Cours de l'IHES 2017-2018
# Mixed Modular Motives and Modular Forms for SL_2 (\Z) (3/4)

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Amphithéâtre Léon Motchane (IHES)
### Amphithéâtre Léon Motchane

#### IHES

Le Bois-Marie
35, route de Chartres
91440 Bures-sur-Yvette

Description

In the `Esquisse d'un programme', Grothendieck proposed studying the action of the absolute Galois group upon the system of profinite fundamental groups of moduli spaces of curves of genus g with n marked points. Around 1990, Ihara, Drinfeld and Deligne independently initiated the study of the unipotent completion of the fundamental group of the projective line with 3 points. It is now known to be motivic by Deligne-Goncharov and generates the category of mixed Tate motives over the integers. It is closely related to many classical objects such as polylogarithms and multiple zeta values, and has a wide range of applications from number theory to physics.

In the first, geometric, half of this lecture series I will explain how to extend this theory to genus one (which generates the theory in all higher genera). The unipotent fundamental groupoid must be replaced with a notion of relative completion, studied by Hain, which defines an extremely rich system of mixed Hodge structures built out of modular forms. It is closely related to Manin's iterated Eichler integrals, the universal mixed elliptic motives of Hain and Matsumoto, and the elliptic polylogarithms of Beilinson and Levin. The question that I wish to confront is whether relative completion stands a chance of generating all mixed modular motives or not. This is equivalent to studying the action of a `motivic' Galois group upon it, and the question of geometrically constructing all generalised Rankin-Selberg extensions.

In the second, elementary, half of these lectures, which will be mostly independent from the first, I will explain how the relative completion has a realisation in a new class of non-holomorphic modular forms which correspond in a certain sense to mixed motives. These functions are elementary power series in $q$ and $\overline{q}$ and $\log |q|$ whose coefficients are periods. They are closely related to the theory of modular graph functions in string theory and also intersect with the theory of mock modular forms.

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