| 7/12 (Mon) | 7/13 (Tue) | 7/14 (Wed) | 7/15 (Thu) | 7/16 (Fri) | |
|---|---|---|---|---|---|
| 9:00--10:20 | Welcome! (*) | Chia-Fu Yu | Sophie Morel | Laurent Fargues | Sug Woo Shin |
| 10:40--12:00 | Jing Yu | Florian Herzig | Kai-Wen Lan | Zhiwei Yun | Jonathan Pottharst |
| 12:00--14:00 | Lunch Break | ||||
| 14:00--15:20 | David Geraghty | Tetsushi Ito | Yoichi Mieda | Vincent Pilloni | Claus Sørensen |
| 15:40--17:00 | Atsushi Ichino | Ming-Lun Hsieh | Ruochuan Liu | Teruyoshi Yoshida | Jiu-Kang Yu |
(*) Registration: 7/12 (Mon), 10:00, first floor of New Math Building
Speaker: Laurent Fargues
(Centre National de la Recherche Scientifique, France)
Title: Curves and vector bundles in p-adic Hodge theory
Abstract:
Given an algebraically closed complete valued field of characteristic p, we construct a curve over Qp and classify vector bundles on it. To some objects in p-adic Hodge theory we associate Galois equivariant vector bundles on this curve. As a particular case of the classification of vector bundles on this curve we find back the two main theorems of p-adic Hodge theory: weakly admissible is equivalent to admissible and De Rham implies potentially semi-stable. This is joint work with Jean-Marc Fontaine.
Speaker: David Geraghty
(Princeton University /
Institute for Advanced Study, USA)
Title: Potential automorphy for compatible systems of l-adic Galois representations
Abstract:
I will discuss a potential automorphy result for compatible systems of l-adic representations of the Galois group of a totally real field. The result applies to compatible systems which are regular, self-dual in an appropriate sense and sufficiently irreducible. This is joint work with Thomas Barnet-Lamb, Toby Gee and Richard Taylor.
Speaker: Florian Herzig
(Northwestern University, USA)
Title: Explicit Serre weight conjectures
Abstract:
We will discuss a generalisation of Serre's conjecture on the possible weights of modular mod p Galois representations for a broad class of reductive groups. In good cases (essentially when the Galois representation is tamely ramified at p) the predicted weight set can be made explicit and compared to previous onjectures. This is joint work with Toby Gee and David Savitt.
Speaker: Ming-Lun Hsieh
(謝銘倫;
National Taiwan University, Taiwan)
Title: On the vanishing of the μ-invariant of anticyclotomic p-adic L-functions for CM fields.
Abstract:
In this talk, we prove the vanishing of Iwasawa μ-invariant of anticycltomic Katz p-adic functions for CM fields if the branch character has gobal root number one and is primitive modulo p. This was proved by Hida if the conductor of the brach character is a product of split primes. We remove the assumption on the conductor using Hida's original approach together with a new input from the local theta dichotomy for (U(1),U(1)).
Speaker: Atsushi Ichino
(市野篤史;
Osaka City University, Japan)
Title: On the Kottwitz-Shelstad normalization of transfer factors for automorphic induction for GL(n)
Abstract:
Cyclic automorphic induction for GL(n) over a p-adic field is an example of endoscopic transfer and its character identity was established by Henniart and Herb, up to a constant. We discuss a relation of this constant to the Kottwitz-Shelstad transfer factor, in particular, to the epsilon factor normalization. This is joint work with Kaoru Hiraga.
Speaker: Tetsushi Ito
(伊藤哲史;
Kyoto University, Japan)
Title: On the order of vanishing of automorphic L-functions for GL(2) over global fields at the center of functional equation
Abstract:
This talk is based on a joint work with Masataka Chida. We study the order of vanishing of automorphic L-functions for GL(2) over global fields from several viewpoints. We propose several different methods (using many quadratic twists, known cases of the BSD conjecture, geometry of Shimura curves and the global Langlands conjecture for GL(2) over function fields (proved by Drinfeld), etc.) to construct automorphic representations of GL(2) over global fields whose L-functions have large order of vanishing at the center of functional equation.
