Number Theory Seminar

Seminar information archive ~12/05Next seminarFuture seminars 12/06~

Date, time & place Wednesday 17:00 - 18:00 056Room #056 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Naoki Imai, Yoichi Mieda

Seminar information archive


17:00-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Jean-Francois Dat (Sorbonne University)
Moduli space of l-adic Langlands parameters and the stable Bernstein center (English)
[ Abstract ]
Motivated by the description of the integral l-adic cohomology of certain Shimura varieties in middle degree, Emerton and Helm have conjectured the existence of a certain local Langlands correspondence for l-adic families of n-dimensional Galois representations. The proof of this conjecture by Helm and Moss relies on a beautiful isomorphism between the ring of functions of the moduli space of l-adic representations and the integral Bernstein center of GL_n(F). We will present a work in progress with Helm, Korinczuk and Moss towards a generalization of this result for arbitrary (tamely ramified) reductive groups.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Joseph Ayoub (University of Zurich)
P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes (English)
[ Abstract ]
A^1-localisation is a universal construction which produces "cohomology theories" for which the affine line A^1 is contractible. It plays a central role in the theory of motives à la Morel-Voevodsky. In this talk, I'll discuss the analogous construction where the affine line is replaced by the projective line P^1. This is the P^1-localisation which is arguably an unnatural construction since it produces "cohomology theories" for which the projective line P^1 is contractible. Nevertheless, I'll explain a few positive results and some computations around this construction which naturally lead to a definition of Kodaira-Spencer classes of arithmetic nature. (Unfortunately, it is yet unclear if these classes are really interesting and nontrivial.)


17:00-18:00   Room #122 (Graduate School of Math. Sci. Bldg.)
Teppei Takamatsu (University of Tokyo)
On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Zongbin Chen (Yau Mathematical Sciences Center, Tsinghua University)
The geometry of the affine Springer fibers and Arthur's weighted orbital integrals (English)
[ Abstract ]
The affine Springer fibers are geometric objects conceived for the study of orbital integrals. They have complicated geometric structures. We will explain our work on the geometry of affine Springer fibers, with emphasize on the construction of a fundamental domain, and show how the study of the affine Springer fibers can be reduced to that of its fundamental domain. As an application, we will explain how to calculate Arthur's weighted orbital integrals via counting points on the fundamental domain.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Lei Fu (Yau Mathematical Sciences Center, Tsinghua University)
p-adic Gelfand-Kapranov-Zelevinsky systems (ENGLISH)
[ Abstract ]
Using Dwork's trace formula, we express the exponential sum associated to a Laurent polynomial as the trace of a chain map on a twisted de Rham complex for the torus over the p-adic field. Treating the coefficients of the polynomial as parameters, we obtain the p-adic Gelfand-Kapranov-Zelevinsky (GKZ) system, which is a complex of D^\dagger-modules with Frobenius structure.


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Laurent Berger (ENS de Lyon)
Formal groups and p-adic dynamical systems (ENGLISH)
[ Abstract ]
A formal group gives rise to a p-adic dynamical system. I will discuss some results about formal groups that can be proved using this point of view. I will also discuss the theory of p-adic dynamical systems and some open questions.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Jean-Stefan Koskivirta (University of Tokyo)
Cohomology vanishing for automorphic vector bundles (ENGLISH)
[ Abstract ]
A Shimura variety carries naturally a family of vector bundles parametrized by the characters of a maximal torus in the attached group. We want to determine which of these vector bundles are ample, and also show cohomology vanishing results. For this we use generalized Hasse invariants on the stack of G-zips of Moonen-Pink-Wedhorn-Ziegler. It is a group-theoretical counterpart of the Shimura variety and carries a similar family of vector bundles. This is joint work with Y.Brunebarbe, W.Goldring and B.Stroh.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Gaëtan Chenevier (CNRS, Université Paris-Sud)
A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem (ENGLISH)
[ Abstract ]
I will show that for any integer N, there are only finitely many cuspidal algebraic automorphic representations of GL_m over Q whose Artin conductor is N and whose "weights" are in the interval {0,...,23} (with m varying). Via the conjectural yoga between geometric Galois representations (or motives) and algebraic automorphic forms, this statement may be viewed as a generalization of the classical Hermite-Minkowski theorem in algebraic number theory. I will also discuss variants of these results when the base field Q is replaced by an arbitrary number field.


