第２１回仙台広島整数論集会 アブストラクト
2022 年 7 月 12 日（火）〜7 月 15 日（金）東北大学大学院理学研究科・数理科学記念館
ハイブリッド開催（対面参加は講演者のみ）

7 月 12 日（火）9:50 – 10:40

高梨悠吾（東京大学）/ Yugo Takanashi (University of Tokyo)p-進体上の GLn の内部形式の共役自己双対表現の偶奇性
Parity of conjugate self-dual representations of inner forms of GLn over p-adic fields
There are two parametrizations of discrete series representations of GLn over p-adic fields. One is the local
Langlands correspondence, and the other is the local Jacquet–Langlands correspondence. The compositeof these two maps the discrete series representations of an inner form of GLn to Galois representations
called discrete L-parameters. On the other hand, we can define the parity for each selfdual representation
depending on whether the representation is orthogonal or symplectic. The composite preserves the notion ofself-duality, and it transforms the parity in a nontrivial manner. Prasad and Ramakrishnan computed thetransformation law, and Mieda proved its conjugate self-dual analog under some conditions on groups andrepresentations. 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.
10:55 – 11:45

大橋

亮 （横浜国立大学）/ Ryo Ohashi (Yokohama National University)

On the maximality of genus-3 Howe curves
It is well-known that all maximal or minimal curves defined over Fp2 are superspecial, while superspecial
curves defined over Fp2 are not necessarily maximal nor minimal. In this paper, we study the maximality
or minimality of Howe curves of genus 3. A Howe curve is defined as the desingularization of the fiber
product over P1 of two elliptic curves. We will show that if a Howe curve of genus 3 is superspecial, its
standard form is maximal or minimal over Fp2 without taking its Fp2 -form.
13:45 – 14:35

田島凌太（九州大学）/ Ryota Tajima (Kyushu University)

Shadow が虚数乗法を持つモックモジュラー形式の p 進的な性質について
A p-adic property of mock modular forms whose shadows have complex multiplication
Let f be a mock modular form and g denote the shadow of f . Then there exists a unique p-adic number
α and we can obtain a p-adic modular form from f and α. When g has complex multiplication by K and p
is split in K, it is known that α is not zero. On the other hand, we don’t know much about α for an inert
prime p. The speaker proved that α is not zero when p is inert in K under certain conditions. In this talk,I will explain the result.

14:50 – 15:40

中山

裕大（東京大学）/ Yuta Nakayama (University of Tokyo)

The canonicity of the integral models of RSZ Shimura varieties
Recently, Pappas formulated canonicity, namely universal properties to characterize integral modelsof Hodge type Shimura varieties defined by Kisin and himself. The formulation modifies more classicalcanonicity of Milne, and applies to the cases of parahoric levels. An example of Hodge type Shimuravarieties was introduced by Rapoport, Smithling and Zhang. They have conjectured relationship, suchas a version of Arithmetic Gan–Gross–Prasad conjecture, of automorphic representation theory and theintersection theory on the Shimura varieties and on their moduli theoretic integral modes. We prove thatthese integral models satisfy the canonicity of Pappas, becoming isomorphic to Kisin–Pappas models as aresult.
16:00 – 16:50

前田洋太（京都大学・ソニーグループ） /Yota Maeda (Kyoto University・Sony Group Corporation)
On the geometry of higher dimensional ball quotients
Determination of the birational types of modular varieties is an important problem. In this talk, I willcompute the Hirzebruch–Mumford volume of unitary groups and prove that, as a consequence, when thediscriminant of imaginary quadratic fields or the rank of unitary groups is sufficiently large, the line bundle,
whose sections are modular forms vanishing on branch divisors, is big on ball quotients. As an application,
under this condition, I show that ball quotients are of general type, assuming that there exists a low-weightcusp form. As background, modular varieties, especially Siegel modular varieties, by Tai, Freitag andMumford, and orthogonal modular varieties, by Gritsenko–Hulek–Sankaran and Ma, are known to be ofgeneral type when their dimension is large enough, and this result is their unitary analog.
17:05 – 17:55

ミュレー

ジョゼフ（Université Sorbonne Paris Nord・東京大学） /Joseph Muller (Université Sorbonne Paris Nord・University of Tokyo)
On the cohomology of the unramified PEL unitary Rapoport-Zink space of signature (1, n − 1)
Rapoport-Zink (RZ) spaces are moduli spaces which classify the deformations of a p-divisible groupwith additional structures. It is equipped with compatible actions of p-adic and Galois groups, and theircohomology 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 bothof them are RZ spaces of EL type. In this talk, we consider the unramified PEL unitary RZ space withsignature (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 unitarygroup. We compute the cohomology of these strata and we draw some consequences on the cohomologyof the RZ space at hyperspecial level. In particular, we prove that it is not admissible in general. Whenn = 3, 4 we deduce an automorphic description of the cohomology of the basic stratum in the corresponding
Shimura variety via p-adic uniformization.

