No Theory Seminar -- Fall 2014

Logistics:
We meet Thursday, 1:30-3:00pm in Eckhart 312. If you're interested, meet at 12:10pm in front of 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 (at least for Fall 2014) -- 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.

Schedule
2014-10-09 -- Sean Howe, tau=logq for ordinary elliptic curves
Abstract: For an elliptic curve over a p-adic field with good ordinary reduction, we can attach two natural objects -- the p-adic Tate module, which is an extension of Z_p by Z_p(1), and the de Rham cohomology, which is a filtered vector space with a semi-linear Frobenius action (coming from the deRham-cristalline isomorphism). This gives us two natural parameters -- a "q" classifying the extension of the Tate module, and a "tau" describing the position of the filtration relative to the Frobenius eigenvalues. We will explain how to obtain both of these parameters and the computation of tau=log q using p-adic Hodge theory. Time permitting, I will also say a few words about Katz's proof that tau=log q by computing the relationship between Gauss-Manin connection and the canonical lift of Frobenius on the universal deformation of an ordinary curve.

2014-10-16 -- Tianqi Fan, Introduction to Arithmetic D-modules
Abstract: In this talk, we will first discuss about examples of rigid analytic varieties over Qp and their integral structures. The definition of Berthelot's D^{\dagger}-module on smooth formal schemes will be briefly explained, and we will compute an example (due to Laumon) of an overconvergent isocrystal on P^1 \ {0, \infty}, and realize it as a coherent D^{\dagger}-module. If time permits, coadmissible D^{\infty}-modules on smooth rigid analytic varieties will be introduced, and a local computation will be done to demonstrate Kashiwara's equivalence for this new category.
I apologize if there are too many strange terminologies in this abstract, but I believe that the talk is self-contained.

2014-10-23 -- Preston Wake, Lots of Witt vectors (and one display)
Abstract: Displays are some kind of semi-linear algebra object that classify p-divisible formal groups over many rings, generalizing Dieudonne and Carter theory. The definition involves taking Witt vectors of rings, not just perfect fields of characteristic p, and so our usual strategy of thinking W(k) = (Z_p-ish thingy) doesn't work.

2014-10-30 -- Yun Cheng, Computing some CM j-invariants
Abstract: j-invariant for an elliptic curve is a good thing. It helps us to determine, for example, the Hilbert class field. We also have explicit formulas for j(z) in terms of z. But it's still hard to calculate explicitly. I'll talk about a method calculating certain j-invariants for some elliptic curves with endomorphism of certain degree, using uniformizer of X_0(N), for not many N though. It's a simple method that only works for a few cases, but at least it gives some non-trivial Hilbert class fields. It's probably of pizza seminar difficulty and length, but without pizza :p

2014-11-6 -- Jack Shotton, The Brauer-Manin obstruction for a family of conics over P^1
Abstract: Families of conics over the projective lines provide examples where the Hasse principle fails -- that is, they have solutions over every completion of Q but not over Q. I will show this 'by hand' in a particular example and reinterpret the example using the Brauer-Manin obstruction (which I shall define). The talk will be elementary and quite short.

2014-11-13 -- Chang Mou Lim, How I learned to stop worrying about the conductor
Abstract: The conductor of an elliptic curve over a local field is some representation theoretic invariant that measures the ramification of the Galois group of the local field on the Tate module of the elliptic curve. Due to wild ramification, computing it can be tricky. Fortunately, Ogg’s formula relates the conductor to a naturally geometric invariant, its minimal discriminant, thereby reducing the computation to mostly an exercise in patience.

2014-11-20 -- Yiwen Zhou, y^2=x^3-x
Abstract: I will do some elementary computations about this curve tomorrow. I will use this example to review a little bit about the theory of complex multiplication and get a bunch of ray class fields of Q(i). Then I will attach a Grossencharacter to this curve and compare the L-function of the curve and the L-function of the Grossencharacter. I may also discuss about biquadratic reciprocity law if time permits.

2014-11-27 -- No seminar (turkey day!)

2014-12-04 -- No seminar