## Number Theory Seminar

Seminar information archive ～04/22｜Next seminar｜Future seminars 04/23～

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

**Seminar information archive**

### 2020/05/13

17:30-18:30 Online

On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)

https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

**Yifeng Liu**(Yale University)On the Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)

[ Abstract ]

In this talk, we will explain the final outcome on the Beilinson-Bloch-Kato conjecture for motives coming from certain automorphic representations of GL(n) x GL(n+1), of our recent project with Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu. In particular, we show that the nonvanishing of the central L-value of the motive implies the vanishing of the corresponding Bloch-Kato Selmer group. We will also explain the main ideas and ingredients of the proof.

[ Reference URL ]In this talk, we will explain the final outcome on the Beilinson-Bloch-Kato conjecture for motives coming from certain automorphic representations of GL(n) x GL(n+1), of our recent project with Yichao Tian, Liang Xiao, Wei Zhang, and Xinwen Zhu. In particular, we show that the nonvanishing of the central L-value of the motive implies the vanishing of the corresponding Bloch-Kato Selmer group. We will also explain the main ideas and ingredients of the proof.

https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

### 2020/04/22

17:30-18:30 Online

Prismatic Dieudonné theory (ENGLISH)

**Arthur-César Le Bras**(CNRS & Université Paris 13)Prismatic Dieudonné theory (ENGLISH)

[ Abstract ]

I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.

I would like to explain a classification result for p-divisible groups, which unifies many of the existing results in the literature. The main tool is the theory of prisms and prismatic cohomology recently developed by Bhatt and Scholze. This is joint work with Johannes Anschütz.

### 2019/12/04

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

Characteristic epsilon cycles of l-adic sheaves on varieties (ENGLISH)

**Daichi Takeuchi**(University of Tokyo)Characteristic epsilon cycles of l-adic sheaves on varieties (ENGLISH)

[ Abstract ]

For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.

In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.

I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.

For l-adic sheaves on varieties over finite fields, the constant terms of the functional equations of the L-functions, called global epsilon factors, are important arithmetic invariants. When the varieties are curves, Deligne and Laumon show that they admit product formulae in terms of local epsilon factors.

In this talk, I will explain that, attaching some coefficients to irreducible components of singular supports, we can define refinements of characteristic cycles. We will see that, after taking modulo roots of unity, they give product formulae of global epsilon factors for higher dimensional varieties.

I will also explain that these results can be generalized to arbitrary perfect fields of any characteristic.

### 2019/11/27

17:00-18:00 Room #117 (Graduate School of Math. Sci. Bldg.)

The relative Hodge-Tate spectral sequence (ENGLISH)

**Ahmed Abbes**(CNRS & IHÉS)The relative Hodge-Tate spectral sequence (ENGLISH)

[ Abstract ]

It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.

It is well known that the p-adic étale cohomology of a smooth and proper variety over a p-adic field admits a Hodge-Tate decomposition and that it is the abutment of a spectral sequence called Hodge-Tate; these two properties are incidentally equivalent. The Hodge-Tate decomposition was generalized in higher dimensions to Hodge-Tate local systems by Hyodo, and was studied by Faltings, Tsuji and others. But the generalization of the Hodge-Tate spectral sequence to a relative situation has not yet been considered (not even conjectured), with the exception of a special case of abelian schemes by Hyodo. This has now been done in a joint work with Michel Gros. The relative Hodge-Tate spectral sequence that we construct takes place in the Faltings topos, but its construction requires the introduction of a relative variant of this topos which is the main novelty of our work. The relative Hodge-Tate spectral sequence sheds new light on the fact that the relative p-adic étale cohomology is Hodge-Tate, but the two properties are not equivalent in general.

### 2019/11/20

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)

**Vasudevan Srinivas**(Tata Institute of Fundamental Research)Algebraic versus topological entropy for surfaces over finite fields (ENGLISH)

[ Abstract ]

For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.

For an automorphism of an algebraic variety, we consider some properties of eigenvalues of the induced linear transformation on l-adic cohomology, motivated by some results from complex dynamics, related to the notion of entropy. This is a report on joint work with Hélène Esnault, and some subsequent work of K. Shuddhodan.

### 2019/10/16

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On slopes of modular forms (ENGLISH)

**Liang Xiao**(BICMR, Peking University)On slopes of modular forms (ENGLISH)

[ Abstract ]

In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin's crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.

In this talk, I will survey some recent progress towards understanding the slopes of modular forms, with or without level structures. This has direct application to the conjecture of Breuil-Buzzard-Emerton on the slopes of Kisin's crystabelline deformation spaces. In particular, we obtain certain refined version of the spectral halo conjecture, where we may identify explicitly the slopes at the boundary when given a reducible non-split generic residual local Galois representation. This is a joint work in progress with Ruochuan Liu, Nha Truong, and Bin Zhao.

