## 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

### 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)

### 2015/05/20

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Shou-Wu Zhang (Princeton University)
Colmez' conjecture in average (English)
[ Abstract ]
This is a report on a joint work with Xinyi Yuan on a conjectured formula of Colmez about the Faltings heights of CM abelian varieties. I will sketch a deduction of this formula in average of CM types from our early work on Gross-Zagier formula. When combined with a recent work of Tsimerman, this result implies the Andre-Oort conjecture for the moduli of abelian varieties.
Our method is different than a recently announced proof of a weaker form of the average formula by Andreatta, Howard, Goren, and Madapusi Pera: we use neither high dimensional Shimura varieties nor Borcherds' liftings.

### 2015/04/08

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Seidai Yasuda (Osaka University)
Integrality of $p$-adic multiple zeta values and application to finite multiple zeta values.
(English)
[ Abstract ]
I will give a proof of an integrality of p-adic multiple zeta values. I would also like to explain how it can be applied to give an upper bound of the dimension of finite multiple zeta values.

### 2015/02/18

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Piotr Achinger (University of California, Berkeley)
Wild ramification and $K(\pi, 1)$ spaces (English)
[ Abstract ]
A smooth variety in characteristic zero is Zariski-locally a $K(\pi,1)$ space, i.e., has trivial higher homotopy groups. This fact is of crucial importance in Artin's proof that $\ell$-adic cohomology agrees with singular cohomology over $\mathbb{C}$. The characteristic $p$ variant of this is not known --- we do not even know whether the affine plane is a $K(\pi, 1)$ in positive characteristic! I will show how to reduce this question to a Bertini-type’' statement regarding wild ramification of $\ell$-adic local systems on affine spaces, which might be of independent interest. I will verify this statement in the special case of local systems of rank $1$ and speculate on how one might treat the general case.

### 2015/01/21

18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ofer Gabber (CNRS, IHES)
[ Abstract ]
(Joint work with Brian Conrad.) Let $K$ be a complete rank 1 valued field with ring of integers $O_K$, $A$ an adic noetherian ring and $f:A\to O_K$ an adic morphism. If $g:X\to Y$ is a proper flat morphism between rigid analytic spaces over $K$ then locally on $Y$ a flat formal model of $g$ spreads out to a proper flat morphism between formal schemes topologically of finite type over $A$. As an application one can prove that for proper smooth $g$ and $K$ of characteristic 0, the Hodge to de Rham spectral sequence for $g$ degenerates and the $R^q g_* \Omega^p_{X/Y}$ are locally free.

### 2015/01/14

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Laurent Berger (ENS de Lyon)
Iterate extensions and relative Lubin-Tate groups
[ Abstract ]
Let K be a p-adic field, let P(T) be a polynomial with coefficients in K, and let {$u_n$} be a sequence such that $P(u_{n+1}) = u_n$ for all n and $u_0$ belongs to K. The extension of K generated by the $u_n$ is called an iterate extension. I will discuss these extensions, show that under certain favorable conditions there is a theory of Coleman power series, and explain the relationship with relative Lubin-Tate groups.

### 2015/01/07

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Sandra Rozensztajn (ENS de Lyon)
Congruences of modular forms modulo p and a variant of the Breuil-Mézard conjecture (English)
[ Abstract ]
In this talk I will explain how a problem of congruences modulo p in the space of modular forms $S_k(\Gamma_0(p))$ is related to the geometry of some deformation spaces of Galois representations and can be solved by using a variant of the Breuil-Mézard conjecture.

### 2014/12/17

18:00-19:00   Room #117 (Graduate School of Math. Sci. Bldg.)
Konstantin Ardakov (University of Oxford)
Equivariant $\wideparen{\mathcal{D}}$ modules on rigid analytic spaces
(English)
[ Abstract ]
Locally analytic representations of p-adic Lie groups are of interest in several branches of arithmetic algebraic geometry, notably the p-adic local Langlands program. I will discuss some work in progress towards a Beilinson-Bernstein style localisation theorem for admissible locally analytic representations of semisimple compact p-adic Lie groups using equivariant formal models of rigid analytic flag varieties.

