## Number Theory Seminar

Seminar information archive ～04/21｜Next seminar｜Future seminars 04/22～

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**

### 2022/09/28

17:00-18:00 Hybrid

A K-theoretic approach to geometric representation theory (ENGLISH)

**Jens Eberhardt**(University of Wuppertal)A K-theoretic approach to geometric representation theory (ENGLISH)

[ Abstract ]

Perverse sheaves and intersection cohomology are central objects in geometric representation theory. This talk is about their long-lost K-theoretic cousins, called K-motives. We will discuss definitions and basic properties of K-motives and explore potential applications to geometric representation theory. For example, K-motives shed a new light on Beilinson--Ginzburg--Soergel's Koszul duality -- a remarkable symmetry in the representation theory and geometry of two Langlands dual reductive groups. We will see that this new form of Koszul duality does not involve any gradings or mixed geometry which are as essential as mysterious in the classical approaches.

Perverse sheaves and intersection cohomology are central objects in geometric representation theory. This talk is about their long-lost K-theoretic cousins, called K-motives. We will discuss definitions and basic properties of K-motives and explore potential applications to geometric representation theory. For example, K-motives shed a new light on Beilinson--Ginzburg--Soergel's Koszul duality -- a remarkable symmetry in the representation theory and geometry of two Langlands dual reductive groups. We will see that this new form of Koszul duality does not involve any gradings or mixed geometry which are as essential as mysterious in the classical approaches.

### 2022/07/20

15:30-18:00 Hybrid

Completed prismatic F-crystals and crystalline local systems (ENGLISH)

Twisted differential operators in several variables (ENGLISH)

**Koji Shimizu**(UC Berkeley) 15:30-16:30Completed prismatic F-crystals and crystalline local systems (ENGLISH)

[ Abstract ]

Bhatt and Scholze introduced the absolute prismatic site of a p-adic ring and proved the equivalence of categories between prismatic F-crystals and lattices in crystalline representations in the CDVR case with perfect residue field. We will define a wider category of completed prismatic F-crystals in the relative case and explain its relation to the category of crystalline local systems. This is joint work with Heng Du, Tong Liu, and Yong Suk Moon.

Bhatt and Scholze introduced the absolute prismatic site of a p-adic ring and proved the equivalence of categories between prismatic F-crystals and lattices in crystalline representations in the CDVR case with perfect residue field. We will define a wider category of completed prismatic F-crystals in the relative case and explain its relation to the category of crystalline local systems. This is joint work with Heng Du, Tong Liu, and Yong Suk Moon.

**Pierre Houedry**(Université de Caen) 17:00-18:00Twisted differential operators in several variables (ENGLISH)

[ Abstract ]

The aim of my presentation is to give an overview of the results I obtained during the first year of my PhD. The theory of $q$-differences equations appeared a long time ago with the Birkhoff's work. It is well understood in the complex setting. In 2004, Lucia Di Vizio and Yves André, in the paper $q$-differences and p-adic local monodromy, gave an equivalence between certain type of $q$-differences equations and a certain type of classical differential equations in the p-adic setting. Recently, Adolfo Quiros, Bernard Le Stum and Michel Gros have been working on a generalization of this result not looking only for $q$-differences equations but also twisted equations in general. The framework that they develop is working for equations in one variable. The goal of my thesis is to generalize those results in several variables.

The aim of my presentation is to give an overview of the results I obtained during the first year of my PhD. The theory of $q$-differences equations appeared a long time ago with the Birkhoff's work. It is well understood in the complex setting. In 2004, Lucia Di Vizio and Yves André, in the paper $q$-differences and p-adic local monodromy, gave an equivalence between certain type of $q$-differences equations and a certain type of classical differential equations in the p-adic setting. Recently, Adolfo Quiros, Bernard Le Stum and Michel Gros have been working on a generalization of this result not looking only for $q$-differences equations but also twisted equations in general. The framework that they develop is working for equations in one variable. The goal of my thesis is to generalize those results in several variables.

### 2022/07/06

17:00-18:00 Hybrid

The characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves (ENGLISH)

**Peijiang Liu**(University of Tokyo)The characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves (ENGLISH)

[ Abstract ]

$\ell$-adic GKZ hypergeometric sheaves are defined to be étale analogues of GKZ hypergeometric $\mathcal{D}$-modules. We introduce an algorithm of computing the characteristic cycles of certain type of $\ell$-adic GKZ hypergeometric sheaves. We compute the irreducible components by a push-forward formula for characteristic cycles of étale sheaves, and compute the multiplicities by considering a comparison theorem between the characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves and those of non-confluent GKZ hypergeometric $\mathcal{D}$-modules. We also explain the limitation of our algorithm by an example.

