Number theory learning seminar 2016-2017

The seminar will meet Wednesdays 1:30--3:30pm in Room 384H. This year's seminar will focus on etale cohomology, the goal being to understand Laumon's proof of the main theorem of Deligne's Weil II paper that gave a powerful and vast generalization of the Riemann Hypothesis over finite fields. Familiarity with various basic topics in arithmetic geometry (schemes, class field theory, derived categories, etc.) whenever needed to get through a lecture in finite time.

In the fall we will largely focus on understanding key examples and calculations as well as proofs of serious theorems concerning etale sheaf theory, aiming to get through much of Chapter 1 of the book of Freitag and Kiehl. That will get us through the important smooth and proper base change theorems, as well as the basic formalism of l-adic cohomology. In the winter we will delve further into the cohomology theory (especially to duality theorems and Kunneth formulas), and then move on to Laumon's technique of l-adic Fourier transforms in the sheaf setting.

Here are some references relevant to this year's seminar (in approximate order of appearance):

Some notes that Conrad wrote long ago that we will be following as the template for the fall and winter (supplemented by other references for omitted details indicated therein); this is an edited .pdf file, explaining some occasional (irrelevant) blank spaces in the middle of text
[FK] "Etale Cohomology and the Weil Conjectures" by Freitag and Kiehl
[Mi] "Etale Cohomology" by Milne
[KW] "Weil Conjectures, Perverse Sheaves, and l-adic Fourer transform" by Kiehl and Weissauer
[M1] "Analytic etale duality" (preprint) and [M2] "q-crystalline cohomologies" (in preparation) by Masullo



Notes -- use at your own risk.

These are informal notes. They may change without warning.

Fall quarter
1 Oct. 5 Conrad Overview (main goals and etale morphisms) .pdf
2 Oct. 12 Conrad Smooth maps, etale topology/sheaves, sheaf operations, stalks .pdf
3 Oct. 19 Rosengarten Constructibility, fundamental group, henselian rings, and applications (1.1.7, 1.2.1-1.2.5) .pdf
4 Oct 26 Sherman First calculations: Zariski comparison, Kummer/Artin-Schreier sequences, cohomology of curves (1.2.6-1.2.7, [9]) .pdf
5 Nov. 2 Warner Cohomology and limits, and reduction of proper base change to constant coefficients (1.3.1-1.3.4.2, [9])
6 Nov. 9 Venkatesh Artin approximation and proof of proper base change (1.3.4.2, [3]) .pdf
7, 8 Nov. 16, 30 Landesman Smooth base change, local acyclicity, and vanishing cycles (1.3.5, [6]) .pdf
9 Dec. 14 Masullo Formal GAGA (EGA III)
Winter quarter
10 Jan. 18 Tam Cohomology with proper supports and Ehresmann's theorem (1.3.6-1.3.7, omit proof of 1.3.6.4) .pdf
11 Jan. 25 Zavyalov Relative purity (section 10 of [9], 7.4.5 of [2] for slicker method)
12, 13 Feb. 1, 8 Feng Poincare duality (1.3.8, [26])
14 Feb. 15 Silliman Kunneth formula and Artin comparison (1.3.9-1.3.10, [9], [2])
15 Feb. 22 Devadas Basic adic formalism (1.4.1-1.4.4.6) .pdf
16 March 1 Lawrence Advanced adic formalism (1.4.4.7-1.4.6) .pdf
17 March 8 Raksit Adic Artin comparison (1.4.7-1.4.8) .pdf
March 15 Cancelled Arizona Winter School
18 March 22 Ronchetti Sheaf Frobenius, Lefschetz trace formula, and purity (1.5) .pdf
Spring quarter
19 April 12 Lim, Dore Weil sheaves and weights [KW, I.1-I.2.11] .pdf
20 April 19 Feng Proof of Lefschetz trace formula [FK, II, 2-4] .pdf
21 April 26 Kemeny Convergence radius and determinant weights [KW, I.2.12-I.3.2]
22 May 3, 10 Rosengarten Monodromy and real sheaves [KW, I.3.3-I.4] .pdf
May 17 Cancelled Scheduling conflict
23 May 24 Venkatesh l-adic Fourier transform [KW, I.5] with examples .pdf
May 31 Masullo Analytic duality and de Rham cohomologies [M1], [M2]
24 June 7 Sherman, Tam Weil conjectures [KW, I.6-I.7] .pdf