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Agenda de l'ENS de Lyon

Higher Hida theory for unitary group GU(2; 1)

ven 27 mar 2020



Amphi B


Soutenance de thèse de M. Manh Tu NGUYEN de l'UMPA sous la direction de M. Vincent PILLONI

Langue(s) des interventions

Description générale

The modularity is a major theme in a vast network of conjectures in the Langlands program. The starting point is the proof of Taylor-Wiles of the Taniyama-Shimura conjecture. Since then there are a lot of works trying to generalize their method. Around 2012, a true new insight of the inner mechanism of Taylor-Wiles method has been proposed by Calegari-Geraghty[1], this allows us to remove some serious restrictions on the base fields and relevant deformation rings. We can therefore ven-ture to attack more general modularity statements. Their method requires however two important inputs. The first is addressed in a great generality in a recent work of Peter Scholze[2], the second remains widely open. Luckily, in some cases, we might get around this second problem by working with p-adic automorphic forms. How-ever one would need a theory that can p-adically interpolate automotphic forms that contribute to higher cohomology groups of some relevant Shimura variety. For the group GSp4, inspired by the classical Hida theory, Vincent Pilloni has proposed a method consisting of p-adically interpolating the entire complex of coherent sheaves of automorphic forms on Siegel threefold [3]. This plays a crucial role in a recent joint work [4], where they have shown that abelian surface over a totally real field is potentially modular. In this thesis, we apply the method in [3] to construct a Hida complex interpolating classes in higher cohomology groups of Picard modular sur-face (see [5]). We also developed a version of Coleman theory for higher cohomology groups [6]. In a future work, we hope to use this to show some similar modularity statement for abelian three-folds arising as Jacobians of some Picard curves.

Mots clés