Workshop on Automorphic Representations, Geometry, and Arithmetic

Many celebrated advances in number theory in the last few decades involves mysterious yet systematic relations among subjects of entirely disparate natures. Knowledge of analytically defined objects, via approaches coming from geometry, have provided information not obtained otherwise about arithmetic. The main theme of this workshop is to understand some of the latest developments along these lines. We have invited many promising researchers over the world, mostly young ones, to share their results, ideas, and ways of thinking. We hope to create a pleasant atmosphere that would encourage questions and discussions inspiring future progresses in these subjects. Researchers and students interested in number theory or arithmetic geometry are especially welcome to attend.

The workshop is supported by the TIMS. There is now an official website containing some practical information.

• Ching-Li Chai (翟敬立), University of Pennsylvania, USA
• Toby Gee, Harvard University, USA
• Florian Herzig, Northwestern University, USA
• Ming-Lun Hsieh (謝銘倫), Academia Sinica, Taiwan
• Tetsushi Ito (伊藤哲史), Kyoto University, Japan
• Wen-Wei Li (李文威), Institut mathématique de Jussieu, Université Paris 7, France
• Kai-Wen Lan (藍凱文), Princeton University / Institute for Advanced Study, USA
• Yoichi Mieda (三枝洋一), Kyushu University, Japan
• Marc-Hubert Nicole, Institut mathématique de Jussieu, Université Paris 7, France
• Jeehoon Park (朴志訓), McGill University, Canada
• Sug Woo Shin (申皙宇), University of Chicago, USA
• Claus Sørensen, Princeton University, USA
• Teruyoshi Yoshida (吉田輝義), Harvard University, USA / Cambridge University, UK
• Chia-Fu Yu (余家富), Academia Sinica, Taiwan
• Jeng-Daw Yu (余正道), National Taiwan University, Taiwan

(*) Registration: 7/21 (Tue), 10:00, first floor of New Math Building

Speaker: Ching-Li Chai (翟敬立; University of Pennsylvania, USA)
Title: CM liftings up to isogeny
Abstract: We outline the proof of the following theorem: Given an abelian variety B over a finite field k, a CM field E with [E:Q]=2 dim(B), and an E-linear action on B up to isogeny, there exist

Speaker: Toby Gee (Harvard University, USA)
Title: Some results on the weights in Serre's conjecture
Abstract: We discuss a variety of results and conjectures on the weights in Serre's conjecture for Hilbert modular forms and unitary groups.

Speaker: Florian Herzig (Northwestern University, USA)
Title: On some irreducible smooth mod p representations of p-adic reductive groups
Abstract: The so far mostly hypothetical mod p local Langlands correspondence, is supposed to match mod p Galois representations with certain mod p smooth representations. So far almost nothing is known about such smooth representations (except for GL₂). We will discuss some new results in this context. Modulo a Serre-type conjecture, we can give an application to the mod p cohomology of U(3) in the p-power level limit.

Speaker: Ming-Lun Hsieh (謝銘倫; Academia Sinica, Taiwan)
Title: Eisenstein congruence and Iwasawa main conjecture
Abstract: We will talk about Eisenstein congruence on certain unitary groups of degree four and explain its application to one divisibility relation towards the main conjecture for GL₂x K^x, where K is an imaginary quadratic field.

Speaker: Tetsushi Ito (伊藤哲史; Kyoto University, Japan)
Title: Logarithmic geometry and Deligne-Rapoport's integral models
Abstract: In their famous Antwep paper in 1972, Deligne and Rapoport constructed integral models of the modular curve X₀(p), which has semistable reduction over the ring of p-adic integers. Since then, several advanced techniques (crystalline deformation theory, p-adic uniformization, local models, etc.) are introduced in the area, and the theory of integral models is vastly generalized to Shimura varieties and related moduli problems. Nowadays, people might feel the results and the methods of Deligne-Rapoport are outdated. In this old-fashioned talk, I would disagree with such a trend. We show how the Deligne-Rapoport's original approach gives us a finer information on the logarithmic structures of X₀(p) in the sense of Fontaine-Illusie-Kato. This viewpoint was sometimes overlooked in the high-tech approach. We also discuss its possible application to mod p modular forms with logarithmic poles as well as generalization to higher dimensional Shimura varieties of Harris-Taylor-Yoshida type.

Speaker: Wen-Wei Li (李文威; Institut mathématique de Jussieu, Université Paris 7, France)
Title: Transcending endoscopy
Abstract: Let F be a local field of characteristic zero. In a paper in 1998, J. Adams initiated a program to bring the metaplectic group Mp(2n,F), which is a nonalgebraic covering of Sp(2n,F), into the picture of functoriality. This can be viewed as a variant of endoscopy, and is closely related to Howe correspondence for the reductive dual pair (O(2n+1),Sp(2n)). As in the Langlands-Shelstad theory for reductive groups, the crux is the definition of transfer factors. Renard has settled the real case in 1999. We define a transfer factor which works over every F. The transfer of orbital integrals and the fundamental lemma are established by Harish-Chandra's descent method, which reduces the problem to Lie algebras. Thus we reduce our problems to the endoscopy for symplectic and unitary groups as well as the nonstandard endoscopy on Lie algebras. We will also put the theory in a global perspective. This is a work in progress supervised by Waldspurger.

