## 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

### 2021/12/22

17:00-18:00   Online
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.

### 2021/11/24

17:00-18:00   Online
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.

### 2021/10/20

17:00-18:00   Online
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.

### 2021/07/07

17:00-18:00   Online
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.

### 2021/06/30

17:00-18:00   Online
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.

### 2021/06/23

17:00-18:00   Online
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.

### 2021/06/16

17:00-18:00   Online
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.

### 2021/05/26

17:00-18:00   Online
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.

### 2021/03/10

17:00-18:00   Online
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.

### 2021/01/20

17:00-18:00   Online
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.

### 2020/12/16

17:00-18:00   Online
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.

### 2020/11/18

17:00-18:00   Online
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.

### 2020/06/17

17:30-18:30   Online
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 ]
https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

### 2020/05/27

17:30-18:30   Online
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 ]
https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

### 2020/05/13

17:30-18:30   Online
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 ]
https://www.ms.u-tokyo.ac.jp/~t-saito/todai_IHES.html

### 2020/04/22

17:30-18:30   Online
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.

### 2019/12/04

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

### 2019/11/27

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

### 2019/11/20

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

### 2019/10/16

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

### 2019/10/09

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

### 2019/07/03

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

### 2019/05/29

17:00-18:00   Room #056 (Graduate School of Math. Sci. Bldg.)
Yasuhiro Oki (University of Tokyo)
On supersingular loci of Shimura varieties for quaternion unitary groups of degree 2 (Japanese)