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yves André (Université Pierre et Marie Curie)
Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)
[ Abstract ]
We shall theorize and exemplify the problem of torsion values of sections of abelian schemes. This « unlikely intersection problem », which arises in various diophantine and algebro-geometric contexts, can be reformulated in a non-trivial way in terms of Kodaira-Spencer maps. A key tool toward its general solution is then provided by recent theorems of Ax-Schanuel type (joint work with P. Corvaja, U. Zannier, and partly Z. Gao).


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Shuji Saito (University of Tokyo)
A motivic construction of ramification filtrations (ENGLISH)
[ Abstract ]
We give a new interpretation of Artin conductors of characters in the framework of theory of motives with modulus. It gives a unified way to understand Artin conductors of characters and irregularities of line bundle with integrable connections as well as overconvergent F-isocrystals of rank 1. It also gives rise to new conductors, for example, for G-torsors with G a finite flat group scheme, which specializes to the classical Artin conductor in case G = Z/nZ. We also give a motivic proof of a theorem of Kato and Matsuda on the determination of Artin conductors along divisors on smooth schemes by its restrictions to curves. Its proof is based on a motivic version of a theorem of Gabber-Katz. This is a joint work with Kay Rülling.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yichao Tian (Université de Strasbourg)
Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)
[ Abstract ]
In my talk, I will report on my ongoing collaborating project together with Yifeng Liu, Liang Xiao, Wei Zhang, and Xinwen Zhu, which concerns the rank 0 case of the Beilinson-Bloch-Kato conjecture on the relation between L-functions and Selmer groups for certain Rankin--Selberg motives for GL(n) x GL(n+1). I will state the main results with some examples coming from elliptic curves, sketch the strategy of the proof, and then focus on the key geometric ingredients, namely the semi-stable reduction of unitary Shimura varieties of type U(1,n) at non-quasi-split places.


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ippei Nagamachi (University of Tokyo)
Criteria for good reduction of hyperbolic polycurves (JAPANESE)
[ Abstract ]
We give good reduction criteria for hyperbolic polycurves, i.e., successive extensions of families of curves, under mild assumption. These criteria are higher dimensional versions of the good reduction criterion for hyperbolic curves given by Oda and Tamagawa. In this talk, we construct homotopy exact sequences by using intermediate quotient groups of geometric etale fundamental groups of hyperbolic polycurves.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Nicolas Templier (Cornell University)
On the Ramanujan conjecture for automorphic forms over function fields
[ Abstract ]
Let G be a reductive group over a function field of large enough characteristic. We prove the temperedness at unramified places of automorphic representations of G, subject to a local assumption at one place, stronger than supercuspidality. Such an assumption is necessary, as was first shown by Saito-Kurokawa and Howe-Piatetskii-Shapiro in the 70's. Our method relies on the l-adic geometry of Bun_G, and on trace formulas. Work with Will Sawin.


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Daichi Takeuchi (University of Tokyo)
Blow-ups and the class field theory for curves (JAPANESE)


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Sug Woo Shin (University of California, Berkeley)
Endoscopy and cohomology of U(n-1,1) (ENGLISH)
[ Abstract ]
We apply the endoscopic classification of automorphic representations for inner forms of unitary groups to bound the growth of cohomology in congruence towers of locally symmetric spaces associated with U(n-1,1). Our bound is sharper than the bound predicted by Sarnak-Xue for general locally symmetric spaces. This is joint work with Simon Marshall.


16:00-17:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Ildar Gaisin (University of Tokyo)
Fargues' conjecture in the GL_2-case (ENGLISH)
[ Abstract ]
Recently Fargues announced a conjecture which attempts to geometrize the (classical) local Langlands correspondence. Just as in the geometric Langlands story, there is a stack of G-bundles and a Hecke stack which one can define. The conjecture is based on some conjectural objects, however for a cuspidal Langlands parameter and a minuscule cocharacter, we can define every object in the conjecture, assuming only the local Langlands correspondence. We study the geometry of the non-semi-stable locus in the Hecke stack and as an application we will show the Hecke eigensheaf property of Fargues conjecture holds in the GL_2-case and a cuspidal Langlands parameter. This is joint work with Naoki Imai.


17:10-18:10   Room #002 (Graduate School of Math. Sci. Bldg.)
Noriyuki Abe (University of Tokyo)


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Minhyong Kim (University of Oxford)
Non-abelian cohomology and Diophantine geometry (ENGLISH)
[ Abstract ]
This lecture will review the construction of moduli schemes of torsors for sheaves of pro-unipotent groups and their applications to the resolution of Diophantine problems.


