第 20 回広島仙台整数論集会アブストラクト

7 月 13 日 (火)13:30 – 14:30 板東 克之 (東京大学) /Katsuyuki Bando (University of Tokyo)
Geometric Satake equivalence in mixed characteristic and Springer correspondence
The geometric Satake correspondence is an equivalence between the category of equivariant perverse sheaves on the aﬃne Grassmannian and the category of representations ofthe Langlands dual group. It is known that there is a mixed characteristic version of theGeometric Satake correspondence. The Springer correspondence is a correspondence between the category of equivariant perverse sheaves on the nilpotent cone and the categoryof representation of the Weyl group. In this talk, we will explain some relation between
these two correspondences, including the mixed characteristic case.
14:45 – 15:45 島田 了輔 (東京大学) /Ryosuke Shimada (University of Tokyo)
Geometric Structure of Aﬃne Deligne-Lusztig Varieties for GL3
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 thiscontext, geometric and homological properties of aﬃne Deligne-Lusztig varieties have beenused to examine Shimura varieties and the local Langlands correspondence. In this talkwe study the geometric structure of aﬃne Deligne-Lusztig varieties Xλ (b) for GL3 and bbasic. We completely determine the irreducible components of the aﬃne Deligne- Lusztigvariety. In particular, we classify the cases where all of the irreducible components areclassical Deligne-Lusztig varieties times ﬁnite-dimensional aﬃne spaces. If this is the case,then the irreducible components are pairwise disjoint.
16:00 – 17:00 植木 潤 (東京電機大学) /Jun Ueki (Tokyo Denki University)
Non-trivial L-functions of the Whitehead link, triangle groups, and roots of unity
In Iwasawa theory and Hida–Mazur theory, torsion modules associated with group representations play important roles. In our setting, the non-trivial L-function of the universal
deformation implies that the associated module is torsion. We previously determined foreach n ∈ Z all residual representations of the twist knot J(2, 2n) with L ̸= 0, characterized
them by exceptional surgeries, and further proved that L =˙ kn2 in O[[x − α]] for a certainkn ∈ Z[x] related to 3n − 1-th roots of unity. In this talk, we recover previous results for alln at once from a result on the Whitehead link W1 as well as point out a theoretical background for L being square. A key is a surjective homomorphism π1 (S 3 − W1 ) ↠ Γ(3, 3, ∞)1
to a triangle group with a 3 + -dimensional origin. This is a joint work with Léo Bénard
2(GAU, Göttingen), Ryoto Tange (Waseda U.), and Anh T. Tran (U. Texas at Dallas).

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7 月 14 日 (水)10:00 – 11:00 齋藤 光一郎 (名古屋大学) /Koichiro Saito (Nagoya University)
単価 Nielsen ポリログと超幾何関数
Single-valued Nielsen polylogarithms and hypergeometric functions

This talk will focus on a new aspect of Nielsen polylogarithms, which were introducedby Nielsen in 1909 in the context of generalizing classical polylogarithms. We will showthat Nielsen polylogarithms represent the meta-abelian quotient of a fundamental solutionof the one-variable KZ equation, and that the generating series of certain single-valuedanalogues of them can be expressed in terms of Gauss hypergeometric functions. We will
also give an explicit closed formula for each single-valued Nielsen polylogarithm.
11:15 – 12:15 竹平 航平 (東北大学) /Kohei Takehira (Tohoku University)
力学系のゼータ関数の有理性と Woods Hole 固定点定理
Rationality of dynamical zeta function and Woods Hole ﬁxed point formula

For a one variable rational function ϕ over a ﬁeld K, we can deﬁne a discrete dynamicalsystem by regarding ϕ as a self morphism over P1 . There are many studies of the zetafunction associated with a discrete dynamical system such as the Artin-Mazur zeta function
and the Ruelle zeta function. In 1995, Hatjispyros and Vivaldi deﬁned a dynamical zetafunction for rational functions using multipliers of periodic points, that is, an invariant
which indicates local behavior of dynamical systems. In this talk, we prove rationality ofdynamical zeta functions of this type for a large class of rational functions. The proof uses
Woods Hole ﬁxed point formula and the trace of a linear map acting on cohomology of acoherent sheaf on P1 .
13:30 – 14:30 冨田 拓希 (慶應義塾大学) /Takuki Tomita (Keio University)
絶対ゼータ関数の絶対 Euler 積表示について
The absolute Euler product representations of the absolute zeta functions

