its coproduct vanishes due to torsion in the Jacobian.
• The multiple Green function $G(a_1,\dots,a_{m+1})$ arises as the leading term in the asymptotic of a Feynman integral with matrix-valued fields, suggesting a general mechanism for constructing motivic periods.
• The paper conjectures that special values $L(S^n H^1(X), n+m)$ are expressible via depth-$m$ multiple polylogarithms on curves, extending Zagier's conjecture to higher symmetric powers.