The goal of my talk will be to introduce the local and the global
the Class field theory. Though definitions of all basic objects
(inverse limits, p-adic numbers, adeles, ideles, etc.) will be briefly recalled,
it is recommended that the participants review these notions
before the lecture.
The talk will be independent of the first lecture
The main goal of my talk will be to introduce adeles and ideles and to formulate the global class
field theory both for number fields and function fields.
Key words: adeles, ideles, class field theory, algebraic curves.
Abstract: Modular forms are historically the first example of automorphic
forms, and are still studied today as they have many applications. In
this talk I want to introduce modular forms, give some examples, and, if
time permits, explain the connection to elliptic curves, objects we
already met in the first lecture.
This is a new seminar, whose official name is "Topics in number theory and algebraic geometry". At least in the beginning the goal of the seminar will be to give a (relatively) gentle introduction to various topics, which should be accessible to beginning but motivated graduate students. The seminar has a number in the shnaton (80942), so graduate students can get a credit for it. First talk: The goal of this organizational/introductory talk will be to describe areas of mathematics, connected to Langlands program.
The hydrogen atom system is one of the most thoroughly studied examples of a quantum mechanical system. It can be fully solved, and the main reason why is its (hidden) symmetry. In this talk I shall explain how the symmetries of the Schrödinger equation for the hydrogen atom, both visible and hidden, give rise to an example in the recently developed theory of algebraic families of Harish-Chandra modules. I will show how the algebraic structure of these symmetries completely determines the spectrum of the Schrödinger operator and sheds new light on the quantum nature of the system.
Abstract. Let L be a non-archimedean local field of characteristic 0. In this talk we will present a variant of the theory of (\varphi,\Gamma)-modules associated with Lubin-Tate groups, developed by Kisin and Ren, in which we replace the Lubin-Tate tower by the maximal abelian extension \Gamma=Gal (L^ab/L). This variation allows us to compute the functors of induction and restriction for (\varphi,\Gamma)-modules, when the ground field L changes. If time permits, we will also discuss the Cherbonnier-Colmez theorem on overconvergence in our setting.
Joint work with Ehud de Shalit.
The recent work of Abe--Henniart--Herzig--Vigneras gives a classification of irreducible admissible mod-$p$ representations of a $p$-adic reductive group in terms of supersingular representations. However, supersingular representations remain mysterious largely, and in general we know them very little. Up to date, there are only a classification of them for the group $GL_2 (Q_p)$ and a few other closely related cases.
Using formal power series one can define, over any field, a class of functions including algebraic and classical modular functions over C. Under simple conditions the power series will have coefficients in a subring of the field - say Z - and this plays a role in Apery's proof of the irrationality of \zeta(3). Remarkably over a finite field all such functions/power series are algebraic.
I will call attention to a natural - but open - problem in this area.
The celebrated Gan-Gross-Prasad conjectures aim to describe the branching behavior of representations of classical groups, i.e., the decomposition of irreducible representations when restricted to a lower rank subgroup.
The Birch and Swinnerton-Dyer conjecture predicts that the group of rational points on an elliptic curve E over Q has rank equal to the order of vanishing of the L-function of E. Generalisations of this conjecture to motives M were formulated by Belinson and Bloch-Kato. I will explain a proof of a version of the Bloch-Kato conjecture in p-adic coefficients, when M is attached to a p-ordinary Hilbert modular form of any weight and the order of vanishing is 1.
Title: Towards Chabauty-Kim loci for the polylogarithmic quotient over an arbitrary number field
Abstract: Let K be a number field and let S be an open
subscheme of Spec O_K.
Minhyong Kim has developed a method for
bounding the set of S-valued points on a
hyperbolic curve X over S; his method opens
a new avenue in the quest for an "effective
But although Kim's approach has lead to the
construction of explicit bounds in special
cases, the problem of realizing the potential
Let X be a complex manifold and let M be a meromorphic connection on X with
poles along a normal crossing divisor D. Levelt-Turrittin theorem asserts that the pull-back of M to the formal neighbourhood of a codimension 1 point in D decom poses (after ramification) into elementary factors easy to work with.
This decomposition may not hold at some other points of D. When it does, we say
that M has good formal decomposition along D. A conjecture of Sabbah, recently
proved by Kedlaya and Mochizuki independently, asserts roughly the
Higher Etale obstructions are cohomological obstructions introduced by Yonatan Harpaz and Tomer Schlank for solutions of algebraic equations over a field. Their definition is based on the theory of relative etale homotopy type. In my talk I will explain the construction of relative etale homotopy type and the resulting obstruction theory.
I will also present the calculation of these obstructions for quadratic equations of the form a_1x_1^2 + ... + a_nx_n^2 = 1. This is a joint work with Edo Arad and Tomer Schlank.