第 22 回広島仙台整数論集会アブストラクト
7 月 11 日 (火)10:00 – 10:45 青木 謙典 (京都大学) / Kensuke Aoki (Kyoto University)
Ainf -コホモロジーと半安定形式スキームの de Rham-Witt 複体との間の p-進 Cartier 同型 /A p-adic Cartier isomorphism between the Ainf -cohomology and de Rham-Witt complexes
for semistable formal schemes
In the p-adic Hodge theory, Bhatt-Morrow-Scholze constructed the Ainf -cohomology
theory for smooth formal schemes and show p-adic Cartier isomorphisms, which is applied to prove crystalline comparison theorem for the Ainf -cohomology. On the other
hand, Česnavičius-Koshikawa constructed the Ainf -cohomology theory for semistable formal schemes over the ring of integers of Cp . We prove the p-adic Cartier isomorphism
between the Ainf -cohomology and de Rham-Witt complexes for semistable formal schemes,
extending the result of Bhatt-Morrow-Scholze in the smooth case.
11:00 – 11:45 浅山 拓哉 (東京工業大学) / Takuya Asayama (Tokyo Institute of Technology)
関数体の Kummer 忠実性 / Kummer-faithfulness for function ﬁelds

Kummer-faithuful ﬁelds, introduced by Mochizuki, form an important class in anabeliangeometry. They are deﬁned by the triviality of the divisible parts of the Mordell-Weilgroups of semi-abelian varieties. In this talk, the speaker introduces the notion of DrinfeldKummer-faithful ﬁelds, which is a function ﬁeld analogue of the notion of Kummer-faithfulﬁelds, using Drinfeld modules. A suﬃcient condition for a Galois extension of a function
ﬁeld to be Drinfeld-Kummer-faithful is provided in terms of ramiﬁcation theory. Inspiredby Ozeki-Taguchi’s examples of highly Kummer-faithful ﬁelds, the speaker gives some examples of Drinfeld-Kummer-faithful ﬁelds.
12:00 – 12:45 後藤 新裕 (九州大学) / Akihiro Goto (Kyushu University)
The conditions of vanishing central value of Hasse-Weil L-function of a certain family of
CM elliptic curve in Eisenstein integers case
The condition under which the central value of the Hasse-Weil L-function of an elliptic
curve vanishes is very important and is also related to the BSD conjecture. √We gavevarious characterizations for the inﬁnite family {Eλ } of CM elliptic curves on Q( −3) for
which L(1, Eλ ) = 0. It is described by rational numbers, which are analogues of Bernoulli
numbers called Bernoulli-Hurwitz type numbers. We also obtained a congruence relation
between the order of the Tate-Shafarevich group of Eλ and the Bernoulli-Hurwitz type
number under L(1, Eλ ) ̸= 0, which will be also presented as well. This talk is based on the
joint work with Ryota Shii.
14:00 – 14:45 足立 大雅 (九州大学) / Taiga Adachi (Kyushu University)
2 次捻りによる保型形式の L 関数の中心値の代数的部分の 2 進付値の変化について / The
2-adic valuations of the algebraic central L-values of quadratic twists of modular forms
Calculating the algebraic central L-values of quadratic twists of a Hecke eigenform isimportant in number theory. Recently, Shuai Zhai, Chao Li and Li Cai obtained the 2-adic
valuation of the algebraic part under certain conditions by using modular symbols. We
succeeded in calculating the 2-adic valuations for a new class of quadratic twists. This talk1

