Thursday, January 5, 2023

2023-01-12 / 14:00 ~ 15:00

학과 세미나/콜로퀴엄 - 정수론: Nonemptiness of affine Deligne-Lusztig varieties

by 임동규()

Affine Deligne-Lusztig varieties are first defined by Rapoport as the (conjectural) p-part of the so-called Langlands-Rapoport conjecture. It can be understood as a p-adic generalization of the classical Deligne-Lusztig varieties. One of the most basic questions is 'when they are nonempty'. For a certain union, the nonemptiness criterion is completely known (by the so-called Mazur's inequality or B(G,μ)). However, the question about the "individual" ones is moderately open (with no general conjecture). I will discuss old and new nonemptiness results and suggest a new conjecture, for the individual ones, in the basic case. As an application, I will briefly mention a new explicit dimension formula in the rank 2 case (for which no conjectural formula was stated before).

2023-01-06 / 14:00 ~ 15:00

학과 세미나/콜로퀴엄 - 정수론:

by ()

The converse theorem for automorphic forms has a long history beginning with the work of Hecke (1936) and a work of Weil (1967): relating the automorphy relations satisfied by classical modular forms to analytic properties of their L-functions and the L-functions twisted by Dirichlet characters. The classical converse theorems were reformulated and generalised in the setting of automorphic representations for GL(2) by Jacquet and Langlands (1970). Since then, the converse theorem has been a cornerstone of the theory of automorphic representations. Venkatesh (2002), in his thesis, gave new proof of the classical converse theorem for modular forms of level 1 in the context of Langlands’ “Beyond Endoscopy”. In this talk, we extend Venkatesh’s proof of the converse theorem to forms of arbitrary levels and characters with the gamma factors of the Selberg class type. This is joint work with Andrew R. Booker and Michael Farmer.

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