Let X be a scheme of ﬁnite type over Z satisfying a certain condition. The absolutezeta function for X is deﬁned as the limit as p → 1 of the congruent zeta function for X,
which is the exponential of the generating function of the number of Fpm -rational pointsof X. In 2016, Kurokawa suggested that the absolute zeta function for a general scheme
should have an inﬁnite product structure (the absolute Euler product). In this talk, wewill give a formulation of his suggestion using a torsion free Noetherian F1 -scheme deﬁned
by Connes and Consani. Moreover, we will explain that each factor of the absolute Euler
product is explicitly determined in terms of the function which interpolates the number ofFpm -rational points.
14:45 – 15:45 臺信 直人 (慶應義塾大学) /Naoto Dainobu (Keio University)
On ideal class groups of number ﬁelds associated to mod p modular Galois representations
Many people have noticed the existence of various relations between ideal class groupsof number ﬁelds and Tate-Shafarevich groups of elliptic curves. For example, recentlyPrasad and Shekhar proved the following result. Let E be an elliptic curve over Q andK = Q(E[p]) the p-th division ﬁeld of E for an odd prime p. Under some assumptions,they showed that if the p-torsion part of the Tate-Shafarevich group of E has dimension at2

least 2 over Fp , then the ideal class group of K has E[p] as its quotient Gal(K/Q)-module.
In this talk, we introduce a generalization of the above result for number ﬁelds attachedto mod p modular Galois representations using setups on Bloch-Kato’s Selmer groups andTate-Shafarevich groups.
16:00 – 17:00 森 政興 (大阪大学) /Masaoki Mori (Osaka University)
A1 代数トポロジーにおける Galois 理論
Galois theory for A1 -algebraic topology
In Galois theory of the covering spaces, it is known that there exists canonical one-to-one
correspondence between the set of isomorphic classes of coverings of a pointed topological space (X, x) and the set of conjugacy classes of subgroups of the fundamental group
π1 (X, x). In this talk, we give an analogue of this theory in A1 -algebraic topology which
was constructed by F. Morel and V. Voevodsky. First, we explain the notion of simplicial
coverings of simplicial sets. For any ﬁbrant connected pointed simplicial set (B, b), we
present one-to-one correspondence between the set of subgroups of π1 (B, b) and the weakly
equivalent class of simplicial coverings of (B, b). Next, we talk about the generalization
for simplicial sheaves of the above correspondence. Finally, for any ﬁbrant A1 -connected
pointed simplicial sheaf (X , x), we give a bijective correspondence between the A1 -weakequivalence classes of A1 -coverings of (X , x) and subsheaves of A1 -fundamental group of(X , x).

7 月 15 日 (木)10:00 – 11:00 村上 友哉 (東北大学) /Yuya Murakami (Tohoku University)
モジュラー j 関数のサイクル積分の不良近似数への拡張
Extended-cycle integrals of the j-function for badly approximable numbers

Cycle integrals of the j-function are expected to play a role in the real quadratic analogof singular moduli. However, it is not clear how one can consider cycle integrals as a"continuous" function on real quadratic numbers. In this talk, we extend the deﬁnitionof cycle integrals of the j-function from real quadratic numbers to badly approximable
numbers to seek an appropriate continuity. We also give some explicit representations
for extended-cycle integrals in some cases which can be considered as a partial result ofcontinuity of cycle integrals.
11:15 – 12:15 沖 泰裕 (東京大学) /Yasuhiro Oki (University of Tokyo)
On the connected components of Shimura varieties for CM unitary groups in odd variables
Let (G, X) be a Shimura datum. Take a prime number p and a Bruhat–Tits subgroup
Kp of G(Qp ). Consider the projective limit of the sets of connected components of Shimuravarieties for (G, X) whose level at p are given by Kp . It is eqipped with the prime-to-p
Hecke action. Then we discuss the question whether the above action is transitive, which
is motivated by the theory of mod p reductions of Shimura varieties. In this talk, we giveinﬁnitely many projective systems of the Shimura varieties for CM unitary groups in odd
variables for which the considering question is negative. To achieve this goal, we study aquestion related to the weak approximation on certain tori over Q.

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13:30 – 14:30 今井 湖都 (東京大学) /Koto Imai (University of Tokyo)
正標数の局所体上の冪零度最大のある有限次 Galois 拡大の分岐群
Ramiﬁcation groups of some ﬁnite Galois extensions of maximal nilpotency class over local
ﬁelds of positive characteristic

