Professor A.J. de Jong, Columbia university, Department of Mathematics.

Organizational:

- Please email me if you want to be on the associated mailing list.
- The talks will be 2x45 minutes with a short break.
- Time and place: we have Room 307 on Fridays 10:00 -- 12:00 AM, but we will start at
**10:30 AM**and be done by 12:00 AM. - First organizational meeting: Friday, September 11 at 10:30 AM in Room 307. If you are a student and interested in actively participating, then attend this meeting and take a look at Tate's Bourbaki talk as well as the paper by Milne. If you are more interested in attending the lectures and not giving one, then attending this meeting is less important.
- No lecture on Friday, October 2. AGNES
- No lecture on Friday, November 27. Thanksgiving.

**Dates** of the seminar

- Friday, September 11: Organizational meeting
- Friday, September 18: Johan de Jong, Lecture 1 below.
- Friday, September 25: Daniel Litt, Lecture 2 below.
- Friday, October 2: NO LECTURE -- AGNES
- Friday, October 9: Daniel Halpern-Leistner, Lecture 3 below.
- Friday, October 16: Johan de Jong, Lecture 4 below.
- Friday, October 23: David Hansen, Lecture 5 below.
- Friday, October 30: Yiwei She, Lecture 6 below.
- Friday, November 6: Johan de Jong, Lecture 7 below.
- Friday, November 13: NO LECTURE -- Oort conference in NL
- Friday, November 20: Chao Li, Lecture 8 below.
- Friday, November 27: NO LECTURE -- Thanksgiving
- Friday, December 4: TBS, On large rank of elliptic curves
- Friday, December 11: Shizhang Li, On large rank of elliptic curves

**Lectures:** We are going to grow a concept map of Milne's result and nearby mathematics during the seminar. The speakers are requested to email Daniel Halpern-Leistner (preferably just before giving the lecture) a list of concepts/theorems/papers that are important ingredients in their lecture and have a short paragraph for each detailing what is going on.

- Introduction and statement of the conjecture. Here are some things you can talk about
- General statement of Tate's conjecture for algebraic cycles on smooth projective varieties over finite fields.
- Briefly mention the Birch and Swinnerton-Dyer conjecture for elliptic curves over global fields briefly; suggestion: stick to the case of elliptic curves with everywhere good reduction -- this will tremendously simplify the exposition and it is enough for our purposes (namely to link it to the conjecture of Artin and Tate for surfaces: the geometric analogue.
- Section 4 of Tate's Bourbaki talk below. Very briefly mention conjecture (d) and then only to indicate that there is a connection between the conjecture of Artin_Tate and the B_S-D conjecture. Conjecture (d) has turned out to be very hard to prove and only has been esthablished as a consequence of the work Milne (that our seminar is about) and the work on the Birch and Swinnerton-Dyer conjecture (for abelian varieties) by many authors. See references II, E, G
- Optional: explain conjectures (Lichtenbaum, Milne) for other values of the Zeta function of varieties over finite fields, see references H, I, J. Motivation for doing this is that it seems to "explain" or at least make less random the at first somewhat bizarre p-adic term occuring in Artin_Tate conjecture.

- Explain the proof of Theorem 5.1 of Tate's Bourbaki talk. Hints and outline:
- Avoid thinking about what happens with 2. In other words, try not to worry about alternating versus skew symmetric
- At each step, please, clearly list the properties of l-adic \'etale cohomology used.
- Talk about Kummer sequence in \'etale cohomology
- Construct diagram (5.1) of the paper
- Talk about Picard variety of a smooth projective variety and divisibility?
- A bit about Galois cohomology for the profinite completion of Z?
- Poincare duality in etale cohomology. Actually, the perfectness in (5.3) of the paper takes a bit of explaining.
- Deduce perfectness of (5.4) from (5.3). This should be fun!
- Finish the proof of Theorem 5.1

- Explain the proof of Theorem 5.2 of Tate's Bourbaki talk. Hints and outline:
- At each step, please, clearly list the properties of l-adic \'etale cohomology used.
- Do some of the elementary lemmas about the Herbrandt quotient.
- Construct diagram (5.6).
- Finish the proof of Theorem 5.2

- Prove Theorem 2.1 of Milne's paper for \mu_p. Hints and outline:
- From this point on we focus on p-adic cohomology of a smooth projective surface over a finite field of characteristic p.
- Explain the statement for n is a power of p. Introduce the sheaves \mu_{p^n} for the flat topology and maybe also O^*/(O^*)^{p^n}.
- Explain what is meant by the pairing being compatible with intersections of divisors? Taken at face value this does not make any sense?
- The theorem in question is Corollary 1.10 in Milne's paper on duality in flat cohomology, see reference A. The proof seems failry self-contained at first glance. You may also want to look at the rewrite by Berthelot reference B.
- Explain the difficulty in doing the duality over an algebraically closed field: some cohomology becomes infinite.
- Talk about de Rham complex in characteristic p and the Cartier operator.
- Etc...

- Explain the mod p^n case of Theorem 2.1.
- Use some parts of Bloch or Illusie. Try to isolate what it is one needs about the de Rham-Witt complex?
- One should make the point that the 5-lemma tells one it suffices to develop enough machinery and the result will follow automatically from the mod p case?
- suggest looking at Milne's paper 1986a Section 1 which explains in some detail where to look for references and how the overall strategy works.

- Reduce Theorem 4.1 of Milne's paper to "(b), (c) or (d) implies (a)".
- Section 5 of Milne's paper (end of the proof of Theorem 4.1).
- Section 6 and 7 of Milne's paper (end of the proof of the main theorem).
- Fun talk about Ulmer's results, nr 1.
- Fun talk about Ulmer's results, nr 2.

Main papers:

- On a conjecture of Artin and Tate by Milne
- useful notes on the above article by Milne
- On the conjectures of Birch and Swinnerton-Dyer and a geometric analogue by Tate

Additional papers:

- Duality in the flat cohomology of a surface by Milne
- Le Th\'eor\`eme de Dualit\'e Plate Pour les Surfaces (d'apres J.S. Milne) by Berthelot
- Values of Zeta functions of varieties over finite fields by Milne
- On the conjecture of Birch and Swinnerton-Dyer for abelian varieties over function fields in characteristic p > 0 by Werner Bauer
- On the Brauer group of a surface by Qing Liu, Dino Lorenzini, Michel Raynaud
- On the conjectures of Birch and Swinnerton-Dyer in characteristic p > 0 by Kazuyo Kato and Fabien Tirhan
- Neron models, Lie algebras, and reduction of curves of genus one by Qing Liu, Dino Lorenzini, Michel Raynaud
- Values of Zeta functions of varieties over finite fields by James Milne
- On the values of the zeta function of a variety over a finite field by Peter Schneider
- Motivic cohomology and values of zeta functions by James Milne
- Galois cohomology of complete discrete valuation fields by Kazuya Kato
- Nilpotent connections and the monodromy theorem: applications of a result of Turrittin by Nick Katz
- Curves and Jacobians over function fields by Douglas Ulmer; contains an overview of the connections between BSD and Artin-Tate

Papers on cohomology theories:

Fun papers, mostly by Douglas Ulmer, about large rank elliptic curves (not really related):

- Ellliptic curves with large rank over function fields
- Rational curves on elliptic surfaces
- L-functions with large analytic rank and abelian varieties with large algebraic rank over function fields
- Towers of surfaces dominated by products of curves and elliptic curves of large rank over function fields by Lisa Berger
- On Mordell-Weil groups of Jacobians over function fields

You can find a video of Ulmer's talk at Utah by going to this link and clicking on the link to the videos of the lectures.