Séminaire de Mathématique

# l-adic representations of fundamental groups: Some results toward potential automorphy

## by Prof. Gebhard BÖCKLE (University Heidelberg)

Europe/Paris
Amphithéâtre Léon Motchane (IHES)

### Amphithéâtre Léon Motchane

#### IHES

Le Bois Marie 35, route de Chartres 91440 Bures-sur-Yvette
Description
Avec le soutien de :


ERC Advanced Grant : AAMOT (Arithmetic of Automorphic Motives)

PI : Michael HARRIS

Let G be a split reductive group over a finite field F_q and let K be a global field with constant field F_q. By fundamental work of Vincent Lafforgue any cuspidal automorphic representation of G(A_K) gives rise to a compatible system of Galois representation of Gal(K^sep/K) valued in the dual group \hat G of G. In joint work with M. Harris, C. Khare and J. Thorne, we investigate the question of when a \hat G-valued continuous l-adic representation of Gal(K^sep/K) is potentially automorphic, i.e. arises potentially from V. Lafforgue’s construction. After an introduction and the statement of a first potential modularity result, I will focus on the aspect of compatible systems and the use of a result of Moret-Bailly in the present context.
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