$\ell$-adic GKZ hypergeometric sheaves are defined to be étale analogues of GKZ hypergeometric $\mathcal{D}$-modules. We introduce an algorithm of computing the characteristic cycles of certain type of $\ell$-adic GKZ hypergeometric sheaves. We compute the irreducible components by a push-forward formula for characteristic cycles of étale sheaves, and compute the multiplicities by considering a comparison theorem between the characteristic cycles of non-confluent $\ell$-adic GKZ hypergeometric sheaves and those of non-confluent GKZ hypergeometric $\mathcal{D}$-modules. We also explain the limitation of our algorithm by an example.

### 2022/06/22

17:00-18:00 Hybrid

Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields (JAPANESE)

**Yugo Takanashi**(University of Tokyo)Parity of conjugate self-dual representations of inner forms of $\mathrm{GL}_n$ over $p$-adic fields (JAPANESE)

[ Abstract ]

There are two parametrizations of discrete series representations of $\mathrm{GL}_n$ over $p$-adic fields. One is the local Langlands correspondence, and the other is the local Jacquet-Langlands correspondence. The composite of these two maps the discrete series representations of an inner form of $\mathrm{GL}_n$ to Galois representations called discrete L-parameters. On the other hand, we can define the parity for each self-dual representation depending on whether the representation is orthogonal or symplectic. The composite preserves the notion of self-duality, and it transforms the parity in a nontrivial manner. Prasad and Ramakrishnan computed the transformation law, and Mieda proved its conjugate self-dual analog under some conditions on groups and representations. We will talk about the proof of the general case of this analog. We use the globalization method, as in the proof of Prasad and Ramakrishnan.

There are two parametrizations of discrete series representations of $\mathrm{GL}_n$ over $p$-adic fields. One is the local Langlands correspondence, and the other is the local Jacquet-Langlands correspondence. The composite of these two maps the discrete series representations of an inner form of $\mathrm{GL}_n$ to Galois representations called discrete L-parameters. On the other hand, we can define the parity for each self-dual representation depending on whether the representation is orthogonal or symplectic. The composite preserves the notion of self-duality, and it transforms the parity in a nontrivial manner. Prasad and Ramakrishnan computed the transformation law, and Mieda proved its conjugate self-dual analog under some conditions on groups and representations. We will talk about the proof of the general case of this analog. We use the globalization method, as in the proof of Prasad and Ramakrishnan.

### 2022/06/20

15:00-16:00 Hybrid

p-adic weight-monodromy conjecture for complete intersections (Japanese)

**Hiroki Kato**(Paris-Saclay University)p-adic weight-monodromy conjecture for complete intersections (Japanese)

### 2022/06/15

17:00-18:00 Hybrid

Steinberg symbols and reciprocity sheaves (JAPANESE)

**Junnosuke Koizumi**(University of Tokyo)Steinberg symbols and reciprocity sheaves (JAPANESE)

[ Abstract ]

The norm residue symbol and the differential symbol are known to satisfy the common relation $(a,1-a)=0$ which is called the Steinberg relation. Hu-Kriz showed that the Steinberg relation can be understood as a relation between certain morphisms in the stable motivic homotopy category. On the other hand, there is also an “additive variant” of the Steinberg relation, namely $(a,a)+(1-a,1-a)=0$, for which the classical motivic theory is no longer applicable. In this talk we will explain how the theory of reciprocity sheaves due to Kahn-Saito-Yamazaki can be utilized to generalize the theory of Hu-Kriz to include the additive Steinberg relation.

The norm residue symbol and the differential symbol are known to satisfy the common relation $(a,1-a)=0$ which is called the Steinberg relation. Hu-Kriz showed that the Steinberg relation can be understood as a relation between certain morphisms in the stable motivic homotopy category. On the other hand, there is also an “additive variant” of the Steinberg relation, namely $(a,a)+(1-a,1-a)=0$, for which the classical motivic theory is no longer applicable. In this talk we will explain how the theory of reciprocity sheaves due to Kahn-Saito-Yamazaki can be utilized to generalize the theory of Hu-Kriz to include the additive Steinberg relation.

### 2022/05/25

17:00-18:00 Hybrid

Torsion points of elliptic curves over cyclotomic fields (JAPANESE)

**Koji Matsuda**(University of Tokyo)Torsion points of elliptic curves over cyclotomic fields (JAPANESE)

[ Abstract ]

By Mordell--Weil theorem, the Mordell--Weil groups of elliptic curves over number fields are finitely generated, and in particular their torsion subgroups are finite. For a fixed elliptic curve, it is easy to compute its torsion subgroups. Conversely using modular curves, we can study the possible torsion subgroups of elliptic curves. More precisely, the existence of an elliptic curve with certain torsion points is essentially equivalent to the existence of certain rational points of a modular curve. In this talk, in order to study the rational points of modular curves over cyclotomic fields, we compute the Mordell--Weil ranks of their Jacobian varieties over cyclotomic fields.

