Title: Some examples of the Bloch-Kato conjecture.
Abstract: In this talk I will talk about the Bloch-Kato conjecture relating the Milnor K-groups of a field K with the etale cohomology of Spec(K) with coefficients in the roots of unity. Everything will be build up from the Kummer theory. I will define the Milnor K-groups and verify the conjecture in 5 simple cases by doing explicit computations.
Title: Emerton's proof of the Mazur-Tate-Teitelbaum conjecture.
Abstract: I will (roughly) discuss Emerton's paper "p-adic L-functions and unitary completions of representations of p-adic reductive groups", where he gave a simple proof of the Mazur-Tate-Teitelbaum conjecture.
Suppose f is a newform of even weight with trivial nebentypus, and p divides the level once. The semi-stable Dieudonne module of the p-adic Galois representation at p is a two dimensional vector space with a filtration. The position of the filtration is called the "L-invariant" of f. [According to Breuil's p-adic local Langlands philosophy, this "L-invariant" also classifies certain quotients of p-adic Banach space representations of GL_2(Q_p).] Mazur-Tate-Teitelbaum conjecture predicts that this "L-invariant" can be seen when we compare the derivative of the p-adic L-function of f at central point, and the special value of the classical L-function at central point.
No No Theory Seminar this week.
Title: Motives and Power Structures
Abstract: In this talk, Sean explains some ways to raise a power series to the power of a variety (e.g., (1+t)^(PP^2) ). But before attending, you should ask yourself -- why is he doing this, and which entrenched authorities are being reinforced?
Title: Galois theory of the real numbers
Abstract: In this talk, we will discuss the relation between 2-dim representations of the Weil group of the real numbers and irreducible representations of GL2(R). We will explicitly describe the Harish-Chandra modules of these irreducible representations of GL2(R) using the geometry of P^1. If time permits, we will mention how the geometry (Hodge theory!) may help in determining unitarizability of Harish-Chandra modules conjectured by Schmid and Vilonen.
Title: Multiplicity One Theorem
Abstract: Hecke showed the Euler product formula for L-functions of cusp forms that are eigenfunctions for all Hecke operators T_n. The multiplicity one theorem tries to deal with the problem remaining after Hecke's work. In this talk, I will view this theorem in the perspective of analysis on groups. I will give the idea of its proof, and consider only toy models whenever there is a risk to fall into theory.
Title: Stickelberger's theorem
Abstract: I will give an elementary proof of Stickelberger's theorem using Gauss sums and then discuss Herbrand's theorem as an application. After this, I will talk about a more modern viewpoint that lends itself to Iwasawa theory and conclude with a theorem of Kurihara that implies (though I will not talk about this) the main conjecture of Iwasawa theory.
No No Theory Seminar this week.
A norm compatible family of units of a number theoretic object often provides useful information about that object. Two well-known examples are cyclotomic units of the cyclotomic field Q(zeta_Np^r) and elliptic units of a CM elliptic curve (both can be used to bound Selmer groups and construct p-adic L-functions). In my talk, I will describe another class of norm compatible units: the so called Siegel units which are functions on the open modular curves Y(N) or Y_1(N). I also hope to explain why I am interested in those units.
Title: Differentials of the Second Kind, the Picard Fuchs Equation, and the Gauss Manin Connection
Abstract: My main goal for the talk will be to introduce the Gauss Manin Connection via the Picard Fuchs equation. I will first derive the Picard Fuchs equation by differentiating the periods of the Legendre family of elliptic curves. Then I will shift to the algebraic setting, and differentiate de Rham cohomology classes (for this same example) using the Gauss Manin connection - these will be literally the same computations, just stated in different contexts.
Outline of talk:
1. Differentials of the Second Kind
2. "Analytic" Derivation of the Picard Fuchs Equation for the Legendre Family
3. Overview of de Rham Cohomology/Hypercohomology
4. Relative de Rham Cohomology and the Gauss Manin Connection
5. "Algebraic" Derivation of the Picard Fuchs Equation for the Legendre Family