7 月 13 日（水）9:30 – 10:20

高潤宣 （名古屋大学）/ Runxuan Gao (Nagoya University)

A Zariski dense exceptional set in Manin’s Conjecture: Dimension 2
Manin’s Conjecture predicts that the asymptotic behavior of rational points on Fano varieties is controlled
by its geometric invariants. The exceptional set in Manin’s conjecture was supposed to be a proper closed
subset under the Zariski topology and it has been proved for several special families of varieties, such as
toric varieties. But counterexamples were discovered then, so the conjecture was modified to assume theexceptional set to be a thin set. Nevertheless, all of these counterexamples are higher dimensional.
Based on a conjectural construction of the exceptional set by Lehmann, Sengupta, and Tanimoto, weconstructed a del Pezzo surface with a Zariski dense exceptional set, which gave a negative answer to aquestion suggested by Lehmann and Tanimoto. As a consequence, we obtained the first counterexample tothe original version of Manin’s Conjecture in dimension 2.
10:30 – 11:20

肖歓（熊本大学）/ Huan Xian (Kumamoto University)

Campana points of the Heisenberg group
Recently Pieropan, Smeets, Tanimoto and Varilly–Alvarado introduced a log Manin conjecture for Campana points on Fano log pairs and they proved the log Manin conjecture for vector group compactifications.
In this talk we consider Campana points on biequivariant compactications of the Heisenberg group andconfirm the log Manin conjecture in this case.
11:35 – 12:25

松田光智（東京大学）/ Koji Matsuda (Tokyo University)

Torsion points of elliptic curves over cyclotomic fields
The Mordell–Weil group of an elliptic curve over an algebraic number field is a finitely generated abeliangroup, hence its torsion subgroup is finite. It is easy to compute the torsion subgroup of the Mordell–Weilgroup of individual given elliptic curve. Conversely for given finite abelian group, it is a hard problem
to determine if it is the torsion subgroup of the Mordell–Weil group of an elliptic curve or not. In 1977
Mazur solved this problem over Q by studying arithmetic properties of modular curves. Roughly speaking,modular curves are algebraic curves parametrizing elliptic curves endowed with certain extra structure. In
this talk, we determine all modular curves whose Jacobian varieties have Mordell–Weil rank zero among
those defined over cyclotomic number fields, not only over the rational number field, and using it we show
a certain classification result of possible torsion subgroups of elliptic curves over cyclotomic fields.
14:00 – 14:50

小泉淳之介（東京大学） / Junnosuke Koizumi (University of Tokyo)

A motivic version of the Hasse–Arf theorem
As one direction to generalize Voevodsky’s theory of A1 -invariant sheaves, the theory of reciprocity
sheaves was introduced by Kahn–Saito–Yamazaki. One important aspect of the theory is that it gives
a unified framework for abelian ramification theory. We generalized the theory of reciprocity sheaves togeneral base schemes and proved a motivic analogue of the Hasse-Arf theorem for relative curves. This isa joint work with Hiroyasu Miyazaki.

15:05 – 15:55

川邊

大貴（東北大学）/ Daiki Kawabe (Tohoku University)

種数 1 のファイバー曲面のモチーフ
Chow motives of genus one fibrations
Genus 1 fibrations play a central role in the classification of surfaces over algebraically closed fields. Let
f : X → C be a genus 1 fibration from a surface to a curve, that is, the generic fiber of f has genus 1.
Then there exists a Jacobian genus 1 fibration j : J → C that satisfies some good properties. In this talk,we prove that the motive of X is isomorphic to the motive of J. Using this result, we also prove motivic
finite-dimensionality for surfaces not of general type with geometric genus 0.
16:05 – 16:55