By Mordell--Weil theorem, the Mordell--Weil groups of elliptic curves over number fields are finitely generated, and in particular their torsion subgroups are finite. For a fixed elliptic curve, it is easy to compute its torsion subgroups. Conversely using modular curves, we can study the possible torsion subgroups of elliptic curves. More precisely, the existence of an elliptic curve with certain torsion points is essentially equivalent to the existence of certain rational points of a modular curve. In this talk, in order to study the rational points of modular curves over cyclotomic fields, we compute the Mordell--Weil ranks of their Jacobian varieties over cyclotomic fields.

### 2022/05/18

17:00-18:00 Hybrid

Local Langlands correspondence for non-quasi-split odd special orthogonal groups (JAPANESE)

**Hiroshi Ishimoto**(University of Tokyo)Local Langlands correspondence for non-quasi-split odd special orthogonal groups (JAPANESE)

[ Abstract ]

In 2013, Arthur established the endoscopic classification of representations of quasi-split symplectic and orthogonal groups, and Mok analogously proved the similar classification for quasi-split unitary groups. In 2014, Kaletha-Minguez-Shin-White established the classification for non-quasi-spilt unitary groups assuming Mok's results. Similarly, we can prove that for non-quasi-split odd orthogonal groups assuming Arthur's results. In this talk, I will explain the local Langlands correspondence for non-quasi-split odd special orthogonal groups, which is a part of the classification of representations.

In 2013, Arthur established the endoscopic classification of representations of quasi-split symplectic and orthogonal groups, and Mok analogously proved the similar classification for quasi-split unitary groups. In 2014, Kaletha-Minguez-Shin-White established the classification for non-quasi-spilt unitary groups assuming Mok's results. Similarly, we can prove that for non-quasi-split odd orthogonal groups assuming Arthur's results. In this talk, I will explain the local Langlands correspondence for non-quasi-split odd special orthogonal groups, which is a part of the classification of representations.

### 2022/05/11

17:00-18:00 Hybrid

Cohomology of the unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ (ENGLISH)

**Joseph Muller**(University of Tokyo)Cohomology of the unramified PEL unitary Rapoport-Zink space of signature $(1,n-1)$ (ENGLISH)

[ Abstract ]

Rapoport-Zink (RZ) spaces are moduli spaces which classify the deformations of a $p$-divisible group with additional structures. It is equipped with compatible actions of $p$-adic and Galois groups, and their cohomology is believed to play a role in the local Langlands program. So far, the cohomology of RZ spaces is entirely known only in the cases of the Lubin-Tate tower and of the Drinfeld space ; in particular both of them are RZ spaces of EL type. In this talk, we consider the unramified PEL unitary RZ space with signature $(1,n-1)$. In 2011, Vollaard and Wedhorn proved that it is stratified by generalized Deligne-Lusztig varieties, whose incidence relations mimic the combinatorics of the Bruhat-Tits building of a unitary group. We compute the cohomology of these strata and we draw some consequences on the cohomology of the RZ space. When $n = 3, 4$ we deduce

an automorphic description of the cohomology of the basic stratum in the corresponding Shimura variety via p-adic uniformization.

Rapoport-Zink (RZ) spaces are moduli spaces which classify the deformations of a $p$-divisible group with additional structures. It is equipped with compatible actions of $p$-adic and Galois groups, and their cohomology is believed to play a role in the local Langlands program. So far, the cohomology of RZ spaces is entirely known only in the cases of the Lubin-Tate tower and of the Drinfeld space ; in particular both of them are RZ spaces of EL type. In this talk, we consider the unramified PEL unitary RZ space with signature $(1,n-1)$. In 2011, Vollaard and Wedhorn proved that it is stratified by generalized Deligne-Lusztig varieties, whose incidence relations mimic the combinatorics of the Bruhat-Tits building of a unitary group. We compute the cohomology of these strata and we draw some consequences on the cohomology of the RZ space. When $n = 3, 4$ we deduce

an automorphic description of the cohomology of the basic stratum in the corresponding Shimura variety via p-adic uniformization.

### 2022/04/27

17:00-18:00 Hybrid

The Kaneko-Zagier conjecture on finite and symmetric multiple zeta values for general integer indices (JAPANESE)

**Shuji Yamamoto**(University of Tokyo)The Kaneko-Zagier conjecture on finite and symmetric multiple zeta values for general integer indices (JAPANESE)

[ Abstract ]

Kaneko and Zagier introduced two variants of multiple zeta values, which we call A-MZVs and S-MZVs, and conjectured that the algebraic structures of them are isomorphic. While these values were originally defined for positive integer (multi-)indices, recently, Komori extended the definition of S-MZVs to general integer indices. Since A-MZVs can also be defined for general integers, Komori's work suggests a generalization of the Kaneko-Zagier conjecture, from positive to general integers. In this talk, we will show how this generalization is reduced to the original conjecture. This is a joint work with Masataka Ono.

Kaneko and Zagier introduced two variants of multiple zeta values, which we call A-MZVs and S-MZVs, and conjectured that the algebraic structures of them are isomorphic. While these values were originally defined for positive integer (multi-)indices, recently, Komori extended the definition of S-MZVs to general integer indices. Since A-MZVs can also be defined for general integers, Komori's work suggests a generalization of the Kaneko-Zagier conjecture, from positive to general integers. In this talk, we will show how this generalization is reduced to the original conjecture. This is a joint work with Masataka Ono.

