## Number Theory Seminar

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

Seminar information archive

### 2017/05/17

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.

### 2017/05/10

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

### 2017/04/12

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)

### 2017/04/11

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.

### 2017/03/30

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.

### 2017/01/11

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.

### 2016/12/14

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.

### 2016/11/09

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.

### 2016/11/02

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.

### 2016/10/12

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.

### 2016/06/08

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.

### 2016/05/18

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)

### 2016/05/11

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.

### 2016/04/27

16:30-17:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Masao Oi (University of Tokyo)
On the endoscopic lifting of simple supercuspidal representations (Japanese)

### 2016/04/20

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Hoto Bekki (University of Tokyo)
On periodicity of geodesic continued fractions (Japanese)

### 2016/04/13

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Akio Tamagawa (RIMS, Kyoto University)
Semisimplicity of geometric monodromy on etale cohomology (joint work with Anna Cadoret and Chun Yin Hui)

(English)
[ Abstract ]
Let K be a function field over an algebraically closed field of characteritic p \geq 0, X a proper smooth K-scheme, and l a prime distinct from p. Deligne proved that the Q_l-coefficient etale cohomology groups of the geometric fiber of X --> K are always semisimple as G_K-modules. In this talk, we consider a similar problem for the F_l-coefficient etale cohomology groups. Among other things, we show that if p=0 (resp. in general), they are semisimple for all but finitely many l's (resp. for all l's in a set of density 1).

### 2016/03/29

17:30-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Matthew Morrow (Universität Bonn)
Motivic cohomology of formal schemes in characteristic p
(English)
[ Abstract ]
The logarithmic Hodge-Witt sheaves of Illusie, Milne, Kato, et al. of a smooth variety in characteristic p provide a concrete realisation of its p-adic motivic cohomology, thanks to results of Geisser-Levine and Bloch-Kato-Gabber which link them to algebraic K-theory. I will explain an analogous theory for formal schemes, as well as applications to algebraic cycles, such as a weak Lefschetz theorem for formal Chow groups.

### 2015/12/09

18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ted Chinburg (University of Pennsylvania & IHES)
Chern classes in Iwasawa theory (English)
[ Abstract ]
Many of the main conjectures in Iwasawa theory can be phrased as saying that the first Chern class of an Iwasawa module is generated by a p-adic L-series. In this talk I will describe how higher Chern classes pertain to the higher codimension behavior of Iwasawa modules. I'll then describe a template for conjectures which would link such higher Chern classes to elements in the K-theory of Iwasawa algebras which are constructed from tuples of Katz p-adic L-series. I will finally describe an instance in which a result of this kind, for the second Chern class of an unramified Iwasawa module, can be proved over an imaginary quadratic field. This is joint work with F. Bleher, R. Greenberg, M. Kakde, G. Pappas, R. Sharifi and M. J. Taylor.

### 2015/11/17

18:00-19:00   Room #117 (Graduate School of Math. Sci. Bldg.)
Dennis Gaitsgory (Harvard University & IHES)
The Tamagawa number formula over function fields. (English)
[ Abstract ]
Let G be a semi-simple and simply connected group and X an algebraic curve. We consider $Bun_G(X)$, the moduli space of G-bundles on X. In their celebrated paper, Atiyah and Bott gave a formula for the cohomology of $Bun_G$, namely $H^*(Bun_G)=Sym(H_*(X)\otimes V)$, where V is the space of generators for $H^*_G(pt)$. When we take our ground field to be a finite field, the Atiyah-Bott formula implies the Tamagawa number conjecture for the function field of X.

The caveat here is that the A-B proof uses the interpretation of $Bun_G$ as the space of connection forms modulo gauge transformations, and thus only works over complex numbers (but can be extend to any field of characteristic zero). In the talk we will outline an algebro-geometric proof that works over any ground field. As its main geometric ingredient, it uses the fact that the space of rational maps from X to G is homologically contractible. Because of the nature of the latter statement, the proof necessarily uses tools from higher category theory. So, it can be regarded as an example how the latter can be used to prove something concrete: a construction at the level of 2-categories leads to an equality of numbers.

### 2015/10/27

18:00-19:00   Room #002 (Graduate School of Math. Sci. Bldg.)
Masanori Asakura (Hokkaido University)
On the period conjecture of Gross-Deligne for fibrations (English)
[ Abstract ]
The period conjecture of Gross-Deligne asserts that the periods of algebraic varieties with complex multiplication are products of values of the gamma function at rational numbers. This is proved for CM elliptic curves by Lerch-Chowla-Selberg, and for abelian varieties by Shimura-Deligne-Anderson. However the question in the general case is still open. In this talk, we verify an alternating variant of the period conjecture for the cohomology of fibrations with relative multiplication. The proof relies on the Saito-Terasoma product formula for epsilon factors of integrable regular singular connections and the Riemann-Roch-Hirzebruch theorem. This is a joint work with Javier Fresan.

### 2015/09/30

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Alan Lauder (University of Oxford)
Stark points and p-adic iterated integrals attached to modular forms of weight one (English)
[ Abstract ]
Given an elliptic curve over Q the only well-understood construction of global points is that of "Heegner points", which are defined over ring class fields of imaginary quadratic fields and are non-torsion only in rank one settings. I will present some new constructions and explicit formulae, in situations of rank one and two, of global points over ring class fields of real or imaginary quadratic fields, cyclotomic fields, and extensions of Q with Galois group A_4, S_4 or A_5. Our constructions and formulae are proven in certain cases - when they can be related to Heegner points - and conjectural, but supported by experimental evidence, otherwise. This is joint work with Henri Darmon and Victor Rotger.

### 2015/09/09

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Emmanuel Ullmo (IHES)
The hyperbolic Ax-Lindemann conjecture (English)
[ Abstract ]
The hyperbolic Ax Lindemann conjecture is a functional transcendental statement which describes the Zariski closure of "algebraic flows" on Shimura varieties. We will describe the proof of this conjecture and its consequences for the André-Oort conjecture. This is a joint work with Bruno Klingler and Andrei Yafaev.

### 2015/07/23

13:00-16:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Lasse Grimmelt (University of Göttingen/Waseda University) 13:00-14:00
Representation of squares by cubic forms - Estimates for the appearing exponential sums (English)
Haoyu Hu (University of Tokyo) 14:15-15:15
Ramification and nearby cycles for $\ell$-adic sheaves on relative curves (English)
[ Abstract ]
I will present a new approach for a formula of Deligne and Kato that computes the dimension of the nearby cycle complex of an $\ell$-adic sheaf on a smooth relative curve over a strictly henselian trait such that $p$ is not one of its uniformizer. Deligne considered the case where the sheaf has no vertical ramification and Kato extended the formula to the general case. My approach is based on ramification theory of Abbes and Saito. It computes the nearby cycle complex in terms of the refined Swan conductor. In fact, I compare Abbes-Saito's refined Swan conductor with Kato's Swan conductor with differential values, which is the key ingredient in Kato's formula; the case of rank one sheaves is due to Abbes and Saito. My approach provides also a new independent proof of Deligne-Kato's formula.
Yasuhiro Wakabayashi (University of Tokyo) 15:30-16:30
Explicit computation of the number of dormant opers and duality (Japanese)

### 2015/06/17

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Norifumi Seki (University of Tokyo)
Hodge-Tate weights of p-adic Galois representations and Banach representations of GL_2(Q_p)
(Japanese)

### 2015/05/27

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ippei Nagamachi (University of Tokyo)
On a good reduction criterion for polycurves with sections (Japanese)