宮崎

弘安（NTT 基礎数学研究センタ） /Hiroyasu Miyazaki (NTT Institute for Fundamental Mathematics)
一般底上のモジュラス付きモチーフ理論について
On motives with modulus over a general base
The theory of motives with modulus, developed in my joint work with Kahn–Saito–Yamazaki, is a
generalization of Voevodsky’s theory of motives over a field. The aim of this generalization is to get a
motivic picture of non-A1 -homotopy invariant phenomena, which were ignored in Voevodsky’s theory. In
this talk, I will explain the main idea of the theory of motives with modulus, and talk about the recent
joint work with Shane Kelly on motives with modulus over a general base.
7 月 14 日（木）9:30 – 10:20

渡辺

業 （広島大学）/ Kosaku Watanabe (Hiroshima University)

二進整数環上の二次の無理数についての Collatz 予想の一般化
Generalization of the Collatz conjecture to Z2
Collatz conjectured that when n ∈ N, then T3k (n) = 1 for some k ∈ N where T3 is the Collatz function.
In this paper, we study the trajectories of Tp where p ∈ Z2 is a quadratic odd number, and Tp (x) = px+1
if
2
x is odd, and Tp (x) = x2 if x is even. When the defining equation of p over Z is p2 + αp + β = 0, there exist
divergent trajectories unless α ≡ 1 (mod 2) and β ≡ 0 (mod 4). For p with p2 + αp + β = 0 with α = 1, 3and β = 4, 8, it seems that all trajectories converge to some cycles. And we got some other results.
10:30 – 11:20

臺信直人（慶應義塾大学）/ Naoto Dainobu (Keio University)

Class groups of the pn -division fields of elliptic curves and everywhere unramified rational points
Let p be a prime number, n be a positive integer, E be an elliptic curve defined over Q, and Q(E[pn ]) be
its pn -division field. On the p-part of the ideal class group of Q(E[pn ]), several lower bounds are obtainedin the literature including recent works by Sairaiji–Yamauchi, Hiranouchi, Ohshita, and Prasad–Shekhar.
Their works imply that the class number of the pn -division field is often divisible by a certain power of p
if the Mordell-Weil rank of E is larger than 1.
In this talk, the speaker presents a new result which improves at least partially the above preceding
works. This result gives a sufficient condition for the p-divisivility of the class number of the pn -division
field including the case of Mordell–Weil rank 1. Since this condition is very easy to verify in practice, we
can apply it to concrete examples as follows.
Let {Es,t } be a family of elliptic curves defined by y 2 = x3 − (s4 + t2 )x with integers s, t. Then, for every
fixed p > 5 and n, we can prove that the above family contains a certain infinite subfamily having thefollowing two novel properties: Their Mordell–Weil ranks are all 1 and the class numbers of their pn -division
fields are all divisible by p2n . The last part of this talk is based on the joint work (arXiv:2205.08946) with
Yoshinosuke Hirakawa and Hideki Matsumura.

11:35 – 12:25

村上和明（東邦大学）/ Kazuaki Murakami (Toho University)

An analogue of Kida’s formula for split prime Zp -extensions
For a given prime number p, Kida’s Riemann–Hurwitz formula describes the variation of the Iwasawa
λ-invariants associated with the cyclotomic Zp -extension in a degree p-extension of CM number fields. In
this talk, we consider a non-cyclotomic Zp -extension and prove a formula which is similar to usual Kida’s
Riemann–Hurwitz formula. More precisely, let p be an odd prime number and k an imaginary quadratic
field in which p splits into p and p∗ . There exists a unique Zp -extension k∞ /k which is unramified outside
p. For an arbitrary abelian extension F over k, we call F∞ = F k∞ the split prime Zp -extension of F . We
describe the variation of the Zp -rank of the p-ramified Iwasawa modules associated with the split prime
Zp -extension in a degree p-extension L/K which are abelian over k.
14:00 – 14:50

王沛鐸 （東京大学）/ Peiduo Wang (University of Tokyo)

On generalized Fuchs theorem over p-adic polyannuli
In this paper, 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 suchdifferential modules on one dimensional p-adic annuli under certain non-Liouvilleness assumption andGachet 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.
keywords: p-adic differential equations; p-adic Fuchs theorem; polyannuli
15:00 – 15:50

三神雄太郎（東京大学） / Yutaro Mikami (University of Tokyo)

