Inexistence, dimension formulas and classification for level one algebraic cusp forms

Abstract: [Please note: there will be 2 talks, 45' each.]

In the first lecture we will explain how improvements on using an old tool in analytic number theory, Riemann-Weil's explicit formula for L-functions, allowed us to prove the non-existence of level one algebraic cusp forms for general linear groups over Q for lots of infinitesimal characters (=sets of Hodge weights). In the second lecture we will explain how these vanishing results yield an "effortless" method to compute the geometric side of Arthur's $L^2$-Lefschetz trace formula for split classical groups with the unit of the unramified Hecke algebra. We obtain dimension formulas as a consequence. Together these results give classification theorems for level one algebraic cusp forms in motivic weight <=23.

number theoryrepresentation theory

Audience: researchers in the topic

( slides | video )

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Automorphic Project & Research Seminar

Series comments: For the rest of the 2022 spring semester, the seminar is switching to a new format, with the goal of preparing the audience for the IHES summer school on the Langlands program. We will have background talks only, with a talk on Monday evening or Tuesday morning ET, followed by a "watch party" on Tuesday morning or Tuesday evening ET, respectively, in order to accommodate participants from any time zone. The "watch parties" will feature a streaming of the recent talk, with the possibility to interrupt for questions, as in the live talks. (Recordings will be posted here after the watch party; click on "Past talks" on the seminar calendar to access them.)

We will also try to update the Automorphic Project, while having these talks. Please contact the organizers if you are interested in contributing to that!

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