Mini-Workshop on
Weil Conjectures
& Fourier Transform
Following up on a PhD Learning Seminar on Etale Cohomology and the Weil Conjectures held in Essen during the last term, there will be a 1 1/2-day mini-workshop on mixed, smooth Weil sheaves, their Fourier Transform and how it can be used to prove Deligne's generalised Riemann Hypothesis concerning zeroes and poles of L-functions of Weil representations of etale fundamental groups of schemes in prime characteristic.
More concretely, our goal is to discuss and understand the first chapter of the book Weil Conjectures, Perverse Sheaves and l'adic Fourier transform by Kiehl and Weissauer, Springer 2001.
When? Wednesday the 20.03.24 in the afternoon and Thursday the 21.03.24 all day
Where? Room: WSC-S-U-3.01
What? - 6 talks given by voluntary junior speakers
- Option to go for joint dinner on Wednesday evening
Who? Everyone who is interested is very welcome to join and/or give a talk!
We are looking forward to seeing you there!
Preliminary Program
14:15-15:15
15:45-16:45
10:00-11:00
11:30-12:30
14:30-15:30
16:00-17:00
Riccardo Tosi - Weil Sheaves [KW01, I.1]
We introduce (l-adic, smooth) Weil sheaves, which are representations of the Weil group, and define their L-series.
Paolo Sommaruga - Deligne's Yoga of Weights [KW01, I.2]
We introduce the notion of weights, purity and mixedness for smooth Weil sheaves and prove two important properties: the Zariski-semicontinuity of weights, Thm. I.2.8., and the apriori-estimate Thm. I.2.16.
Niklas Müller - Apriori Statements for Mixedness [KW01, I.3, I.4]
We learn about two important situations in which smooth Weil sheaves are automatically mixed: Sheaves of rank 1, Thm. I.3.1., and real sheaves, Thm. I.4.3.
Luca Marannino - Fourier Transform [KW01, I.5]
We introduce the Fourier transform for smooth Weil sheaves and discuss its basic properties: The Plancherel formula, the Fourier Inversion formula and the Orthogonality Relations, Thm I.5.7- I.5.9.
Giulio Marazza - Weights and Derived Push-Forward: Mixedness [KW01, I.6, I.7]
We prove Deligne's theorem that the derived push-forward with proper support of a mixed sheaf is again mixed, Thm. I.7.1. Using Duality, this proves the generalised Riemann Hypothesis for pure sheaves on smooth proper variety.
Chirantan Chowdhurry - Weights and Derived Push-Forward: Integrality [KW01, I.8, I.9]
We prove the integrality of weights of derived push-forward with proper support of mixed sheaves, Thm I.9.1. As a consequence, we obtain the generalised Riemann Hypothesis for mixed sheaves on arbitrary varieties.
Organisatorial Board:
Giulio Marazza & Niklas Müller