Workshop on Arithmetic Geometry in Kanazawa
(November 24 - 27, 2009)

Program

Banquet: Kanazawa Daimyo Jaya, Nov 26(Thu), 19:00-

Program (PDF): [English] [Japanese]

	9:30-10:40	10:50-12:00	14:00-15:10	15:40-16:50	17:00-18:10
Nov 24	Hello	Ito	Lan	Taniguchi	Harashita
	9:30-10:40	10:50-12:00	14:00-15:10	15:40-16:50	17:00-18:10
Nov 25	Geraghty	Mieda	Zakharevich	Ichino	Ohshita
	9:30-10:40	10:50-12:00	14:00-15:10	15:40-16:50	17:00-18:10	19:00-
Nov 26	Barnet-Lamb	Gee	Abe	Loeffler	Yoshida	Banquet
	9:30-10:40	10:50-12:00	14:00-15:10	15:40-16:50	17:00
Nov 27	Imai	Pottharst	Sasaki	Shin	Good bye

Noriyuki Abe (Tokyo)

Title: On the dimension of Whittaker vectors

Abstract: The famous multiplicity one theorem says that the dimensions of the space of non-degenerate Whittaker vectors of an irreducible representation of a quasi-split (real) group is less than or equal to one. In this talk, I will discuss what happens if such conditions (non- degenerate, quasi-split) are not satisfied. I will study Whittaker vectors of parabolic induction and give a formula of the dimension.

Thomas Barnet-Lamb (Brandeis)

Title: Potential automorphy and the Sato-Tate conjecture for modular forms over Q

Abstract: I will describe recent joint work with Geraghty, Harris and Taylor in which we make some improvements in the potential automorphy theorems available for Galois representations of any dimension. For ordinary representations, one is now able to prove a moderately general result. For non-ordinary representations, the theorems available are still much more restrictive: but we can prove a result for certain niveau 2 representations with equally spaced Hodge Tate numbers. Together, these results imply the Sato-Tate conjecture for all elliptic modular new forms over the rationals.

Toby Gee (Harvard)

Title: The Sato-Tate conjecture for Hilbert modular forms

Abstract: We discuss joint work with Tom Barnet-Lamb and David Geraghty in which we prove the Sato-Tate conjecture for all non-CM regular algebraic cuspidal automorphic representations of GL_2 over an arbitrary totally real field.

David Geraghty (Harvard)

Title: Automorphy lifting for ordinary Galois representations

Abstract: I will discuss a generalization of the automorphy lifting theorems of Clozel, Harris and Taylor to the case of ordinary Galois representations. The result is obtained by applying the Taylor-Wiles method (with innovations due to Kisin and Taylor) over a Hida family. A key step is to construct an appropriate ordinary lifting ring and determine its irreducible components.

Shushi Harashita (Kobe)

Title: Estimating the Newton polygon of a p-divisible group from its p-kernel

Abstract: In this talk, we give a combinatorial algorithm determining the optimal upper bound of the Newton polygons of p-divisible groups with a given p-kernel type. This can be seen as an unpolarized analogue of Oort's conjecture on the intersections of Newton polygon strata and Ekedahl-Oort strata in the moduli space of principally polarized abelian varieties in positive characteristic.

Atsushi Ichino (Osaka City)

Title: Formal degrees and local theta correspondence

Abstract: The formal degree conjecture, which was formulated with K. Hiraga and T. Ikeda, relates a certain representation-theoretic invariant to an arithmetic invariant. It seems hard to prove it but possible to test its functoriality property. We discuss the case of local theta correspondence. This is joint work with Wee Teck Gan.

Naoki Imai (Tokyo)

Title: Compatibility of global and p-adic Langlands correspondences

Abstract: In this survey talk, I discuss a compatibility of global and p-adic Langlands correspondences after Breuil and Emerton. Main ingredients are a criterion of local splitness of Galois representations and theory of Jacquet modules. I want to explain how these are used, and a proof of a criterion of local splitness if there is time.

