Multiple zeta values and iterated Eisenstein integrals

Thesis

The affine ring of the motivic path torsor ₀Π^mot₁:= π ^mot₁ (P¹\ {0, 1,∞},~1₀, −~1₁) is an ind-object in the Tannakian category MT(Z) of mixed Tate motives over the integers [16]. Its periods are Q[(2πi) ^±]-linear combinations of multiple zeta values (MZVs). Brown showed that O(0^mot₁) generates MT(Z) by exhibiting a specific basis for the Q-vector space of motivic MZVs [5].

Brown also introduced a class of periods of fundamental groups called multiple modular values [7]. They are periods of the relative completion of the fundamental group of the moduli stack M1,1 of elliptic curves [22]. Among such quantities are iterated integrals of Eisenstein series along elements of the topological fundamental group of M1,1 based at the tangential basepoint ∂/∂q at the cusp, which is isomorphic to SL₂(Z).

In this thesis we prove that all motivic MZVs may be expressed as certain Q[2πi]- linear combinations of motivic iterated Eisenstein integrals (Theorem 12.0.1). This uses a construction relating the (relative) de Rham fundamental groups of P¹\ {0, 1,∞} and M_1,1 via the de Rham fundamental group of the fiber E × _∂/∂q of the punctured Tate curve over ∂/∂q. We explain how the coefficients in this linear combination may be partially determined using the Galois coaction on motivic periods.

As a consequence we also obtain a new Tannakian generator for MT(Z) constructed from the universal monodromy representation of the relative fundamental group of M1,1 on the fundamental group of E × _∂/∂q (Theorem 13.1.1).

Authors: Saad, A

• DOI: 10.5287/ora-bmmw84nyy
• Type of award: DPhil
• Level of award: Doctoral
• Awarding institution: University of Oxford
• Language: English
• Keywords: mixed Tate motives; fundamental group; iterated Eisenstein integrals; multiple modular values; Galois theory; periods; multiple zeta values; motives
• Subjects: periods; arithmetic geometry; Galois theory; number theory