### 2022/04/20

17:00-18:00 Hybrid

On generalized Fuchs theorem over p-adic polyannuli (ENGLISH)

**Peiduo Wang**(University of Tokyo)On generalized Fuchs theorem over p-adic polyannuli (ENGLISH)

[ Abstract ]

In this talk, we study finite projective differential modules on p-adic polyannuli satisfying the Robba condition. Christol and Mebkhout proved the decomposition theorem (the p-adic Fuchs theorem) of such differential modules on one dimensional p-adic annuli under certain non-Liouvilleness assumption and Gachets generalized it to higher dimensional cases. On the other hand, Kedlaya proved a generalization of the p-adic Fuchs theorem in one dimensional case. We prove Kedlaya's generalized version of p-adic Fuchs theorem in higher dimensional cases.

In this talk, we study finite projective differential modules on p-adic polyannuli satisfying the Robba condition. Christol and Mebkhout proved the decomposition theorem (the p-adic Fuchs theorem) of such differential modules on one dimensional p-adic annuli under certain non-Liouvilleness assumption and Gachets generalized it to higher dimensional cases. On the other hand, Kedlaya proved a generalization of the p-adic Fuchs theorem in one dimensional case. We prove Kedlaya's generalized version of p-adic Fuchs theorem in higher dimensional cases.

### 2021/12/22

17:00-18:00 Online

Some properties of the Hecke eigenclasses of the mod p-cohomology of Shimura curves (English)

**Stefano Morra**(Paris 8 University)Some properties of the Hecke eigenclasses of the mod p-cohomology of Shimura curves (English)

[ Abstract ]

The mod p local Langlands program, foreseen by Serre and proposed in precise terms by C. Breuil after his p-divisible groups computations in the Breuil-Conrad-Diamond-Taylor proof of the Shimura-Taiyama-Weil conjecture, was realized in the particular case of GL_2(\mathbf{Q}_p) thanks to a vast convergence of new tools: classification of mod p-representations of GL_2(\mathbf{Q}_p), local Galois deformation techniques, local-global compatibility arguments.

When trying to extend these conjectures to more general groups, multiple problems arise (lack of classification results for smooth mod p-representations of p-adic groups, absence of explicit integral models for Galois representations with the relevant p-adic Hodge theory conditions), and the only way to formulate, and test, conjectures on a mod p local Langlands correspondence relies on its expected realization in Hecke eigenclasses of Shimura varieties (or, in other words, the expectation of a local-global compatibility of the Langlands correspondence).

In this talk we describe some properties of Hecke isotypical spaces of the mod p-cohomology of Shimura curves with infinite level at p, when the reflex field F is unramified at p and the Shimura curve arises from a quaternion algebra which is split at p. These Hecke isotypical spaces are expected to be the “good” smooth mod p-representations of GL_2(F_{\mathfrak{p}}) attached to mod p Galois representations of Gal(\overline{\mathbf{Q}_p}/F_{\mathfrak{p}}) via the expected local Langlands correspondence mod p. We will in particular comment on their Gelfand-Kirillov dimension, and their irreducibility (in particular, the finite length of these Hecke eigenspaces as GL_2(F_{\mathfrak{p}})-representations).

This is a report on a series of work joint with C. Breuil, F. Herzig, Y. Hu et B. Schraen.

The mod p local Langlands program, foreseen by Serre and proposed in precise terms by C. Breuil after his p-divisible groups computations in the Breuil-Conrad-Diamond-Taylor proof of the Shimura-Taiyama-Weil conjecture, was realized in the particular case of GL_2(\mathbf{Q}_p) thanks to a vast convergence of new tools: classification of mod p-representations of GL_2(\mathbf{Q}_p), local Galois deformation techniques, local-global compatibility arguments.

When trying to extend these conjectures to more general groups, multiple problems arise (lack of classification results for smooth mod p-representations of p-adic groups, absence of explicit integral models for Galois representations with the relevant p-adic Hodge theory conditions), and the only way to formulate, and test, conjectures on a mod p local Langlands correspondence relies on its expected realization in Hecke eigenclasses of Shimura varieties (or, in other words, the expectation of a local-global compatibility of the Langlands correspondence).

In this talk we describe some properties of Hecke isotypical spaces of the mod p-cohomology of Shimura curves with infinite level at p, when the reflex field F is unramified at p and the Shimura curve arises from a quaternion algebra which is split at p. These Hecke isotypical spaces are expected to be the “good” smooth mod p-representations of GL_2(F_{\mathfrak{p}}) attached to mod p Galois representations of Gal(\overline{\mathbf{Q}_p}/F_{\mathfrak{p}}) via the expected local Langlands correspondence mod p. We will in particular comment on their Gelfand-Kirillov dimension, and their irreducibility (in particular, the finite length of these Hecke eigenspaces as GL_2(F_{\mathfrak{p}})-representations).

