Workshop on Weil conjectures

Date: 31st May - 2nd June 2012

Place: room number 403 (lecture 6 on 1st June will take place in room 321) in IMPAN, ul. Śniadeckich 8, Warsaw, Poland.

Attention: there will be "Wine and cheese" meeting for the participants on 1st June at 17.30 in room 409. Each participant willing to take part in it, should contribute 10 zl.

Description of the workshop:

The Weil conjectures express some natural properties of zeta functions of varieties over finite fields. If X is a non-singular n-dimensional projective variety over the finite field F_q with q elements, then the zeta function of X is a rational function \prod P_i(T) (-1) ^{i+1} (product being taken from 0 to 2n) and writing each polynomial P_i as a product of (1-a_ij T), where T is a variable, the Riemann hypothesis (part of Weil conjectures) says that the absolute value of each a_ij is equal to q^{i/2}. This was proved by Pierre Deligne in [De1] (and the proof is discussed at length in [FK]). Later on, Deligne generalized Weil conjectures largely.

A constructible sheaf on a scheme X of finite type over F_q is called pure of weight b if for all x in X, all the eigenvalues of the Frobenius morphism at x have absolute value N(x) ^{b/2}. It is called mixed of weight <b, if we can write it as repeated extensions of pure sheaves of weight smaller than b. The main theorem of [De2] says that if F is a mixed sheaf of weight <b, then the sheaves R^if_! F are mixed of weight <b+1. We retrieve original Weil conjectures by taking F to be equal to Q_l (l-adic numbers).

The goal of the seminar is to motivate Weil conjectures, show some applications of them and understand [De1] and some parts of [De2]. We will follow in that mostly [Ka] and [KW] who give a simplified proof of [De2] based on [Lau].

References:

[De1] P. Deligne "La conjecture de Weil I" (Weil I)

[De2] P. Deligne "La conjecture de Weil II" (Weil II)

[FK] E. Freitag, R. Kiehl "Etale cohomology and the Weil conjectures", book

[Ka] N. Katz "L-functions and monodromy: four lectures on Weil II"

[KW] R. Kiehl, R. Weissauer "Weil conjectures, perverse sheaves and l-adic Fourier transform", book

[Lau] G. Laumon "Transformation de Fourier, constantes d'équations fonctionnelles et conjecture de Weil"

[Sch] P. Scholze "Perfectoid spaces"

Prerequesities: It is advisable to be acquainted with foundations of etale cohomology: definitions, basic properties on the level of Arcata from SGA 4 1/2.

Timetable:

31st May:

Lecture 1 (Piotr Achinger) 10.15-11.45

Description: Zeta function of a scheme. Proof of the Weil conjectures for elliptic curves. Sketch of a proof for curves. Statement of the Weil conjectures.

Lecture 2 (Jakub Byszewski) 12.00-13.30

Description: Lefschetz trace formula in etale cohomology. How the existence of good cohomology theory implies Weil conjectures (besides Riemann hypothesis).

Lecture 3 (Bartosz Naskręcki) 14.45-16.15

Description: Applications of Weil conjectures. K3 surfaces.

1st June:

Lecture 4 (Przemysław Chojecki) 10.15-11.45

Description: General description of the content of Weil I and Weil II, proof of how Weil II implies Weil I. Brief sketch of the strategy of proof of Weil II.

Lecture 5 (Bartosz Naskręcki) 12.00-13.30

Description: l-adic sheaves, l-adic cohomology, weights and the target theorem. After the first lecture of Katz in [Ka].

Lecture 6 (Jakub Byszewski) 14.45-16.15 (room 321)

Description: The Artin-Schreier sheaf and the purity theorem. Reduction of the target theorem to the purity theorem. After the second lecture of Katz in [Ka].

Wine and cheese at 17.30 in room 409

2nd June:

Lecture 7 (Piotr Achinger) 10.15-11.45

Description: Reduction of the purity theorem to the monodromy theorem. After the third lecture of Katz in [Ka].

Lecture 8 (Piotr Achinger) 12.00-13.30

Description: Proof of the monodromy theorem. Some applications of Weil II. After the fourth lecture of Katz in [Ka].

Lecture 9 (Przemyslaw Chojecki) 14.45-16.15

Description: Analogues of Weil II in characteristic 0, Deligne's weight-monodromy conjecture and perfectoid spaces (after [Sch]).