18:00-19:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Ana Caraiani (Imperial College)
On the vanishing of cohomology for certain Shimura varieties (ENGLISH)
[ Abstract ]
I will prove that the compactly supported cohomology of certain unitary or symplectic Shimura varieties at level Gamma_1(p^\infty) vanishes above the middle degree. The key ingredients come from p-adic Hodge theory and studying the Bruhat decomposition on the Hodge-Tate flag variety. I will describe the steps in the proof using modular curves as a toy model. I will also mention an application to Galois representations for torsion classes in the cohomology of locally symmetric spaces for GL_n. This talk is based on joint work in preparation with D. Gulotta, C.Y. Hsu, C. Johansson, L. Mocz, E. Reineke, and S.C. Shih.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Javier Fresán (École polytechnique)
Exponential motives (ENGLISH)
[ Abstract ]
What motives are to algebraic varieties, exponential motives are to pairs (X, f) consisting of an algebraic variety over some field k and a regular function f on X. In characteristic zero, one is naturally led to define the de Rham and rapid decay cohomology of such pairs when dealing with numbers like the special values of the gamma function or the Euler constant gamma which are not expected to be periods in the usual sense. Over finite fields, the étale and rigid cohomology groups of (X, f) play a pivotal role in the study of exponential sums.
Following ideas of Katz, Kontsevich, and Nori, we construct a Tannakian category of exponential motives when k is a subfield of the complex numbers. This allows one to attach to exponential periods a Galois group that conjecturally governs all algebraic relations among them. The category is equipped with a Hodge realisation functor with values in mixed Hodge modules over the affine line and, if k is a number field, with an étale realisation related to exponential sums. This is a joint work with Peter Jossen (ETH).


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Xin Wan (Morningside Center for Mathematics)
Iwasawa theory and Bloch-Kato conjecture for modular forms (ENGLISH)
[ Abstract ]
Bloch and Kato formulated conjectures relating sizes of p-adic Selmer groups with special values of L-functions. Iwasawa theory is a useful tool for studying these conjectures and BSD conjecture for elliptic curves. For example the Iwasawa main conjecture for modular forms formulated by Kato implies the Tamagawa number formula for modular forms of analytic rank 0.
In this talk I'll first briefly review the above theory. Then we will focus on a different Iwasawa theory approach for this problem. The starting point is a recent joint work with Jetchev and Skinner proving the BSD formula for elliptic curves of analytic rank 1. We will discuss how such results are generalized to modular forms. If time allowed we may also explain the possibility to use it to deduce Bloch-Kato conjectures in both analytic rank 0 and 1 cases. In certain aspects such approach should be more powerful than classical Iwasawa theory, and has some potential to attack cases with bad ramification at p.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Michael Temkin (The Hebrew University of Jerusalem)
Logarithmic resolution of singularities (ENGLISH)
[ Abstract ]
The famous Hironaka's theorem asserts that any integral algebraic variety X of characteristic zero can be modified to a smooth variety X_res by a sequence of blowings up. Later it was shown that one can make this compatible with smooth morphisms Y --> X in the sense that Y_res --> Y is the pullback of X_res --> X. In a joint project with D. Abramovich and J. Wlodarczyk, we construct a new algorithm which is compatible with all log smooth morphisms (e.g. covers ramified along exceptional divisors). We expect that this algorithm will naturally extend to an algorithm of resolution of morphisms to log smooth ones. In particular, this should lead to functorial semistable reduction theorems. In my talk I will tell about main ideas of the classical algorithm and will then discuss logarithmic and stack-theoretic modifications we had to make in the new algorithm.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Kazuya Kato (University of Chicago)
Height functions for motives, Hodge analogues, and Nevanlinna analogues (ENGLISH)
[ Abstract ]
We compare height functions for (1) points of an algebraic variety over a number field, (2) motives over a number field, (3) variations of Hodge structure with log degeneration on a projective smooth curve over the complex number field, (4) horizontal maps from the complex plane C to a toroidal partial compactification of the period domain. Usual Nevanlinna theory uses height functions for (5) holomorphic maps f from C to a compactification of an agebraic variety V and considers how often the values of f lie outside V. Vojta compares (1) and (5). In (4), V is replaced by a period domain. The comparisons of (1)--(4) provide many new questions to study.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Yongquan Hu (Chinese Academy of Sciences, Morningside Center of Mathematics)
Multiplicity one for the mod p cohomology of Shimura curves (ENGLISH)
[ Reference URL ]


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ryotaro Sakamoto (University of Tokyo)
Stark Systems over Gorenstein Rings (JAPANESE)

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