This is a report on a series of work joint with C. Breuil, F. Herzig, Y. Hu et B. Schraen.

### 2021/11/24

17:00-18:00 Online

On the formal degree conjecture for non-singular supercuspidal representations (Japanese)

**Kazuma Ohara**(University of Tokyo)On the formal degree conjecture for non-singular supercuspidal representations (Japanese)

[ Abstract ]

We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein's work proving the formal degree conjecture for regular supercuspidal representations. The main difference between our work and Schwein's work is that in non-singular case, the Deligne--Lusztig representations can be reducible, and the $S$-groups are not necessarily abelian. Therefore, we have to compare the dimensions of irreducible constituents of the Deligne--Lusztig representations and the dimensions of irreducible representations of $S$-groups.

We prove the formal degree conjecture for non-singular supercuspidal representations based on Schwein's work proving the formal degree conjecture for regular supercuspidal representations. The main difference between our work and Schwein's work is that in non-singular case, the Deligne--Lusztig representations can be reducible, and the $S$-groups are not necessarily abelian. Therefore, we have to compare the dimensions of irreducible constituents of the Deligne--Lusztig representations and the dimensions of irreducible representations of $S$-groups.

### 2021/10/20

17:00-18:00 Online

Geometric arc fundamental group (English)

**Alex Youcis**(University of Tokyo)Geometric arc fundamental group (English)

[ Abstract ]

Unlike algebraic geometry, the correct notion for a ‘covering space’ of a rigid analytic variety is non-obvious to define. In particular, the class of finite etale covering spaces doesn’t encompass many real world examples of ‘covering space’-like maps (e.g. Tate’s uniformization of elliptic curves, or period mappings of Rapoport—Zink spaces). In de Jong’s seminal work on the topic he made great strides forward by studying a notion of covering space, suggested by work of Berkovich, which includes many previous ‘covering spaces’ which are not finite etale and is rich enough to support a theory of a fundamental group.

Unfortunately, de Jong’s notion of covering space lacks many of the natural properties one would expect from the notion of a ‘covering space’. In this talk we discuss recent work of Achinger, Lara, and myself which proposes a larger class of ‘covering spaces’ than those considered by de Jong which enjoys the geometric properties missing from de Jong’s picture. In addition, we mention how this larger category is related to work of Scholze on pro-etale local systems as well as work of Bhatt and Scholze on the pro-etale fundamental group of a scheme.

Unlike algebraic geometry, the correct notion for a ‘covering space’ of a rigid analytic variety is non-obvious to define. In particular, the class of finite etale covering spaces doesn’t encompass many real world examples of ‘covering space’-like maps (e.g. Tate’s uniformization of elliptic curves, or period mappings of Rapoport—Zink spaces). In de Jong’s seminal work on the topic he made great strides forward by studying a notion of covering space, suggested by work of Berkovich, which includes many previous ‘covering spaces’ which are not finite etale and is rich enough to support a theory of a fundamental group.

Unfortunately, de Jong’s notion of covering space lacks many of the natural properties one would expect from the notion of a ‘covering space’. In this talk we discuss recent work of Achinger, Lara, and myself which proposes a larger class of ‘covering spaces’ than those considered by de Jong which enjoys the geometric properties missing from de Jong’s picture. In addition, we mention how this larger category is related to work of Scholze on pro-etale local systems as well as work of Bhatt and Scholze on the pro-etale fundamental group of a scheme.

### 2021/07/07

17:00-18:00 Online

On the BSD conjecture for the quadratic twists of the elliptic curve $X_0(49)$ (Japanese)

**Takumi Yoshida**(Keio University)On the BSD conjecture for the quadratic twists of the elliptic curve $X_0(49)$ (Japanese)

[ Abstract ]

The full BSD conjecture (the full Birch-Swinnerton-Dyer conjecture) is the important conjecture, which connects the algebraic invariants and analytic invariants of elliptic curves. When the elliptic curve is defined over $\mathbb{Q}$, these invariants are known to be rational numbers. Now, even when the elliptic curve is defined over $\mathbb{Q}$ and the $L$-function is not $0$ at $s=1$, it is not shown that the $2$-orders of these invariants are equal. Coates, Kim, Liang and Zhao proved the full BSD conjecture for some quadratic twists of $X_0(49)$, by proving that these $2$-orders are same. We extends this result, and prove the full BSD conjecture for more twists.

The full BSD conjecture (the full Birch-Swinnerton-Dyer conjecture) is the important conjecture, which connects the algebraic invariants and analytic invariants of elliptic curves. When the elliptic curve is defined over $\mathbb{Q}$, these invariants are known to be rational numbers. Now, even when the elliptic curve is defined over $\mathbb{Q}$ and the $L$-function is not $0$ at $s=1$, it is not shown that the $2$-orders of these invariants are equal. Coates, Kim, Liang and Zhao proved the full BSD conjecture for some quadratic twists of $X_0(49)$, by proving that these $2$-orders are same. We extends this result, and prove the full BSD conjecture for more twists.

