# Seminar: Cohomology and Analytic Number Theory over Function Fields (Fall 2022)

## Topic: Characteristic cycles and applications

The characteristic cycle of an l-adic sheaf on a smooth variety X is a cycle in T^*X that keeps track of the ramification of the sheaf. Saito proved an index formula expressing the Euler characteristic of a sheaf as the intersection number between its characteristic cycle and the zero section, making the characteristic cycle a useful tool for computing or bounding cohomology groups. This seminar will begin with Saito's theory of characteristic cycles in positive characteristic and then survey some applications to analytic number theory over function fields. We won't go through too many of Saito's proofs, instead focusing on properties as they come up in applications.

- When: Fridays, 5-6:30 PM
- Where: Room 528
- Organizers: Amadou Bah, Kevin Chang, Matthew Hase-Liu
- References:
- Another seminar on characteristic cycles
- Theory
**[Bei]** Beilinson - *Constructible sheaves are holonomic*, link
**[Sai1]** Saito - *The characteristic cycle and the singular support of a constructible sheaf*, link
**[Sai2]** Saito - *Characteristic cycles and the conductor of direct image*, link
- Notes by Arthur Forey
- Notes by Lars Kindler
- Applications
**[Saw1]** Sawin - *Bounds for the stalks of perverse sheaves in characteristic p and a conjecture of Shende and Tsimerman*, link
**[Saw2]** Sawin - *A geometric approach to the sup-norm problem for automorphic forms: the case of newforms on GL_2(F_q(T)) with squarefree level*, link
**[Saw3]** Sawin - *Square-root cancellation for sums of factorization functions over squarefree progressions in F_q[t]*, link
**[SFFK]** Sawin, Forey, Fresán, Kowalski - *Quantitative sheaf theory*, link

#### Schedule

Week 1 (09/09)
Kevin Chang
**Review: Weil II, perverse sheaves**
In 1.5 hours, I will try to summarize most of what we did in Caleb's Friday seminar last semester. I'll start by discussing the formalism of l-adic sheaves, mixed sheaves, and perverse sheaves. Then I'll state the main theorem of Weil II and talk about some of its implications for the structure theory of mixed sheaves and maybe some other cool applications if there's time.
notes
Week 2 (09/16)
Kevin Chang
**The Swan conductor and the Grothendieck-Ogg-Shafarevich formula**
In 1.5 hours, I will try to summarize most of what we did in 1 hour in Caleb's Wednesday seminar last semester. The Grothendieck-Ogg-Shafarevich formula is an analogue of the Riemann-Roch formula for l-adic sheaves, making it useful for computing cohomology of sheaves on curves. In positive characteristic, there are extra terms coming from ramification of sheaves. I'll start by reviewing some ramification theory and defining the Swan conductor. Then I'll state the Grothendieck-Ogg-Shafarevich formula and give some applications to exponential sums.
notes
Week 3 (09/23)
Amadou Bah
**Local acyclicity, nearby/vanishing cycles and the semi-continuity of the Swan conductor**
The goal is to explain the semi-continuity of Swan-conductors, due to Deligne (and Laumon), as formulated in **[Sai1, 2.16]**. To that end, I will introduce the notion of local acyclicity of a morphism f : X → S relative to a complex F ∈ D_b^c(X,Λ), the nearby cycles complex RΨ_f(F) and its cousin the vanishing cycles complex RΦ_f(F), and present some of their properties.
I will also introduce the Swan conductor function φ_{F,f}, state (and sketch the proof of) the result of Deligne-Laumon that φ_{F,f} is constructible and lower semi-continuous, and is locally constant if and only if f is (universally) locally acyclic.
Finally, time permitting, I will also indicate the role this result (or rather its generalization **[Sai1, 2.16]**) plays in the construction of the Characteristic Cycle.
Weeks 4 and 5 (09/30, 10/07)
Matthew Hase-Liu
**Definition and key properties of the characteristic cycle**
I'll try to briefly summarize Saito's theory of the characteristic cycle for \ell-adic sheaves, which generalizes the Grothendieck-Ogg-Shafarevich formula. Since the theory is quite technical, there will absolutely be no proofs in this talk. Instead, we'll try to outline the theory and give some examples. We'll start by looking at Beilinson's theory of the singular support and the Radon transform, which we'll need just to define the characteristic cycle. Once we define the characteristic cycle, we'll look at some of its key properties, like the Milnor formula and the index formula.
Week 6 (10/14)
Kevin Chang
**Bounds on stalks of perverse sheaves**
Characteristic cycles of perverse sheaves are nice: all the components of the singular support appear with positive multiplicity. I'll talk about the first half of **[Saw1]**, where Sawin proves a bound on the stalks of a perverse sheaf in terms of the polar multiplicities of its characteristic cycle. I hope to cover as much of the proof as I can in 1.5 hours.
notes
Week 7 (10/28)
Kevin Chang
**Application to equidistribution on Bun_2(P^1)**
We'll see our first example of an application of the characteristic cycle. In this talk, I'll finish up the paper from last time, where Sawin computes the characteristic cycle of a certain pushforward complex and uses its polar multiplicities to bound its stalks. This bound implies the equidistribution on Bun_2(P^1) of pairs of rank 2 bundles on P^1 obtained by pushing forward certain line bundles on hyperelliptic curves.
Week 8 (11/04)
Amadou Bah
**Characteristic Cycles and the conductor of direct image (after T. Saito)**
The Characteristic cycle of the direct image of a constructible sheaf F under a morphism f: X––>Y of smooth schemes (over a perfect field k) is the "proper direct image" of CC(F), under some dimension assumption. This formula generalizes the index formula (Y=Spec(k)). When Y is a curve, it is equivalent to a formula for the Artin conductor of the direct image at a closed point of Y as the intersection of CC(F) with the section df in T*X. I will explain these relationships and talk about the proof of the latter conductor formula.
Week 9 (11/11)
Amadou Bah
**Bloch's conductor formula; index formula for vanishing cycles (after T. Saito)**
I will explain (maybe) how the proof of the formula for the characteristic cycle of the direct image of F by f: X––>Y reduces to the case of dim(Y)=1 which itself is equivalent to the conductor formula proved last week. The latter implies a conjecture of Bloch on the difference of Euler Characteristics of generic and special fibers, which I will deduce. I will finish with an index formula for the vanishing cycles complex at a closed point y of Y which computes its Euler characteristic as the intersection number of two cycles: one the one hand the 0-section of the fiber X_y, and on the other hand the difference of the specialization of CC(F) at y and the CC of the restriction of F to X_y.
Week 10 (12/02)
Matthew Hase-Liu
**Every connected affine \F_p-sheme is a K(\pi, 1)**
This week, we will take a look at Achinger's fundamental result in etale cohomology about K(\pi, 1)-schemes in etale cohomology. In particular, we will show that affine space is a K(\pi, 1) by using a "Bertini theorem" for lcc sheaves to induct on the dimension. Then, a modification of the proof of Noether normalization will imply the general result. Somehow, characteristic cyles will appear in the picture (namely due to a result of Deligne and Laumon about higher direct images being locally constant given some information about higher ramification).