Speaker: Kai-Wen Lan
(藍凱文;
Princeton University /
Institute for Advanced Study, USA)
Title: Vanishing theorems for torsion automorphic sheaves
Abstract:
In this talk, I will explain my joint work with Junecue Suh on when and why the cohomology of Shimura varieties (with nontrivial integral coefficients) has no torsion, based on certain vanishing theorems we have proved recently. (All conditions involved will be explicit, independent of level, and effectively computable.)
Speaker: Ruochuan Liu
(劉若川;
Institute for Advanced Study, USA)
Title: Locally analytic vectors of unitary principal series of GL2(Qp)
Abstract:
The p-adic local Langlands correspondence for GL2(Qp), which is initiated by Breuil and established by Colmez, attaches an admissible unitary representation Π(V) of GL2(Qp) to any
2-dimensional irreducible representation V of GQp. The unitary principal series of GL2(Qp) are those admissible unitary representations corresponding to trianguline representations. Although the present version of the p-adic local Langlands correspondence for GL2(Qp) is formulated at the level of admissible unitary representations, it is very useful, as in Breuil's original works (and many other examples), to have the information of the space of locally analytic vectors Π(V)an of Π(V). The main result of this talk is a determination of the space of locally analytic vectors for all (non-exceptional) unitary principle series of GL2(Qp); this proves a conjecture of Emerton. This is a joint work with Bingyong Xie and Yuancao Zhang.
Speaker: Yoichi Mieda
(三枝洋一;
Kyushu University, Japan)
Title: Cuspidal representations in the l-adic cohomology of the Rapoport-Zink space for GSp(4)
Abstract:
I will report on my recent work with Tetsushi Ito on the l-adic cohomology of the Rapoport-Zink space for GSp(4). We prove that the smooth representation of GSp4(Qp) obtained as the i-th compactly supported l-adic cohomology of the Rapoport-Zink tower has no quasi-cuspidal subquotient unless i=2,3,4. In the proof, the variants of formal nearby cycle introduced by myself play essential roles.
Speaker: Sophie Morel
(Harvard University, USA)
Title: Combinatorial problems arising in the application of the trace formula to Shimura varieties
Abstract:
To apply Arthur's trace formula to the calculation of the cohomology of Shimura varieties, we need to be able to explicitly evaluate weighted orbital integrals (and their stable versions) on discrete series pseudo-coefficients on real groups. This leads to interesting formulas for stable discrete series characters.
Speaker: Vincent Pilloni
(Columbia University, USA)
Title: Geometric overconvergent modular forms
Abstract:
We give a geometric construction of overconvergent modular forms of any p-adic weight. This construction works over any PEL Shimura variety where the ordinary locus is dense in the special fiber (modular curves, Hilbert and Siegel varieties...). We obtain finite slope families of eigenforms over the total weight space. This has interesting applications. This is a joint work with F. Andreatta and A. Iovita.
Speaker: Jonathan Pottharst
(Boston University, USA)
Title: Iwasawa theory of modular forms at nonordinary primes
Abstract:
The Iwasawa theory of modular forms concerns the p-adic interpolation of their arithmetic data, e.g. their p-adic L-functions. Thanks to the work of many mathematicians, we have a detailed understanding of many parts of this picture. But Galois-theoretic aspects in the non-ordinary case have been particularly resistant to analysis, and understanding this case is a major goal of current reasearch. We will explain a new method for bringing most nonordinary primes onto an equal footing with the ordinary ones, in such a way that much of our intuition generalizes. We make use of recent improvements in p-adic Hodge theory and Galois cohomology.
