Aspects modulaires et elliptiques des relations entre multizêtas

摘要

This thesis deals with the family of numbers called multiple zeta values, and on the relations they satisfy. The first chapter is a general introduction to the field and has the goal of briefly presenting the different settings into which the results of the three other chapters fit, and stating these results. In chapter 2, we study the linear relations between simple and double zeta values, establishing a rigorous connection between these relations, the linear relations between Poisson bracket of depth 1 elements of the free Lie algebra on two generators, and the space of modular forms. The proofs consist mainly in performing elementary linear algebra on explicitly defined matrices. The main result of chapter 3 involves a Lie algebra of derivations derived from the study of the category of mixed elliptic motives introduced by Hain and Matsumoto. We prove the existence of linear relations observed by Pollack in this algebra and which also come from modular forms. The bulk of the proofs rely on applying techniques introduced by Ecalle to the study of the properties of certain non-commutative polynomials. The fourth and last chapter proposes a construction of an elliptic formal multizeta algebra, in analogy with work by Hain and Matsumoto on mixed elliptic motives and Enriquez on elliptic associators. The latter falls within the écallian formalism of moulds, we prove two partial results which support the validity of said construction.