Number Theory Seminar

Seminar information archive ~04/22Next seminarFuture seminars 04/23~

Date, time & place Wednesday 17:00 - 18:00 117Room #117 (Graduate School of Math. Sci. Bldg.)
Organizer(s) Naoki Imai, Shane Kelly

Seminar information archive


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)


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Olivier Fouquet (Université Paris-Sud)
The Equivariant Tamagawa Number Conjecture for modular motives with coefficients in Hecke algebras (ENGLISH)
[ Abstract ]
The Equivariant Tamagawa Number Conjecture (ETNC) of Kato is an awe-inspiring web of conjectures predicting the special values of L-functions of motives as well as their behaviors under the action of algebras acting on motives. In this talk, I will explain the statement of the ETNC with coefficients in Hecke algebras for motives attached to modular forms, show some consequences in Iwasawa theory and outline a proof (under mild hypotheses on the residual representation) using a combination of the methods of Euler and Taylor-Wiles systems.


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiroki Kato (University of Tokyo)
Wild ramification and restrictions to curves (JAPANESE)


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hiraku Atobe (University of Tokyo)
A conjecture of Gross-Prasad and Rallis for metaplectic groups (JAPANESE)


17:30-18:30   Room #117 (Graduate School of Math. Sci. Bldg.)
Peter Scholze (University of Bonn)
The geometric Satake equivalence in mixed characteristic (ENGLISH)
[ Abstract ]
In order to apply V. Lafforgue's ideas to the study of representations of p-adic groups, one needs a version of the geometric Satake equivalence in that setting. For the affine Grassmannian defined using the Witt vectors, this has been proven by Zhu. However, one actually needs a version for the affine Grassmannian defined using Fontaine's ring B_dR, and related results on the Beilinson-Drinfeld Grassmannian over a self-product of Spa Q_p. These objects exist as diamonds, and in particular one can make sense of the fusion product in this situation; this is a priori surprising, as it entails colliding two distinct points of Spec Z. The focus of the talk will be on the geometry of the fusion product, and an analogue of the technically crucial ULA (Universally Locally Acyclic) condition that works in this non-algebraic setting.


16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Haoyu Hu (University of Tokyo)
Logarithmic ramifications via pull-back to curves (English)
[ Abstract ]
Let X be a smooth variety over a perfect field of characteristic p>0, D a strict normal crossing divisor of X, U the complement of D in X, j:U—>X the canonical injection, and F a locally constant and constructible sheaf of F_l-modules on U (l is a prime number different from p). Using Abbes and Saito’s logarithmic ramification theory, we define a Swan divisor SW(j_!F), which supported on D. Let i:C-->X be a quasi-finite morphism from a smooth curve C to X. Following T. Saito’s idea, we compare the pull-back of SW(j_!F) to C with the Swan divisor of the pull-back of j_!F to C. It answers an expectation of Esnault and Kerz and generalizes the same result of Barrientos for rank 1 sheaves. As an application, we obtain a lower semi-continuity property for Swan divisors of an l-adic sheaf on a smooth fibration, which gives a generalization of Deligne and Laumon’s lower semi-continuity property of Swan conductors of l-adic sheaves on relative curves to higher relative dimensions. This application is a supplement of the semi-continuity of total dimension of vanishing cycles due to T. Saito and the lower semi-continuity of total dimension divisors due to myself and E. Yang.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Lei Fu (Tsinghua University)
Deformation and rigidity of $\ell$-adic sheaves (English)
[ Abstract ]
Let $X$ be a smooth connected algebraic curve over an algebraically closed field, let $S$ be a finite closed subset in $X$, and let $F_0$ be a lisse $\ell$-torsion sheaf on $X-S$. We study the deformation of $F_0$. The universal deformation space is a formal scheme. Its generic fiber has a rigid analytic space structure. By studying this rigid analytic space, we prove a conjecture of Katz which says that if a lisse $\overline{Q}_\ell$-sheaf $F$ is irreducible and physically rigid, then it is cohomologically rigid in the sense that $\chi(X,j_*End(F))=2$, where $j:X-S\to X$ is the open immersion.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Luc Illusie (Université Paris-Sud)
On vanishing cycles and duality, after A. Beilinson (English)
[ Abstract ]
It was proved by Gabber in the early 1980's that $R\Psi$ commutes with duality, and that R\Phi preserves perversity up to shift. It had been in the folklore since then that this last result was in fact a consequence of a finer one, namely the compatibility of $R\Phi$ with duality. In this talk I'll give a proof of this, using a method explained to me by A. Beilinson.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Emmanuel Ullmo (Institut des Hautes Études Scientifiques)
Flows on Abelian Varieties and Shimura Varieties (English)
[ Abstract ]
I will discuss several questions and some results about algebraic flows, o-minimal flows and holomorphic flows on abelian varieties and Shimura varieties.