is based on the joint work with Keiichiro Nomoto (Koden Electronics Co., Ltd.) and Ryota
Shii (Kyushu university).
15:00 – 15:45 Abhinandan (University of Tokyo)
Prismatic F -crystals and Wach modules
For an unramiﬁed extension F/Qp with perfect residue ﬁeld, by works of Fontaine,
Colmez, Wach and Berger it is known that the category of Wach modules over a certain
integral period ring A+
F is equivalent to the category of lattices inside crystalline representations of GF , i.e. the absolute Galois group of F . Moreover, by recent work of Bhatt and
Scholze, we also know that lattices inside crystalline representations of GF are equivalent
to the category of prismatic F -crystals over OF , the ring of integers of F . In this talk, wewill present a direct construction of the categorical equivalence between Wach modules overA+
F and prismatic F -crystals over OF . If time permits, we will also mention generalizationof our construction to the imperfect residue ﬁeld case as well as to the relative case.
16:00 – 16:45 池田 香凜 (九州大学) / Karin Ikeda (Kyushu University)
On real zeros of the Hurwitz zeta function
In this talk, I will present a solution to the problem of real zeros of the Hurwitz zetafunction, which has remained unsolved in previous studies. After reviewing the works of
Spira, Nakamura, Matsuzaka, and Endo-Suzuki, I discuss the remaining case, namely thezeros in the interval (−4, 0). I also present an observation of a curious behavior of a familyof polynomials used in the proof.
17:00 – 17:45 井上 絢太郎 (京都大学) / Kentarou Inoue (Kyoto University)
Degeneration of p-divisible groups in terms of logarithmic geometry
The moduli space of abelian varieties are compactiﬁed by Faltings-Chai. Recently,
Kajiwara-Kato-Nakayama realized this compactiﬁcation as the moduli space of log abelianvarieties which are degenerated objects in the world of logarithmic geometry. In the studyof the mod p-ﬁber of this compactiﬁed moduli space, log p-divisible groups occur as degenerated objects of p-divisible groups. In this talk, we prove that log p-divisible groups overlog regular bases admit slope ﬁltrations up to isogeny.

7 月 12 日 (水)10:00 – 10:45 石塚 康介 (東北大学) / Kosuke Ishizuka (Tohoku University)
The Goldstine theorem in the non-archimedean caseIn functional analysis over R or C (which we call classical analysis ), the Goldstine
theorem says that the closed unit ball of a Banach space is dense in the closed unit ballof the bidual space for the weak star topology. On the other hand, Schikhof proved that
the Goldstine theorem also holds in the non-archimedean case, if we replace the closed
unit ball with the open unit ball. Then the question arises whether the exactly samestatement as the classical Goldstine theorem holds in the non-archimean case. In this talk,we discuss the Goldstine theorem from the perspective of the theory of edged sets and thatof compactoids which are analogous to compactness. As a result, we obtain a new result
about the Goldstine theorem when the coeﬃcient ﬁeld is spherically complete.

2

11:00 – 11:45 石塚 伶 (東京工業大学) / Ryo Ishizuka (Tokyo Institute of Technology)
On the commutative ring-theoretic structure of the perfectoidization of semiperfectoid rings
The perfectoidization was introduced by Bhatt-Scholze as a mixed characteristic analogueof the perfection of rings of positive characteristic. In general, this is only a complex ofsome derived category, but for example the perfectoidization of semiperfectoid rings gives
an honest ring. On the other hand, the notion of p-root closure was introduced by Roberts
in commutative ring theory. This is a simple closure operation that existed before perfectoid
rings appeared and is closely related to the structure of the Fontaine ring (now often calledtilting). In this talk, we show that the perfectoidization of the semiperfectoid ring S is(almost) isomorphic to the p-adic completion of the p-root closure of S.
12:00 – 12:45 石井 竣 (慶應義塾大学) / Shun Ishii (Keio University)
On pro-p outer Galois representations associated to once-punctured CM elliptic curves
Let p be an odd prime. Shariﬁ proved that, under the Deligne-Ihara conjecture (nowa theorem of Brown), the kernel of the pro-p outer Galois representation associated tothe thrice-punctured projective line corresponds to the maximal pro-p extension of Q(µp )unramiﬁed outside p if p is regular. In this talk, we discuss an analogue of his result for
imaginary quadratic ﬁelds of class number one by considering once-punctured CM elliptic
curves instead of the thrice-punctured projective line.
14:00 – 14:45 川村 花道 (東京理科大学) / Hanamichi Kawamura (Tokyo University of Science)
1/2 多重ゼータ値の重み付き和公式とその類似 / Weighted sum formula for half multiple
zeta values and its variants
The weighted sum formula for multiple zeta values has been found in various forms asa weighted analogue of Granville’s sum formula. In this talk, we consider a new typeof formulas weighted by powers of 2 that have been discovered for the special cases of
Yamamoto’s interpolated multiple zeta values using some formulas of the generalized hypergeometric series. This is a joint work with Takumi Maesaka (Kanazawa Institute ofTechnology) and Masataka Ono (Waseda University).
15:00 – 15:45 角野 裕太 (東北大学) / Yuta Kadono (Tohoku University)
川島関数の積分表示と Taylor 係数 / An integral representation of Kawashima function andits Taylor coeﬃcients

