ℓ-adic Fourier transform and the Weil conjecture (Number theory working seminar)

The purpose of this working-seminar is to understand Laumon’s proof of Weil II via the ℓ-adic Fourier transform, following the proof in the first 60 pages of the book by Kiehl-Weissauer [KW], and also [Lau].

Our goal will be the following special case of the main result of Weil II (cf. [KW], Thm. 6.1):

Theorem 0.1 (Deligne) Let X_{0} be a smooth, geometrically irreducible, projective curve over 𝔽_{q}. Let

j_{0} : U_{0} ↪ X_{0}

be an open nonempty subscheme. Let F_{0} be a smooth sheaf, pure of weight w on U_{0}. Then the cohomology groups

H^{i}(X, j_{∗}(F_{0}))

are pure of weight w + i, for i = 0, 1, 2.

This is the special case (the curve case) of the main theorem of Weil II, and in fact the proof of the latter proceeds (via Noetherian induction) by reduction to the curve case (cf. section I.7 of [KW]). Laumon used the ℓ-adic Fourier transform to give a new proof of the above theorem, and since our goal is to understand how this works, we will focus mostly on the curve case.

Here is an outline for the talks in the seminar. These are intended as general guidelines for the talks, and need not be strictly followed (speaker’s choice).

Lecture 1: Sheaves and derived categories.

Review of smooth sheaves as representations of the algebraic fundamental group. Constructible sheaves and Deligne’s category D_{c}^{b}(X, ℚ̄_{ℓ}) (see [KW], Appendix A). The formalism of the (derived) functors f^{∗}, f_{∗}, f^{!}, f_{!} on this category (just statements, no proofs). Statement of the technical tool of duality: Lemma 6.2 [KW] (don’t give proof, but give the easy derivation of this from the more general fact that the Goresky-MacPherson extension j_{!∗} commutes with Verdier duality (cf. [KW] III.5, Cor. 5.3).

Lecture 2: Weil sheaves and theory of weights.

This material is mostly the first two sections of [KW]. Grothendieck L-series for ℚ̄_{ℓ}-sheaves. Weil sheaves, as representations of the Weil group. The fact that smooth Weil sheaves are “twists” of smooth ℚ̄_{ℓ}-sheaves (Theorem 1.4 – state without proof). The consequence that Grothendieck’s L-series is defined for Weil sheaves. Weights, and semi-continuity of weights (Theorem 2.8). The goal of the talk: Theorem 2.16 of [KW].

Lecture 3: Real sheaves and Rankin-Selberg method.

Determinental weights (via Zariski-closure of Monodromy): state and prove [KW] Theorem 3.8, using results from section I.3 on Zariski-closure of monodromy as needed (without proof). Real sheaves, and Rankin-Selberg Method (Lemma 4.2 [KW]). The goal is Theorem 4.3: any real sheaf is mixed.

Lecture 4: ℓ-adic Fourier transform.

Artin-Schreier coverings and sheaves. The definition of the Fourier transform and its basic properties. Goals: Theorems 5.5, 5.7, 5.8 of [KW].

Lecture 5: Proof of the Weil conjecture.

Putting all the preliminaries together, prove Theorem 6.1 (Curve Case), following section I.6 of [KW]. Sketch the deduction of the General Case: Theorem 7.1 and in particular Cor. 7.3 (cf. [KW], section I.7).

Schedule of Talks: during the Number Theory seminar, 3:00-4:15 Wednesdays (we expect each talk to last a little over an hour, but we should allow extra time for questions. The tentative dates for the talks are Nov. 13,20, 27(?), Dec. 4,11. (If Nov. 27 is bad, we could use Dec. 18 instead, or some Friday in December...)

Volunteers: I volunteer to give Lecture 4, and Lecture 5 if short of volunteers. Therefore at this time I am requesting volunteers to deliver Lectures 1-3 (and possibly 5). Please contact Tom Haines at tjh@math.umd.edu to sign up!

References

[Del1] P. Deligne, La conjecture de Weil I, Publications Mathématiques IHÉS 43 (1974).

[Del2] P. Deligne, La conjecutre de Weil II, Publications Mathématiques IHÉS 52 (1980).

[Lau] G. Laumon, Transformation de Fourier constantes déquations fonctionelles et conjecture de Weil, Publications Mathématiques IHÉS 65 (1987).

[KW] R. Keihl, R. Weissauer, Weil Conjectures, Perverse Sheaves and ℓ-adic Fourier Transform, Ergebnisse der Mathematik und ihrer Grenzgebiete; 3. Folge Bd 42, Springer Verlag (2001).