18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yves André (CNRS, Institut de Mathématiques de Jussieu)
Direct summand conjecture and perfectoid Abhyankar lemma: an overview (English)
[ Abstract ]
According to Hochster's direct summand conjecture (1973), a regular ring R is a direct summand, as an R-module, of every finite extension ring. We shall outline our recent proof which relies on perfectoid techniques. Similar arguments also establish the existence of big Cohen-Macaulay algebras for complete local domains of mixed characteristics.


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Uwe Jannsen (Universität Regensburg, The University of Tokyo)
Filtered de Rham Witt complexes and wildly ramified higher class field theory over finite fields (joint work with Shuji Saito and Yigeng Zhao) (English)
[ Abstract ]
We will consider abelian coverings of smooth projective varieties over finite fields which are wildly ramified along a divisor D with normal crossings, and will describe the corresponding abelianized fundamental group via modified logarithmic de Rham-Witt sheaves.


16:00-18:30   Room #16:00-17:00は002, 17:30-18:30は056 (Graduate School of Math. Sci. Bldg.)
Bruno Kahn (Institut de mathématiques de Jussieu-Paris Rive Gauche) 16:00-17:00
Torsion order of smooth projective surfaces (English)
[ Abstract ]
To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbb{Q}$-universally trivial one can associate its torsion order ${\mathrm{Tor}}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition à la Bloch-Srinivas. We show that ${\mathrm{Tor}}(X)$ is the exponent of the torsion in the Néron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.
Xu Shen (Morningside Center of Mathematics) 17:30-18:30
Local and global geometric structures of perfectoid Shimura varieties (English)
[ Abstract ]
In this talk, we will investigate some geometric structural properties of perfectoid Shimura varieties of abelian type. In the global part, we will construct some minimal and toroidal type compactifications for these spaces, and we will describe explicitly the degeneration of Hodge-Tate period map at the boundaries. In the local part, we will show that each Newton stratum of these perfectoid Shimura varieties can be described by the related (generalized) Rapoport-Zink space and Igusa variety. As a consequence of our local and global constructions, we can compute the stalks of the relative cohomology under the Hodge-Tate period map of the intersection complex (on the minimal compactification), in terms of cohomology of Igusa varieties at the boundary with truncated coefficients.


17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Takenori Kataoka (University of Tokyo)
A consequence of Greenberg's generalized conjecture on Iwasawa invariants of Z_p-extensions (Japanese)


17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Wiesława Nizioł (CNRS & ENS de Lyon)
Syntomic complexes and p-adic nearby cycles (English)
[ Abstract ]
I will present a proof of a comparison isomorphism, up to some universal constants, between truncated sheaves of p-adic nearby cycles and syntomic cohomology sheaves on semistable schemes over a mixed characteristic local rings. This generalizes the comparison results of Kato, Kurihara, and Tsuji for small Tate twists (where no constants are necessary) as well as the comparison result of Tsuji that holds over the algebraic closure of the field. This is a joint work with Pierre Colmez.

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