It is known that there are many Q-linear relations among multiple zeta values. In
particular, the Kawashima relation is one of families of relations that is expected to generate
all Q-linear relations among multiple zeta values and can be described using the Taylor
coeﬃcients of the Kawashima function. In this talk, we generalize the multiple integrals,
which have been studied for a long time in transcendental number theory, to give a multiple
integral representation of the Kawashima function. Furthermore, we express the Taylor
coeﬃcients of the Kawashima function in terms of multiple zeta values with explicit indices.
16:00 – 16:45 隈部 哲 (九州大学) / Satoshi Kumabe (Kyushu University)
Supercongruences for the Appell functions and elliptic curves with complex multiplication
There are congruence relations modulo a power of a prime number p arising from thecoeﬃcients of a power series expansion of an invariant diﬀerential on an elliptic curve over3

Q. These congruence relations are called Atkin and Swinnerton-Dyer congruences. Furthermore, these congruence relations hold modulo a higher power of p in a particular case.
Then such congruence relations are called supercongruences. In this talk, I will introduce
supercongruences for certain hypergeometric functions of two variables and elliptic curves
with complex multiplication.
17:00 – 17:45 松月 大知 (名古屋大学) / Daichi Matsuzuki (Nagoya University)
On non-vanishing of higher genus multiple zeta values
Thakur introduced multiple zeta values associated with the rational function ﬁeld overa ﬁnite ﬁeld and proved that they are non-zero by using a result of Sheats. In the talk,we prove that multiple zeta values associated with the function ﬁeld of a higher genus
curve, which were introduced by Lara Rodriguez and Thakur, are non-zero under a certain
condition on the Weierstrass gap sequence at a ﬁxed rational point ∞. By arguments
similar to those performed by Thakur in the proof for the case of the rational function
ﬁeld, we explicitly calculate the ∞-adic valuation of a given multiple zeta value and we
obtain the result by the calculation.
7 月 13 日 (木)09:00 – 09:45 小林 雅人 (神奈川大学) / Masato Kobayashi (Kanagawa University)
Applications of Ramanujan’s work on Eisenstein series
Ramanujan (1916) expressed quotients of certain q-series as polynomials of the Eisensteinseries P, Q, R and derived what we call Ramanujan’s diﬀerential equations. We continue
this research with the variants of Eisenstein-type series which Hahn (2007) recently introduced. We also prove new formulas of convolution sums for divisor sum functions as
subsequent work of Cheng-Williams (2004) and Huard-Ou-Spearman-Williams (2002).
10:00 – 10:45 松本 圭峰 (大阪大学) / Keiho Matsumoto (Osaka University)
非可換 p-進 Hodge 理論について / On non-commutative p-adic Hodge theory

In this talk, we will discuss the crystalline conjecture for non-commutative algebraic
varieties, and explain our results about the non-commutative version of Bhatt-MorrowScholze’s Breuil-Kisin cohomology theory. Time permitting, as an application to arithmetic
geometry, we will talk about the relation between our results and the crystalline comparison
theorem for open varieties proved by Yamashita.
11:00 – 11:45 松本 晃二郎 (東京大学) / Kojiro Matsumoto (University of Tokyo)
On the potential automorphy and the local-global compatibility for the monodromy operators at p ̸= l over CM ﬁelds
Let F be a CM ﬁeld and n be a positive integer. About the global Langlands correspondence for GLn,F , there are many general results in regular essentially conjugate self-dualcases. For example, we have the complete local-global compatibility and the potential
automorphy for many general Galois representations. On the other hand, for any regularC-algebraic cuspidal automorphic representation π of GLn (AF ), Harris-Lan-Taylor-Thorne
constructed the Galois representation corresponding to π, Ila Varma proved the local-global
compatibility up to semisimpliﬁcation at p ̸= l and many people proved some automorphy
lifting theorems and potential automorphy theorems for some special Galois representations. But we haven’t proved the local-global compatibility for the monodromy operators4

at p ̸= l and the potential automorphy for general Galois representations, unlike regularessentially conjugate self-dual cases. In this talk, we observe some relations between thepotential automorphy of l-adic Galois representations of F and the local-global compatibility for the monodromy operators at p ̸= l. As applications, we prove the local-global
compatibility for the monodromy operators at p ̸= l in many cases and the Ramanujan
conjecture for regular C-algebraic cuspidal automorphic representations of GL2 (AF ) byusing some potential automorphy theorems.
12:00 – 12:45 村上 友哉 (九州大学) / Yuya Murakami (Kyushu University)
3 次元多様体の量子不変量, 偽テータ関数, L 関数の特殊値 / Quantum invariants of 3manifolds, false theta functions and special values of L-functions
Quantum invariants of 3-manifolds are topological objects deﬁned from a mathematicalphysics viepoint. They also have deep connections with number theory. For instance,
Zagier and others had been pointed out relation between the asymptotic expansions ofquantum invariants and modular forms. Under this perspective, mathematical physicists
Gukov-Pei-Putrov-Vafa conjectured a certain relationship between quantum invariants. In
this talk, I will discuss a proof of this conjecture. The methods of proof yields relation andvanishing property of special values of certain L-functions.
14:00 – 14:45 室谷 岳寛 (東京工業大学) / Takahiro Murotani (Tokyo Institute of Technology)
有限体の無限次代数拡大体上の遠アーベル幾何学 / Anabelian geometry over inﬁnite algebraic extensions of ﬁnite ﬁelds