### 2021/06/30

17:00-18:00 Online

Prismatic and q-crystalline sites of higher level (Japanese)

**Kimihiko Li**(University of Tokyo)Prismatic and q-crystalline sites of higher level (Japanese)

[ Abstract ]

Two new p-adic cohomology theories, called prismatic cohomology and q-crystalline cohomology, were defined for generalizing crystalline cohomology and they recover most known integral p-adic cohomology theories. On the other hand, higher level crystalline cohomology was defined for constructing p-adic cohomology theory over a ramified base. In this talk, for a positive integer m, we will give a construction of the level m primastic and q-crystalline sites and prove a certain equivalence between the category of crystals on the m-prismatic site or the m-q-crystalline site and that on the usual prismatic site or the usual q-crystalline site, which can be regarded as the prismatic analogue of the Frobenius descent. We will also prove the equivalence between the category of crystals on the m-prismatic site and that on the (m-1)-q-crystalline site.

Two new p-adic cohomology theories, called prismatic cohomology and q-crystalline cohomology, were defined for generalizing crystalline cohomology and they recover most known integral p-adic cohomology theories. On the other hand, higher level crystalline cohomology was defined for constructing p-adic cohomology theory over a ramified base. In this talk, for a positive integer m, we will give a construction of the level m primastic and q-crystalline sites and prove a certain equivalence between the category of crystals on the m-prismatic site or the m-q-crystalline site and that on the usual prismatic site or the usual q-crystalline site, which can be regarded as the prismatic analogue of the Frobenius descent. We will also prove the equivalence between the category of crystals on the m-prismatic site and that on the (m-1)-q-crystalline site.

### 2021/06/23

17:00-18:00 Online

Ramification groups of some finite Galois extensions of maximal nilpotency class over local fields of positive characteristic (Japanese)

**Koto Imai**(University of Tokyo)Ramification groups of some finite Galois extensions of maximal nilpotency class over local fields of positive characteristic (Japanese)

[ Abstract ]

Galois extensions of local fields is one of the most important subjects in the field of number theory. A ramification filtration is a filtration of a Galois group used to investigate the ramification of the extension. It is particularly useful when the extension is wildly ramified. In this talk, we examine the ramification groups of finite Galois extensions over complete discrete valuation fields of characteristic $p>0$. Brylinski calculated the ramification groups in the case where the Galois groups are abelian. We extend the results of Brylinski to some non-abelian cases where the Galois groups are of order $\leq p^{p+1}$ and of maximal nilpotency class.

Galois extensions of local fields is one of the most important subjects in the field of number theory. A ramification filtration is a filtration of a Galois group used to investigate the ramification of the extension. It is particularly useful when the extension is wildly ramified. In this talk, we examine the ramification groups of finite Galois extensions over complete discrete valuation fields of characteristic $p>0$. Brylinski calculated the ramification groups in the case where the Galois groups are abelian. We extend the results of Brylinski to some non-abelian cases where the Galois groups are of order $\leq p^{p+1}$ and of maximal nilpotency class.

### 2021/06/16

17:00-18:00 Online

Hecke eigensystems of automorphic forms (mod p) of Hodge type and algebraic modular forms (Japanese)

**Yasuhiro Terakado**(National Center for Theoretical Sciences)Hecke eigensystems of automorphic forms (mod p) of Hodge type and algebraic modular forms (Japanese)

[ Abstract ]

In a 1987 letter to Tate, Serre showed that the prime-to-p Hecke eigensystems arising in the space of mod p modular forms are the same as those appearing in the space of automorphic forms on a quaternion algebra. This result is regarded as a mod p analogue of the Jacquet-Langlands correspondence. In this talk, we give a generalization of Serre's result to the Hecke eigensystems of mod p automorphic forms on a Shimura variety of Hodge type with good reduction at p. This is joint work with Chia-Fu Yu.

In a 1987 letter to Tate, Serre showed that the prime-to-p Hecke eigensystems arising in the space of mod p modular forms are the same as those appearing in the space of automorphic forms on a quaternion algebra. This result is regarded as a mod p analogue of the Jacquet-Langlands correspondence. In this talk, we give a generalization of Serre's result to the Hecke eigensystems of mod p automorphic forms on a Shimura variety of Hodge type with good reduction at p. This is joint work with Chia-Fu Yu.

### 2021/05/26

17:00-18:00 Online

Geometric Structure of Affine Deligne-Lusztig Varieties for $\mathrm{GL}_3$ (Japanese)

**Ryosuke Shimada**(University of Tokyo)Geometric Structure of Affine Deligne-Lusztig Varieties for $\mathrm{GL}_3$ (Japanese)

[ Abstract ]

The Langlands correspondence, which contains class field theory as a special case, is one of the most important topics in number theory. Shimura varieties have been used, with great success, towards applications in the realm of the Langlands program. In this context, geometric and homological properties of affine Deligne-Lusztig varieties have been used to examine Shimura varieties and the local Langlands correspondence.

