Valeurs multiples de fonctions L de formes modulaires

TL;DR

This thesis explores the connections between modular forms, period polynomials, and multiple zeta values, extending classical theories and investigating relations among multiple L-values and MZVs.

Contribution

It generalizes Eichler-Shimura-Manin relations and links multiple L-values to multiple zeta values, advancing understanding of their interrelations.

Findings

Extended classical relations to multiple L-values.

Connected multiple L-values with multiple zeta values.

Provided new insights into the structure of modular form-related values.

Abstract

This doctoral thesis studies the overlap between two well-known collections of results in number theory: the theory of periods and period polynomials of modular forms as developed by Eichler, Shimura and Manin and its extensions by K\"ohnen and Zagier, and the theory of 'multiple zeta values' (MZV's) as initied by Euler and studied by many authors in the last two decades. These two theories had been linked by Manin, who introduced 'multiple L-values' (MLV's). We propose to study those families of number and especially the relations among them, generalizing the Eichler-Shimura-Manin relations, and also linked some of those MLV's to MZV's.