The Grothendieck conjecture for hyperbolic curves over ﬁnite ﬁelds was solved aﬃrmatively by Tamagawa and Mochizuki. Moreover, Tamagawa also proved (a “weak version”of) the Grothendieck conjecture for hyperbolic curves of genus zero over algebraic closuresof ﬁnite ﬁelds. So, it is natural to consider anabelian geometry over (inﬁnite) algebraic extensions of ﬁnite ﬁelds. In this talk, we consider to what extent we can develop anabeliangeometry over these ﬁelds, and give some results (e.g., a generalization of Tamagawa’s
result).
15:00 – 15:45 中川 彬雄 (千葉大学) / Akio Nakagawa (Chiba University)
Multi-variable hypergeometric functions and algebraic varieties over ﬁnite ﬁelds
Hypergeometric functions over C satisfy many formulas and have integral representations. For one-variable hypergeometric functions over ﬁnite ﬁelds, ﬁnite ﬁeld analoguesof the integral representations are known. Through the integral representations and theiranalogues, hypergeometric functions over C and over ﬁnite ﬁelds are related with algebraic
varieties. In this talk, we prove ﬁnite ﬁeld analogues of some formulas and integral representations for Appell-Lauricella multi-variable hypergeometric functions. Furthermore,
we express the numbers of rational points on some algebraic varieties in terms of AppellLauricella functions over ﬁnite ﬁelds. The speaker will also talk about the recent joint workwith Ryojun Ito, Satoshi Kumabe and Yusuke Nemoto on Kampé de Fériet hypergeometricfunctions over ﬁnite ﬁelds.
16:00 – 16:45 沖 泰裕 (北海道大学) / Yasuhiro Oki (Hokkaido University)
On Tamagawa numbers of tori attached to CM algebras
The notion of Tamagawa numbers is one of the most important subject in arithmetic of5tori over number ﬁelds. In 1963, Takashi Ono predicted that all positive rational numbers
are realized as the Tamagawa numbers of tori over number ﬁelds. This conjecture wasresolved by Shin-ichi Katayama in 1985. In this talk, we deﬁne “tori attached to CM
algebras”, which are deﬁned over the ﬁeld of rational numbers, and give an analogue of
Ono’s conjecture/Katayama’s theorem for such tori. More precisely, we explain that every
power of 2 can be written as the Tamagawa number of a torus attached to a CM algebra.
This talk is based on the joint work with Pei-Xin Liang, Hsin-Yi Yang and Chia-Fu Yu.
17:00 – 17:45 大塚 瑛介 (東北大学) / Eisuke Otsuka (Tohoku University)
Iterated integrals over some algebraic curves and their motivic interpretations
Multiple zeta values (MZVs for short) can be represented as iterated integrals of Qrational diﬀerential forms on P1 (C) \ {0, 1, ∞}. This representation allows us to consider
MZVs geometrically, and this is one of the motivations for Deligne, Goncharov, Terasoma
et al. to give a motivic interpretation of MZVs by using the fundamental group and mixed
Tate motive theory. In this talk, we consider the values given by iterated integrals on otherconcrete algebraic curves over Q and explain some results. In particular, we give motivic
interpretations of them by investigating relations with MZVs with levels. At this point, it
is important to consider the base expansion and the Galois invariant part of the space of
motivic MZVs. I will also explain it.