In this talk we study the geometric structure of affine Deligne-Lusztig varieties $X_{\lambda}(b)$ for $\mathrm{GL}_3$ and $b$ basic.

We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.

The Langlands correspondence, which contains class field theory as a special case, is one of the most important topics in number theory. Shimura varieties have been used, with great success, towards applications in the realm of the Langlands program. In this context, geometric and homological properties of affine Deligne-Lusztig varieties have been used to examine Shimura varieties and the local Langlands correspondence.

In this talk we study the geometric structure of affine Deligne-Lusztig varieties $X_{\lambda}(b)$ for $\mathrm{GL}_3$ and $b$ basic.

We completely determine the irreducible components of the affine Deligne-Lusztig variety. In particular, we classify the cases where all of the irreducible components are classical Deligne-Lusztig varieties times finite-dimensional affine spaces. If this is the case, then the irreducible components are pairwise disjoint.

### 2021/03/10

17:00-18:00 Online

Geometric Satake equivalence in mixed characteristic and Springer correspondence (Japanese)

**Katsuyuki Bando**(University of Tokyo)Geometric Satake equivalence in mixed characteristic and Springer correspondence (Japanese)

[ Abstract ]

The geometric Satake correspondence is an equivalence between the category of equivariant perverse sheaves on the affine Grassmannian and the category of representations of the Langlands dual group. It is known that there is a mixed characteristic version of the geometric Satake correspondence. The Springer correspondence is a correspondence between the category of equivariant perverse sheaves on the nilpotent cone and the category of representation of the Weyl group. In this talk, we will explain some relation between these two correspondences, including the mixed characteristic case.

The geometric Satake correspondence is an equivalence between the category of equivariant perverse sheaves on the affine Grassmannian and the category of representations of the Langlands dual group. It is known that there is a mixed characteristic version of the geometric Satake correspondence. The Springer correspondence is a correspondence between the category of equivariant perverse sheaves on the nilpotent cone and the category of representation of the Weyl group. In this talk, we will explain some relation between these two correspondences, including the mixed characteristic case.

### 2021/01/20

17:00-18:00 Online

Overconvergent Lubin-Tate $(\varphi, \Gamma)$-modules for different uniformizers (Japanese)

**Yuta Saito**(University of Tokyo)Overconvergent Lubin-Tate $(\varphi, \Gamma)$-modules for different uniformizers (Japanese)

[ Abstract ]

$(\varphi, \Gamma)$-modules are used for investigating p-adic Galois representations, which has an important role in constructing the p-adic local Langlands correspondence for GL_2(Q_p). When we try to construct the p-adic local correspondence for GL_2(F) for a general local field F, we want more useful and more suitable $(\varphi, \Gamma)$-modules. Lubin-Tate $(\varphi, \Gamma)$-modules are the candidates for such $(\varphi, \Gamma)$-modules. Lubin-Tate extensions are used for defining Lubin-Tate $(\varphi, \Gamma)$-modules. However, these extensions depend on the choice of uniformizers and the behavior of Lubin-Tate $(\varphi, \Gamma)$-modules for different uniformizers has not been discussed so much. We focus on overconvergency and discuss the coincidence for 2-dimensional triangulable $(\varphi, \Gamma)$-modules for different uniformizers.

$(\varphi, \Gamma)$-modules are used for investigating p-adic Galois representations, which has an important role in constructing the p-adic local Langlands correspondence for GL_2(Q_p). When we try to construct the p-adic local correspondence for GL_2(F) for a general local field F, we want more useful and more suitable $(\varphi, \Gamma)$-modules. Lubin-Tate $(\varphi, \Gamma)$-modules are the candidates for such $(\varphi, \Gamma)$-modules. Lubin-Tate extensions are used for defining Lubin-Tate $(\varphi, \Gamma)$-modules. However, these extensions depend on the choice of uniformizers and the behavior of Lubin-Tate $(\varphi, \Gamma)$-modules for different uniformizers has not been discussed so much. We focus on overconvergency and discuss the coincidence for 2-dimensional triangulable $(\varphi, \Gamma)$-modules for different uniformizers.

### 2020/12/16

17:00-18:00 Online

Rigid analytic Hyodo--Kato theory with syntomic coefficients (Japanese)

**Kazuki Yamada**(Keio University)Rigid analytic Hyodo--Kato theory with syntomic coefficients (Japanese)

[ Abstract ]

The Hyodo—Kato theory is the study of comparison between Hyodo—Kato cohomology and de Rham cohomology associated to semistable schemes over complete discrete valuation rings of mixed characteristic $(0,p)$.

In this talk, we will give a rigid analytic reconstruction of Hyodo—Kato theory and study coefficients of cohomology.

Our construction is useful for explicit computation and treatment of base extension, because it gives us a natural interpretation of the dependence of Hyodo—Kato theory on the choice of a branch of the $p$-adic logarithm.