### 2019/10/09

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Twisted doubling integrals for classical groups (ENGLISH)

**Yuanqing Cai**(Kyoto University)Twisted doubling integrals for classical groups (ENGLISH)

[ Abstract ]

In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.

In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.

In the 1980s, Piatetski-Shapiro and Rallis discovered a family of Rankin-Selberg integrals for the classical groups that did not rely on Whittaker models. This is the so-called doubling method. It grew out of Rallis' work on the inner products of theta lifts -- the Rallis inner product formula.

In this talk, we present a family of Rankin-Selberg integrals (the twisted doubling method, in joint work with Friedberg, Ginzburg, and Kaplan) for the tensor product L-function of a pair of automorphic cuspidal representations, one of a classical group, the other of a general linear group. This can be viewed as a generalization of the doubling integrals of Piatetski-Shapiro and Rallis. Time permitting, we will discuss the twisted doubling integrals for Brylinski-Deligne covers of classical groups.

### 2019/07/03

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Explicit calculation of values of the regulator maps on a certain type of Kummer surfaces (Japanese)

**Ken Sato**(University of Tokyo)Explicit calculation of values of the regulator maps on a certain type of Kummer surfaces (Japanese)

### 2019/06/12

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On extension of overconvergent log isocrystals on log smooth varieties (Japanese)

**Kazumi Kasaura**(University of Tokyo)On extension of overconvergent log isocrystals on log smooth varieties (Japanese)

### 2019/06/05

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)

**Shin Hattori**(Tokyo City University)Duality of Drinfeld modules and P-adic properties of Drinfeld modular forms (English)

[ Abstract ]

Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

Let p be a rational prime, q>1 a p-power and P a non-constant irreducible polynomial in F_q[t]. The notion of Drinfeld modular form is an analogue over F_q(t) of that of elliptic modular form. Numerical computations suggest that Drinfeld modular forms enjoy some P-adic structures comparable to the elliptic analogue, while at present their P-adic properties are less well understood than the p-adic elliptic case. In 1990s, Taguchi established duality theories for Drinfeld modules and also for a certain class of finite flat group schemes called finite v-modules. Using the duality for the latter, we can define a function field analogue of the Hodge-Tate map. In this talk, I will explain how the Taguchi's theory and our Hodge-Tate map yield results on Drinfeld modular forms which are classical to elliptic modular forms e.g. P-adic congruences of Fourier coefficients imply p-adic congruences of weights.

### 2019/05/29

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)

**Yasuhiro Oki**(University of Tokyo)On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)

### 2019/05/08

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

On the types for supercuspidal representations of inner forms of GL_n (Japanese)

**Yuki Yamamoto**(University of Tokyo)On the types for supercuspidal representations of inner forms of GL_n (Japanese)

### 2019/04/30

17:00-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

Moduli space of l-adic Langlands parameters and the stable Bernstein center (English)

**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.

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.

### 2019/04/24

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

P^1-localisation and a possible definition of arithmetic Kodaira-Spencer classes (English)

**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.)

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.)

### 2019/04/17

17:00-18:00 Room #122 (Graduate School of Math. Sci. Bldg.)

On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)

**Teppei Takamatsu**(University of Tokyo)On the Shafarevich conjecture for minimal surfaces of Kodaira dimension 0 (Japanese)

### 2019/04/10

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

The geometry of the affine Springer fibers and Arthur's weighted orbital integrals (English)

**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.

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.

### 2019/01/16

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

p-adic Gelfand-Kapranov-Zelevinsky systems (ENGLISH)

**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.

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.

### 2019/01/09

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Formal groups and p-adic dynamical systems (ENGLISH)

**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.

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.

### 2018/12/19

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

Cohomology vanishing for automorphic vector bundles (ENGLISH)

**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.

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.

### 2018/12/12

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A higher weight (and automorphic) generalization of the Hermite-Minkowski theorem (ENGLISH)

**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.

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.

### 2018/11/21

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Poncelet games, confinement of algebraic integers, and hyperbolic Ax-Schanuel (ENGLISH)

**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).

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).

### 2018/11/14

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

A motivic construction of ramification filtrations (ENGLISH)

**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.

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.

### 2018/10/10

18:00-19:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Beilinson-Bloch-Kato conjecture for Rankin-Selberg motives (ENGLISH)

**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.

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.

### 2018/06/20

17:00-18:00 Room #056 (Graduate School of Math. Sci. Bldg.)

Criteria for good reduction of hyperbolic polycurves (JAPANESE)

**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.

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.

### 2018/06/06

17:30-18:30 Room #056 (Graduate School of Math. Sci. Bldg.)

On the Ramanujan conjecture for automorphic forms over function fields

**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.

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.