7 月 14 日 (金)09:00 – 09:45 齋藤 陽平 (慶應義塾大学) / Yohei Saito (Keio University)
Theta lifts to certain cohomological representations of indeﬁnite orthogonal groups
Howe and Tan investigated a degenerate principal series representation of indeﬁnite orthogonal groups and completely described its composition series. Their works imply that
there exists a unique unitalizable irreducible submodule, which is isomorphic to a nontempered derived functor module. We study the theta liftings to this irreducible representation. In this talk, we introduce a new observation of the liftings of holomorphic
automorphic forms on Mp2 (R) and, by combining it with the Borcherds’ method, explicitly compute the Fourier expansion of resulting automorphic forms. As an application, we
prove the square integrability of the liftings.
10:00 – 10:45 椎井 亮太 (九州大学) / Ryota Shii (Kyushu University)
On non-trivial Λ-submodules with ﬁnite index of the plus/minus Selmer group over anticyclotomic Zp -extension at inert primes
In Iwasawa theory, it is a basic problem that a Selmer group of elliptic curves has no nontrivial Λ-submodules with ﬁnite index. Regarding this problem, there is some research by
R. Greenberg, B. D. Kim, and T. Kitajima–R. Otsuki. Recently, we solved this problem
under some assumptions for the anticyclotomic Zp -extension of an imaginary quadraticﬁeld in which p is inert. In this talk, we present the background of this research and prove
this result.
11:00 – 11:45 島田 了輔 (東京大学) / Ryosuke Shimada (University of Tokyo)
Crystal bases and aﬃne Deligne-Lusztig varieties

6

The notion of aﬃne Deligne-Lusztig variety was ﬁrst introduced by Rapoport, whichplays an important role in understanding geometric and arithmetic properties of Shimuravarieties. The geometric properties of aﬃne Deligne-Lusztig varieties have been studied bymany people, and one of the most interesting results is an explicit description of the setof irreducible components. Surprisingly, irreducible components of aﬃne Deligne-Lusztig
varieties are parametrized by a certain weight space of crystal bases. In this talk, I will talkabout my recent work on the explicit construction of irreducible components from crystal
bases.
12:00 – 12:45 高谷 悠太 (東京大学) / Yuta Takaya (University of Tokyo)
Equidimensionality of aﬃne Deligne-Lusztig varieties in mixed characteristic
Aﬃne Deligne-Lusztig varieties are a generalization of the special ﬁber of local Shimuravarieties, such as Rapoport-Zink spaces. It is expected to have geometric properties similar to those of Newton strata of Shimura varieties. In this talk, we prove one of its
expected properties, equidimensionality. In the equicharacteristic case, equidimensionality
was proven by Hartl and Viehmann. However, aﬃne Deligne-Lusztig varieties are perfectschemes in mixed characteristic, which makes it harder to study their geometric properties.
We will explain obstacles to applying Hartl and Viehamann’s arguments in mixed characteristic and present a way of resolving those issues.
14:00 – 14:45 山口 永悟 (東京工業大学) / Naganori Yamaguchi (Tokyo Institute of Technology)
遠アーベル幾何学の m 次可解化を使った新たな展開について / New developments in anabelian geometry by using m-step reconstruction

In anabelian geometry, there is an important conjecture known as the Grothendieckconjecture. This conjecture states that the geometric properties of hyperbolic (algebraic)
curves can be determined group-theoretically by studying their arithmetic fundamentalgroups. H. Nakamura, A. Tamagawa, S. Mochizuki, and J. Stix successfully proved thisconjecture for ﬁelds of any characteristic (until 2009). Nevertheless, several open problems
remain related to this conjecture. This talk focuses on one such problem, called the m-stepsolvable Grothendieck conjecture, which serves as the central theme. The m-step solvable Grothendieck conjecture concerns the group-theoretical reconstruction of geometricproperties of hyperbolic (algebraic) curves by "near" abelian quotients of their fundamental groups – more precisely, by the maximal geometrically m(≥ 2)-step solvable quotientof their arithmetic fundamental groups. In this talk, we will present the m-step solvable
Grothendieck conjecture and a part of its proof as obtained by the speaker.
15:00 – 15:45 山本 寛史 (東京大学) / Hirofumi Yamamoto (University of Tokyo)
p-通常的半整数重さ次数 2 ジーゲルモジュラー形式の空間の次元について / On the dimensionof spaces of p-ordinary half-integral weight Siegel modular forms of degree 2
Hecke eigenforms whose eigenvalues at p are p-adic units are called p-ordinary. The
dimension of the space spanned by p-ordinary eigenforms over many reductive groups,
including Siegel modular forms, is known to be bounded regardless of levels and weights.
Furthermore, Hida showed that the dimension is bounded in the case of half-integral weight
modular forms as well, using Shimura correspondence. We will show the similar resultfor half-integral weight Siegel modular forms of degree 2, using an analogy of Shimura
correspondence for Siegel modular forms (Ibukiyama conjecture).

7