### 2014/11/19

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Fabien Pazuki (Univ Bordeaux and Univ Copenhagen)
Bad reduction of curves with CM jacobians (English)
[ Abstract ]
An abelian variety defined over a number field and having complex multiplication (CM) has potentially good reduction everywhere. If a curve of positive genus which is defined over a number field has good reduction at a given finite place, then so does its jacobian variety. However, the converse statement is false already in the genus 2 case, as can be seen in the entry $[I_0-I_0-m]$ in Namikawa and Ueno's classification table of fibres in pencils of curves of genus 2. In this joint work with Philipp Habegger, our main result states that this phenomenon prevails for certain families of curves.

We prove the following result: Let F be a real quadratic number field. Up to isomorphisms there are only finitely many curves C of genus 2 defined over $\overline{\mathbb{Q}}$ with good reduction everywhere and such that the jacobian Jac(C) has CM by the maximal order of a quartic, cyclic, totally imaginary number field containing F. Hence such a curve will almost always have stable bad reduction at some prime whereas its jacobian has good reduction everywhere. A remark is that one can exhibit an infinite family of genus 2 curves with CM jacobian such that the endomorphism ring is the ring of algebraic integers in a cyclic extension of $\mathbb{Q}$ of degree 4 that contains $\mathbb{Q}(\sqrt{5})$, for example.

### 2014/11/12

18:00-19:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Ruochuan Liu (BICMR)
Relative (φ, Γ)-modules (English)
[ Abstract ]
In this talk, we will introduce the theory of (φ, Γ)-modules for general adic spaces. This is a joint work with Kedlaya.

### 2014/10/28

16:40-18:50   Room #002 (Graduate School of Math. Sci. Bldg.)
Judith Ludwig (Imperial college) 16:40-17:40
[ Abstract ]
Let B be a definite quaternion algebra over the rationals, G the algebraic group defined by the units in B and H the subgroup of G of norm one elements. Then the classical transfer of automorphic representations from G to H is well understood thanks to Labesse and Langlands, who proved formulas for the multiplicity of irreducible admissible representations of H(adeles) in the discrete automorphic spectrum.
The goal of this talk is to prove a p-adic version of this transfer. By this we mean an extension of the classical transfer to p-adic families of automorphic forms as parametrized by certain rigid analytic spaces called eigenvarieties. We will prove the p-adic transfer by constructing a morphism between eigenvarieties, which agrees with the classical transfer on points corresponding to classical automorphic representations.
Jan Nekovar (Université Paris 6) 17:50-18:50
Plectic cohomology (English)

### 2014/10/14

17:30-18:30   Room #002 (Graduate School of Math. Sci. Bldg.)
Fabrizio Andreatta (Università Statale di Milano)
A p-adic criterion for good reduction of curves (ENGLISH)
[ Abstract ]
Given a curve over a dvr of mixed characteristic 0-p with smooth generic fiber and with semistable reduction, I will present a criterion for good reduction in terms of the (unipotent) p-adic étale fundamental group of its generic fiber.

### 2014/06/25

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Masahiko Takiguchi (University of Tokyo)
Periods of some two dimensional reducible p-adic representations and non-de Rham B-pairs (JAPANESE)

### 2014/06/17

17:30-18:30   Room #056 (Graduate School of Math. Sci. Bldg.)
Bao Châu Ngô (University of Chicago, VIASM)
Vinberg's monoid and automorphic L-functions (ENGLISH)
[ Abstract ]
We will explain a generalisation of the construction of the local factors of Godement-Jacquet's L-functions, based on Vinberg's monoid.

### 2014/05/28

16:40-17:40   Room #056 (Graduate School of Math. Sci. Bldg.)
Gantsooj Batzaya (University of Tokyo)
On simultaneous approximation to powers of a real number by rational numbers (ENGLISH)