Speaker: Sug Woo Shin
(申皙宇;
University of Chicago, USA)
Title: A remark on the cohomology of some unitary Shimura varieties
Abstract:
There is a conjectural counting point formula for the special fibers of Shimura varieties (with no level structure at p), which has been established for PEL Shimura varieties of unitary or symplectic type. (There are partial results in some other cases.) I will explain a technique (in progress) which will eventually enable us to identify the Eisenstein part in the alternating sum of the compact support cohomology (or the usual cohomology) starting from the counting point formula. As we usually don't know yet enough about the automorphic representations of the relevant group, the results would often be partial or conditional.
Speaker: Claus Sørensen
(Princeton University, USA)
Title: Weight-raising without Hasse invariants
Abstract:
A modular form of weight two and level N, prime-to-p, is congruent mod p to a weight p+1 form of the same level N. This is proved by multiplication by a classical Eisenstein series E, a characteristic zero lift of the Hasse invariant. A different proof by Edixhoven and Khare avoids the use of E, and gives analogous congruences for GL(2) over imaginary quadratic fields and over certain totally real fields, in which p is inert. We will apply this trick to definite unitary groups U(n), by invoking a result of Bardoe and Sin.
Speaker: Teruyoshi Yoshida
(吉田輝義;
Cambridge University, UK)
Title: The Lubin-Tate space and affine Hecke algebra
Abstract:
The Lubin-Tate space for Iwahori level has semistable reduction, and has an action of affine Hecke algebra for GL(n) via local Hecke correspondences. We describe this action geometrically, and analyze the induced action on the l-adic etale cohomology.
Speaker: Chia-Fu Yu
(余家富;
Academia Sinica, Taiwan)
Title: Kottwitz-Rapoport strata on Hilbert-Blumenthal moduli spaces with parahoric level structure
Abstract:
The Kottwitz-Rapoport stratification of Siegel moduli spaces of Iwahori level structure was introduced by Ngo and Gennstier about 9 years ago, following work of Kottwitz and Rapoport. In this talk we consider the Hilbert-Blumenthal moduli space with Iwahori level structure where p is unramified in the totally real number field. We shall describe the relationships with the EO and NP strata.
Speaker: Jing Yu
(于靖;
National Taiwan University, Taiwan)
Title: On algebraic independence of special zeta values in positive characteristic
Abstract:
Special zeta values at positive integers are usually expressed in terms of arithmetic invariants, algebraic as well as transcendental. A basic problem is to determine all the algebraic relations among such special values. In particular to prove their algebraic independence when natural algebraic relations could not be found. There is a motivic program of Grothendieck from half century ago toward this end. We shall report on recent progresses of this program in the positive characteristic world.
Speaker: Jiu-Kang Yu
(于如岡;
Purdue University, USA)
Title: Invariant theory of theta-groups, Weyl group elements, and supercuspidal representaitons
Abstract:
We generalize Gross-Reeder's simple supercuspidal representations by using Vinberg's invariant theory of theta-groups and Springer's theory of regular elements in the Weyl group. This is a joint work with B. Gross and M. Reeder.
Speaker: Zhiwei Yun
(惲之瑋;
Massachusetts Institute of Technology, USA)
Title: Towards an arithmetic fundamental lemma
Abstract:
Jacquet and Rallis proposed a relative trace formula approach to the Gross-Prasad conjecture for unitary groups. A key step in their approach is a fundamental-lemma-like orbital integral identity. This identity has been proved by the speaker.
Wei Zhang proposed an approach to the arithmetic Gross-Prasad conjecture and a key step is an arithmetic version of the Jacquet-Rallis fundamental lemma (conjecture). The conjecture relates a derivative form of an orbital integral for GLn to an intersection number in the unitary Rapoport-Zink space. The speaker is working on a function-field analog of this conjecture.
We will introduce the moduli spaces whose motives realize the relevant orbital integrals, and explain how the two sides of the arithmetic fundamental lemma show up naturally in this geometric context.