Winter 2016 No Theory Seminar

1/14/2016 Weiyan

Title: Analytic number theory for zero cycles.

Abstract: Integers and polynomials over finite fields are siblings. In this talk, I will explain why this well-known kinship should also include effective zero cycles on varieties over finite fields, and many more objects. I will talk about results on the "prime factorization" of zero cycles, in analogy to classical theorems in analytic number theory. This talk will contain lots of elementary computations, lots of one-line proofs, and no theory (despite the bold appearance of "theory" in the title).

1/21/2016 Yiwen

Title: Congruences between modular forms.

Abstract: I will first provide several examples about congruences between modular forms of different weights and levels. We could notice that for each pair of congruence modulo some prime p, the ratio of the levels are often a power of p.

Then I will use several ways to explain these congruences. For example, some of them can be proved by using elementary methods, and some can be explained with the help of Igusa curves. We could also write down the Galois representations attached to some of these modular forms explicitly, so that the congruences could be seen. In the end, I will explain these congruences following some papers by Serre and Hida.

This talk might spend a lot of time reviewing the materials covered in the student number theory seminar organized by Sean last summer with the same title, as well as some results introduced by several speakers during the previous No Theory Seminar talks.

1/28/2016 Phil

Title: Periods for the projective line minus three points

Abstract: A period is a number that can be written as an integral of a rational function over an algebraic domain. Periods appear in many different disciplines in mathematics and physics, notably including special values of L-functions and Feynman amplitudes. Well, for No theory seminar, we will discuss neither, but focus on one example, which is the fundamental group of the projective line minus three points. We aim to discuss a tiny bit of Deligne's paper on the subject, but in a more explicit way using iterated integrals.

2/4/2016 Sean

Title: Analytic Galois theory, or everything I need to know about Galois extensions I learned from Professor Souganidis.

Abstract: If L/K is a Galois extension of p-adic fields then the Galois group G acts by unitary transformations on the normed K-vector space L. You might not be impressed yet, but if L is an infinite extension then that means G also acts on the completion \widehat{L}, an infinite dimensional Banach space. We'll study this Banach representation in the case that L/K is a Lubin-Tate extension (but really mostly just when it's the p-power cyclotomic extension of Q_p). In this case the Galois group G is a p-adic Lie group and an important role is played by the locally analytic vectors. We'll explain why it's important (turns out you can use this computation to, e.g., produce invariants of boring old finite dimensional representations of the absolute Galois group) and why there's a big difference between the cyclotomic extension and other Lubin-Tate extensions. Maybe you don't think points are interesting? It's ok, after all we're really just studying complete vector bundles on rigid analytic spaces via descent from a perfectoid cover.

2/11/2016 Mathilde

Title: Ray Class fields of Q(i): what they didn't teach you in high school.

Abstract: In this talk, we'll play "20th century number theorist" and do some computations in explicit class field theory. I'll construct the Ray class field of Q(i) of conductor 5, and then meticulously check that I didn't break math by introducing ramification at (1+i). Come for the local Artin maps, stay for the explicit formula for cos(pi/10).

2/18/2016 Jack

Title: Hodge versus Newton

Abstract: If X is a smooth proper variety over Z_p, then we have its Hodge polygon (from geometry) and its Newton polygon (from arithmetic). Who will come out on top and be crowned king of the polygons? Come to No Theory to find out! There will be examples and applications to point-counting.

2/25/2016 Zhiyuan

Title: Elliptic Functions According to Eisenstein and Kronecker

Abstract: I will discuss the book by Weil in the title. The main focus will be the part by Kronecker, including the real analytic Eisenstein series, its analytic continuation and functional equation, the limit formula, and applications to number theory. We shall only use complex analysis in the 19th century.

3/3/2016 Drew

Title: Elliptic Curves with Complex Multiplication

Abstract: I will give you my topic talk, emphasizing the examples I prepared for the exam. My topic aimed to treat Elliptic Curves with CM more arithmetically than the treatment found in Silverman. For example, Silverman emphasizes the Main Theorem of Complex Multiplication and CM curves over C, while I will define the Grossencharacter by lifting Frobenius, and will prove explicit class field theory (for quadratic imaginary fields) using the connection between the Grossencharacter and the Tate module.

Notes.

3/10/2016 Karl

Title: The Greenberg-Wiles Selmer group formula isn't as scary as you think

Abstract: When I first came across the Greenberg-Wiles Selmer group formula, I thought it was scary. It turns out it's not as scary as I thought. I'll try to demonstrate this by showing how to apply the formula to prove two classical results about number fields: First the Kronecker-Weber theorem (which I discussed very briefly during my FLT talk last quarter but I will flesh out more here) and then a reflection theorem of Kummer on class numbers of cyclotomic extensions.

3/17/2016 Joel

Title: A Naive Approach to the Moduli of Elliptic Curves

Abstract: No Lattices, No Upper Half Plane, No Theory.