Tetsushi Ito (Kyoto, Math Dept)

Title: On the history of the Sato-Tate conjecture

Abstract: This is an expository talk. We will show a number of handwritten letters and numerical tables written by Mikio Sato in the spring of 1963, when he formulated a striking conjecture, nowadays called ``the Sato-Tate conjecture'', on elliptic curves and Fourier coefficients of modular forms based on computer experiments. We also review how ``Non-abelian Class Field Theory'', which is a special case of Langlands' functoriality principle, explains the conjecture via analytic properties of symmetric power $L$-functions. If time permits, we explain what we can and we cannot say about the Sato-Tate conjecture over number fields which are not necessarily totally real.

Kai-Wen Lan (Princeton/IAS)

Title: Vanishing theorems for torsion automorphic sheaves

Abstract: Given a compact PEL-type Shimura variety, a sufficiently regular weight (defined by mild effective conditions), and a prime number p unramified in the linear data and larger than an effective bound given by the weight, we show that the etale cohomology with Z_p-coefficients of the given weight vanishes away from the middle degree, and hence has no p-torsion. We do not need any other assumption (such as ones on the images of the associated Galois representations). (This is joint work with Junecue Suh.)

David Loeffler (Cambridge)

Title: P-adic families of automorphic forms for reductive groups

Abstract: Coleman's work on overconvergent p-adic modular forms shows that finite slope modular eigenforms can be interpolated p-adically, forming rigid-analytic families of eigenforms parametrised by their weights. This has been generalised by Emerton to cohomological automorphic representations of a wide range of reductive groups, where the local factor at p is principal series. I shall give an exposition of Emerton's construction and of recent work in which I have extended this to certain non-principal-series cases.

Yoichi Mieda (Kyushu)

Title: A comparison result in rigid geometry

Abstract: In this talk, I will discuss a comparison result for l-adic cohomology of rigid spaces. First I will give a result for Fujiwara spaces and next explain how we derive from it an analogous result for adic spaces.

Tatsuya Ohshita (Kyoto, Math Dept)

Title: Iwasawa theory and higher Fitting ideals

Abstract: Kurihara proved that all the higher Fitting ideals of the minus-part of the Iwasawa modules associated to the cyclotomic Z_p-extention of CM-fields coincide with the higher Stickelberger ideals. In this talk, we study the plus-part of the Iwasawa modules associated to the cyclotomic Z_p-extention of Q(\mu_p). We define the "higher cyclotomic ideals" C_i, which are ideals of the Iwasawa algebra defined by the Euler system of cyclotomic units, and we prove that they give upper bounds of the higher Fitting ideals. Our result is a refinement of the Iwasawa main conjecture.

Jonathan Pottharst (Boston College)

Title: A nonordinary control theorem

Abstract: We present an analogue of Mazur's control theorem for elliptic curves at nonordinary primes, and indicate some applications, such as to Selmer parity and growth.

Sug Woo Shin (Chicago)

Title: Plancherel density theorem

Abstract: Let S be a finite set of finite primes. Let G be a connected reductive group over Q such that G(R) has a discrete series. I prove that the S-components of discrete automorphic representations of G(A) are equidistributed with respect to the Plancherel measure on the unitary dual of G(Q_S). I'll discuss an application to the existence theorem for automorphic representations.

Shu Sasaki (King's College London)

Title: On Artin representations and nearly ordinary Hecke algebras over totally real fields

Abstract: I will explain how to prove an analogue in the ``completely split'' Hilbert case of a result of Buzzard and Taylor about two-dimensional Artin representations and weight one forms. Time permitting, I will talk about the unramified (inert) case.

Takashi Taniguchi (Kobe)

Title: Survey on Higher composition laws

Abstract: In this talk, I will give a survey of Bhargava's famous work "Higher composition laws" and its applications. The theory gives clear insight of integer-orbits structures of certain specific representations, such as the space of pairs of ternary quadratic forms. Many of these are closely related to exceptional groups.

Inna Zakharevich (MIT)

Title: Topological Modular Forms

Abstract: Given a generalized cohomology theory we often get a formal group defined by that cohomology theory. In this talk we will attempt to classify formal groups in the image of this functor which also arise as the formal groups of elliptic curves. In the course of doing this we discover a cohomology theory whose values on the even-dimensional spheres are precisely the modular forms (after inverting 6).