Galois extensions of local ﬁelds is one of the most important subjects in the ﬁeld ofnumber 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 theramiﬁcation groups in the case where the Galois groups are abelian. We extend the resultsof Brylinski to some non-abelian cases where the Galois groups are of order ≤ pp+1 and of
maximal nilpotency class.
14:45 – 15:45 金井 和貴 (新潟大学) /Kazuki Kanai (Niigata University)
Norm one tori and Hasse norm principle
Let k be a global ﬁeld, K/k be a ﬁnite extension and A×
K be the idele group of K. We
say that the Hasse norm principle (HNP) holds for K/k if Obs(K/k) := (NK/k (A×K) ∩××k )/NK/k (K ) = 1 where NK/k is the norm map. In 1931, Hasse proved that if K/k√√is cyclic then the HNP holds but does not hold for bicyclic extension Q( −39, −3)/Q.
The HNP for Galois extensions was investigated by Tate, Gerth, Gurak, etc. However, for
non-Galois extensions, very little is known about the HNP. In this talk, we give a methodfor computing H 1 (k, Pic X) where X is a smooth k-compactiﬁcation of the norm one torus(1)
T = RK/k (Gm ) by the computer algebra system GAP. Using this, we determine a necessary
and suﬃcient condition for the HNP for K/k with [K : k] ≤ 15 by combining the resultsof Ono and Voskresenskii in the 1960s, and of Drakokhrust and Platonov in the 1980s. As
applications of the results, we get the group T (k)/R of R-equivalence classes over a local
ﬁeld k via Colliot-Thélène and Sansuc’s formula and the Tamagawa number τ (T ) over anumber ﬁeld k via Ono’s formula τ (T ) = |H 1 (k, Tb)|/|X(T )|. This is the joint work withAkinari Hoshi (Niigata University) and Aiichi Yamasaki (Kyoto University).
16:00 – 17:00 室谷 岳寛 (京都大学) /Takahiro Murotani (Kyoto University)
完備離散付値体の遠アーベル幾何学
A study on anabelian geometry of complete discrete valuation ﬁelds

Anabelian geometry has been developed over a much wider class of ﬁelds than Grothendieck,who is the originator of anabelian geometry, conjectured. More precisely, he consideredthat anabelian geometry should be developed over ﬁelds ﬁnitely generated over prime ﬁelds.
However, Mochizuki gave some anabelian results over p-adic ﬁelds. On the other hand, algebraically closed ﬁelds of characteristic 0 are typical examples of ﬁelds which are notsuitable for the base ﬁelds of anabelian geometry. So, it is natural to ask the following
question: What kinds of ﬁelds are suitable for the base ﬁelds of anabelian geometry? In
this talk, we consider this problem for higher local ﬁelds and mixed-characteristic complete
discrete valuation ﬁelds with perfect residue ﬁelds.

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7 月 16 日 (金)9:30 – 10:30 李 公彦 (東京大学) /Kimihiko Li (University of Tokyo)
Prismatic and q-crystalline sites of higher level
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 thistalk, for a positive integer m, we will give a construction of the level m primastic andq-crystalline sites and prove a certain equivalence between the category of crystals on them-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 siteand that on the (m − 1)-q-crystalline site.
10:45 – 11:45 石田 哲也 (佐賀大学) /Tetsuya Ishida (Saga University)
Local ε-conjecture for de Rham (φ, Γ)-modules and p-adic diﬀerential equations(φ, Γ)-modules are deﬁned by Fontaine as a generalization of p-adic representations of
the absolute Galois group of a p-adic number ﬁeld. Nakamura formulated a conjecture for(φ, Γ)-modules which generalizes Kato’s local ε-conjecture for p-adic representations. On
the other hand, for a (φ, Γ)-module satisfying the de Rham condition, Berger constructeda simpler p-adic diﬀerential equation. In this talk, we explain that the conjecture for thecyclotomic deformation of any de-Rham (φ, Γ)-module can be deduced from the one for thecyclotomic deformation of the corresponding p-adic diﬀerential equation. This is a joint
work with Kentaro Nakamura.
12:00 – 13:00 町出 智也 (国立情報学研究所) /Tomoya Machide (National Institute of Informatics)
ブール多項式の連立方程式に関する公式について
On a formula for systems of Boolean polynomial equations

A Boolean polynomial is a congruence class of the polynomial ring over the ﬁnite ﬁeld of
characteristic 2, and identiﬁed with a Boolean function on the two-element set consistingof true and false. The Boolean polynomial is preferred in mathematics, and appears inalgebraic geometry, Grobner basis, and cryptography. The Boolean function is preferredin computer science, and used to discuss the computational complexity exempliﬁed by
NP problem. In any areas, solving a system (or ideal) of Boolean polynomial equations
is a main subject. In this talk, we prove a formula for systems of Boolean polynomialequations. The formula has both operations of conjunction (logical AND) and disjunction(logical OR) recursively, and conforms to De Morgan’s duality. The formula comes fromthe distributivity of AND and OR. We also give its applications to the computational
complexity using a parameter which is similar to the bandwidth of the matrix.

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