The results of this talk are based on a joint work with Veronika Ertl, which deals with the case of trivial coefficient.

The Hyodo—Kato theory is the study of comparison between Hyodo—Kato cohomology and de Rham cohomology associated to semistable schemes over complete discrete valuation rings of mixed characteristic $(0,p)$.

In this talk, we will give a rigid analytic reconstruction of Hyodo—Kato theory and study coefficients of cohomology.

Our construction is useful for explicit computation and treatment of base extension, because it gives us a natural interpretation of the dependence of Hyodo—Kato theory on the choice of a branch of the $p$-adic logarithm.

The results of this talk are based on a joint work with Veronika Ertl, which deals with the case of trivial coefficient.

### 2020/11/18

17:00-18:00 Online

Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic (Japanese)

**Daichi Takeuchi**(University of Tokyo)Symmetric bilinear forms and local epsilon factors of isolated singularities in positive characteristic (Japanese)

[ Abstract ]

For a function on a smooth variety with an isolated singular point, we have two invariants. One is a non-degenerate symmetric bilinear form (de Rham), and the other is the vanishing cycles complex (\'etale). The latter is a Galois representation of a local field measuring a complexity of the singularity.

In this talk, I will give a formula which expresses the local epsilon factor of the vanishing cycles complex in terms of the bilinear form. In particular, the sign of the local epsilon factor is determined by the discriminant of the bilinear form. This can be regarded as a refinement of Milnor formula in SGA 7, which compares the rank of the bilinear form and the total dimension of the vanishing cycles.

In characteristic 2, we find a generalization of Arf invariant, which can be regarded as an invariant for a non-degenerate quadratic singularity, to a general isolated singularity.

For a function on a smooth variety with an isolated singular point, we have two invariants. One is a non-degenerate symmetric bilinear form (de Rham), and the other is the vanishing cycles complex (\'etale). The latter is a Galois representation of a local field measuring a complexity of the singularity.

In this talk, I will give a formula which expresses the local epsilon factor of the vanishing cycles complex in terms of the bilinear form. In particular, the sign of the local epsilon factor is determined by the discriminant of the bilinear form. This can be regarded as a refinement of Milnor formula in SGA 7, which compares the rank of the bilinear form and the total dimension of the vanishing cycles.

In characteristic 2, we find a generalization of Arf invariant, which can be regarded as an invariant for a non-degenerate quadratic singularity, to a general isolated singularity.

### 2020/06/17

17:30-18:30 Online

On modular representations of GL_2(L) for unramified L (ENGLISH)

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

**Christophe Breuil**(CNRS, Université Paris-Sud)On modular representations of GL_2(L) for unramified L (ENGLISH)

[ Abstract ]

Let p be a prime number and L a finite unramified extension of Q_p. We give a survey of past and new results on smooth admissible representations of GL_2(L) that appear in mod p cohomology. This is joint work with Florian Herzig, Yongquan Hu, Stefano Morra and Benjamin Schraen.

[ Reference URL ]Let p be a prime number and L a finite unramified extension of Q_p. We give a survey of past and new results on smooth admissible representations of GL_2(L) that appear in mod p cohomology. This is joint work with Florian Herzig, Yongquan Hu, Stefano Morra and Benjamin Schraen.

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

### 2020/05/27

17:30-18:30 Online

Shintani generating class and the p-adic polylogarithm for totally real fields (ENGLISH)

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

**Kenichi Bannai**(Keio University/RIKEN)Shintani generating class and the p-adic polylogarithm for totally real fields (ENGLISH)

[ Abstract ]

In this talk, we will give a new interpretation of Shintani's work concerning the generating function of nonpositive values of Hecke $L$-functions for totally real fields. In particular, we will construct a canonical class, which we call the Shintani generating class, in the cohomology of a certain quotient stack of an infinite direct sum of algebraic tori associated with a fixed totally real field. Using our observation that cohomology classes, not functions, play an important role in the higher dimensional case, we proceed to newly define the p-adic polylogarithm function in this case, and investigate its relation to the special value of p-adic Hecke $L$-functions. Some observations concerning the quotient stack will also be discussed. This is a joint work with Kei Hagihara, Kazuki Yamada, and Shuji Yamamoto.

[ Reference URL ]In this talk, we will give a new interpretation of Shintani's work concerning the generating function of nonpositive values of Hecke $L$-functions for totally real fields. In particular, we will construct a canonical class, which we call the Shintani generating class, in the cohomology of a certain quotient stack of an infinite direct sum of algebraic tori associated with a fixed totally real field. Using our observation that cohomology classes, not functions, play an important role in the higher dimensional case, we proceed to newly define the p-adic polylogarithm function in this case, and investigate its relation to the special value of p-adic Hecke $L$-functions. Some observations concerning the quotient stack will also be discussed. This is a joint work with Kei Hagihara, Kazuki Yamada, and Shuji Yamamoto.

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