No Theory Seminar -- Winter 2015

Logistics:
We meet Thursdays at 1:30pm in Eckhart 312. If you're interested, meet at 12:30pm at the Eckhart tea room to join us beforehand for No Theory Lunch.

If you want to be added to the mailing list, e-mail me (Sean).

What is No Theory Seminar?
No Theory Seminar is the student number theory seminar at UChicago -- the name, inspired by a typo in the common abbreviation "No."=Number, reflects the focus on short examples, computations, or exercises, rather than proofs and theorems.

In the seminar, students will present on specific and concrete topics that have come up in their own work or reading. While the level of the talks may vary wildly with the speaker, all speakers should strive to make their talks:
* Self-contained -- recall notations used, and don't assume knowledge from prior talks.
* Accessible -- even if a beginning graduate student can't follow an entire talk, they should still be able to get something useful out of it.
* Concrete -- examples and computations instead of theorems and proofs.
* Interesting -- try to explain a computation you know not everyone in the audience has done (feel free to ask your fellow students if they think something will be a good topic!)

Speakers are also encouraged to make notes available after their talks. E-mail typed notes or a scan to the seminar organizer (Sean) to be put on the website.

Past No Theory Seminars:
Fall 2014

Schedule
2015-01-08 -- Valia, Weil reciprocity and Milnor K-groups.
Abstract: In this talk I will mostly speak about Weil's reciprocity. I will follow a proof given in the book of Serre, "Algebraic Groups and Class Fields". In more detail, I will start with an introduction about generalized Jacobians of curves with respect to rational maps to algebraic groups. We will define the modulus condition and see that it is equivalent to the existence of a local symbol. I will then give a detailed computation of the local symbol in the case of the group G_m (G_a will also be done in less detail). All this means lots of valuations. Other than that, the talk will be self-contained. You only need to be a bit familiar with divisors on curves. If time permits at the end of the talk, I will discuss a relation to Milnor K-theory.

2015-01-15 -- Sean, Overconvergence of the canonical subgroup -- an example
Abstract: Any elliptic curve over Q_p with ordinary reduction admits a "canonical" order p subgroup of its p-torsion -- the kernel of the reduction map to the special fiber. If the reduction is supersingular, then the special fiber doesn't have any non-trivial p-torsion points, and so we cannot distinguish a subgroup this way. Nonetheless, by studying the valuations of the coordinates, it turns out that for supersingular curves lying close enough to the ordinary locus, there is a way to distinguish a canonical subgroup. I'll work this out in a specific example, and maybe discuss some related ideas.

2015-01-22 -- Dan, Taylor-Wiles
Abstract: I'll discuss how the Taylor-Wiles method can be used to prove the Kronecker-Weber theorem. Hopefully the talk will not be circular.

2015-01-29 -- Jeff, Modular Forms: a concrete example of the Langlands correspondence.
Abstract: The Langlands conjectures roughly say that (sufficiently nice) representations of the absolute Galois group Gal(\bar{Q}/Q) should be in some sort of natural bijection with certain analytic objects called automorphic forms. I won't really say much about this. What I will do is describe a specific example of this correspondence.
The most famous example of an automorphic form is the modular form:
q\prod_{n=1}^\infty(1-q^n)^2(1-q^{11n})^2 = q-2q^2-q^3+2q^4+q^5+\cdots
My goal is to describe the Langlands correspondence for this modular form. That is, I will construct a Galois representation which "corresponds to" (I'll say what this means) this modular form.
Along the way, I'll introduce modular curves, modular forms and Hecke operators. I'll also say what modular curves look like in characteristic p. As a corollary of this, I will also count the number of supersingular elliptic curves over \bar{F_p}.
I will try to make the talk fairly self-contained. You will need some familiarity with elliptic curves, and with algebraic curves and their Jacobians. Some familiarity with Galois groups and Frobenius elements will also help.

2015-02-05 -- Preston, Finite flat group schemes
Abstract: I'll talk about explicit examples of finite flat groups schemes, working roughly around the example of E[p] where E is a supersingular elliptic curve. Yes, it's an extension of \alpha_p by \alpha_p, but which one? And how many such extensions are there anyhow?

2015-02-12 --No seminar

2015-02-19 -- Vaidehee, Dealing with Defect
Abstract: The classical ramification theory is about complete discrete valued field extensions with several nice properties. In this talk, I will discuss more general valuations and defect extensions. The focus will be on Artin-Schreier extensions $L|K$ given by $T^p-T=f \in K$; where $K$ is a henselian valued field of characteristic $p>0$. Warning: Might contain some theory.

2015-02-26 -- Sean, Finding the cuspidal representation of S_3 in geometry
Abstract: The representations of GL_2(F_q) are split into two families -- the principal series, which are parabolically induced, and the cuspidals, which are very mysteriousish! Some groups are bigger than other groups, and so, I prefer to take the smallest. In this talk, I will explain how to find the cuspidal representation of GL_2(F_2) = S_3 in the first etale cohomology of an elliptic curve, and then maybe I'll talk a bit about the general setup.

2015-03-05 -- TBA, TBA
Abstract:

2015-03-12 -- Fan, Casselman's submodule theorem for SL2
Abstract: Casselman's submodule theorem says that if M is a finitely generated (g, K)-module, then M/nM is non-zero, where n is a maximal nilpotent subalgebra of the reductive Lie algebra g. In particular, we see that an irreducible Harish-Chandra module can be embedded into a principal series.
In this talk, we will follow Beilinson and Bernstein: using the geometry of the flag variety to study representations of a reductive linear algebraic group. The n-coinvariant of a g-module is related to the fiber of some twisted D-modules on the flag manifold, and if the twist is correct, the D-modules will have full support. We will see some explicit computations in the case of SL2, which will hopefully give an idea of what is happening.