Speaker: Kai-Wen Lan (藍凱文; Princeton University / Institute for Advanced Study, USA)
Title: Elevators for degenerations of PEL structures
Abstract: We explain how the conjecture made by Yasuo Morita in 1975, concerning potential good reductions of abelian varieties with PEL structures, and its generalization concerning degenerations of higher torus ranks, can be proved as an exercise in the theory of degeneration of Mumford and Faltings-Chai, without any assumptions on the local properties of the associated reductive algebraic group.

Speaker: Yoichi Mieda (三枝洋一; Kyushu University, Japan)
Title: Nearby cycle functor for adic spaces
Abstract: In this talk, I will talk on my attempt to construct the nearby cycle functor for adic spaces over the adic spectrum of a discrete valuation ring. It is motivated by non-abelian Lubin-Tate theory. I will propose a definition of nearby cycles and give some results on comparison and finiteness.

Speaker: Marc-Hubert Nicole (Institut mathématique de Jussieu, Université Paris 7, France)
Title: Traverso's truncation conjectures and refinements of the Newton polygon stratification
Abstract: Circa 1979, C. Traverso conjectured sharp quantitative bounds to determine a p-divisible group up to isogeny (resp. up to isomorphism). The isogeny conjecture was proved in an elementary fashion by the speaker and Vasiu in 2006. This talk will report on the thornier isomorphism conjecture, which is wrong in general. As we shall see, a corrected and optimal bound for the isomorphism conjecture is given, roughly speaking, by twice the bound for the isogeny conjecture. In particular, the original isomorphism conjecture of Traverso is true when the codimension of the p-divisible group is equal to its dimension (for example, for the p-divisible group of an abelian variety). We will also describe the behaviour of the isogeny cutoff and isomorphism number in families, giving rise to natural stratifications of Newton polygon strata. Joint work with E. Lau and A. Vasiu.

Speaker: Jeehoon Park (朴志訓; McGill University, Canada)
Title: Iwasawa main conjecture for CM elliptic curves at supersingular primes
Abstract: We generalize the Pollack-Rubin proof of the Iwasawa main conjecture for CM elliptic curves over Q at supersingular primes to CM elliptic curves over an abelian extension of the imaginary quadratic field given by CM. We will explain the precise set-up, how to construct plus/minus algebraic p-adic L-functions and plus/minus analytic p-adic L-functions and how they coincide. This is a joint work with Byoung Du Kim and Bei Zhang.

Speaker: Sug Woo Shin (申皙宇; University of Chicago, USA)
Title: Finding automorphic representations of prescribed type
Abstract: Let G be a connected reductive group over a number field F. Let S be a finite set of places of F. Assuming certain facts in representation theory including the local Langlands classification, we will explain how the simple trace formula (together with the stable trace formula formalism) allows us to find an automorphic representation of G(A_F) which has prescribed types on S. In fact we prove that there are many representations of any prescribed type. (This kind of result may be well-known to experts in the trace formula.) If we restrict to the case of discrete series types, the result is unconditional due to Clozel (1980's). If G is an inner form of a product of general linear groups on S, the result is also unconditional. If time permits, we interpret this in the world of Galois representations.

Speaker: Claus Sørensen (Princeton University, USA)
Title: Oddness and polarized regular motives
Abstract: We report on work in progress on associating motives (for absolute Hodge cycles) to RAESDC automorphic representations of GL_n over a totally real field. Among other things, this should explain the oddness (in the sense of Gross and Ash-Sinnott) of the book project Galois representations. The basic idea is to pass to a CM field, satisfying the usual special hypotheses, construct motives there (guided by Harris-Taylor) and patch up representations of the motivic Galois group.

Speaker: Teruyoshi Yoshida (吉田輝義; Harvard University, USA / Cambridge University, UK)
Title: On arithmetic geometry of Lubin-Tate spaces
Abstract: Lubin-Tate spaces are an example of local symmetric spaces for p-adic groups (Rapoport-Zink spaces), whose etale cohomology groups realize local Langlands correspondence for GL(n) (proven via global theory of Shimura varieties). We discuss moduli-theoretic stratifications and resolutions of integral models of Lubin-Tate spaces with Drinfeld level structures.

Speaker: Chia-Fu Yu (余家富; Academia Sinica, Taiwan)
Title: Geometry of the Siegel threefold with paramodular level structure
Abstract: In this talk we will describe the geometry of the Siegel threefold with paramodular level structure at p. The singularity that is local complete intersection is well-known, while we describe it explicitly in this case. We also determine some higher terms of the defining equation, which are needed in description of singularities of subvarieties of the special fiber.

Speaker: Jeng-Daw Yu (余正道; National Taiwan University, Taiwan)
Title: Ordinary crystals with logarithmic poles
Abstract: We study the abstract formalism of ordinary crystals with logarithmic poles over a smooth affine base and give some of their properties. In particular, the canonical coordinates of an ordinary crystal are obtained.