Faithfully flat descent of quasi-coherent complexes on rigid analytic varieties over non-archimedean
local fields via condensed mathematics
In contrast to scheme theory, it is difficult to construct fundamental theories of rigid geometry such as atheory of quasi-coherent sheaves because of issues of topological algebraic systems such as topological rings
and topological modules. Recently Clausen and Scholze have developed a new approach to treat these,
which is called condensed mathematics. By using this, we generalize faithfully flat descent of pseudocoherent complexes on rigid analytic varieties to that of quasi-coherent complexes in the case where thebase field is a complete non-archimedean field. In this talk, we will introduce condensed mathematics
briefly, then we will explain the difference from fppf-descent in scheme theory and state how it will be
resolved.
16:05 – 16:55

根本裕介（千葉大学）/ Yusuke Nemoto (Chiba University)

On the K2 -regulator of the Hesse cubic curve and hypergeometric functions
It is known that the Beilinson regulators of certain classes of algebraic varieties are described in terms ofgeneralized hypergeometric functions. In this talk, we discuss the Hesse cubic curve defined by x30 +y03 +z03 =3tx0 y0 z0 . It is known that its periods are written in terms of Gauss’s hypergeometric functions. We
construct explicitly some elements in the K2 -group of the Hesse cubic curve by using the tangent lines at
the flex points. We compute their Beilinson regulators via the dlog map and express them in terms ofgeneralized hypergeometric functions 3 F2 and Kampé de Fériet hypergeometric functions. Our elements
are integral for some t, and we verify numerically the Beilinson conjecture on the special value of the
L-function.

17:05 – 17:55

Peijiang Liu（東京大学）/ Peijiang Liu (University of Tokyo)

The characteristic cycles of non-confluent `-adic GKZ hypergeometric sheaves`-adic GKZ hypergeometric sheaves are defined to be étale analogues of GKZ hypergeometric D-modules.
We introduce an algorithm of computing the characteristic cycles of certain type of `-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 thecharacteristic cycles of `-adic GKZ hypergeometric sheaves and those of GKZ hypergeometric D-modules.
We also explain the limitation of our algorithm.

7 月 15 日（金）9:30 – 10:20

齋藤

耕太（筑波大学）/ Keita Saito (University of Tsukuba)

Topological properties and algebraic independence of sets of prime-representing constants
Let (ck )∞k=1 be a sequence of positive integers. We investigate the set of A > 1 such that the integer
part of Ac1 ···ck is always a prime number for every positive integer k. Let W(ck ) be this set. The firstgoal of this talk is to determine the topological structure of W(ck ). Under some conditions on (ck )∞k=1 , wereveal that W(ck ) ∩ [0, a] is homeomorphic to the Cantor middle third set for some a. The second goal is to
propose an algebraically independent subset of W(ck ) if ck is rapidly increasing. As a corollary, we disclosethat the minimum of W(k) is transcendental. In addition, we apply the main result to W(ck ) in the casewhen c1 · · · ck = 3k! . As a consequence, we give an algebraically independent and countably infinite subsetof W(ck ). This research is joint work with Wataru Takeda (Tokyo University of Science).
10:30 – 11:20

加藤裕基（宇部高専）/ Yuki Kato (National Institute of Technology, Ube College)

Almost mathematics and its algebraic K-theory from non-unital algebras viewpoint
While almost mathematics has various applications to arithmetic geometry, Quillen mentioned linear
algebra over non-unital rings which is the same as almost mathematics introduced by Faltings; and Gabberand Ramero. Quillen’s work is more conceptional in the sense of using categorical language: in his work,
almost mathematics is characterized as bilocalization of an abelian category of modules. By using thetheory of bilocalization, we define the derived category (or stable ∞-category) of almost perfect complexes,
enabling us to obtain the K-theory of almost modules, and compare the Thomason–Troubauth K-theoryof almost perfect complexes and perfect complex whose weak equivalences are almost quasi-isomorphisms.
Furthermore, by modifying K-theory of almost algebra, we prove that the modified K-groups decompose
into the almost algebraic K-theory and the almost acyclic part of them.
11:35 – 12:25

後藤

新裕（九州大学）/ Akihiro Goto (Kyushu University)

Congruence relations connecting Tate-Shafarevich groups with Bernoulli-Hurwitz numbers by
elliptic Gauss sums in Eisenstein integers case√
There are classical congruences between the class number of the imaginary quadratic field Q( −p)for a rational prime p > 3 and a Bernoulli number or an Euler number. Under the BSD conjecture on
the 2-parts of the leading term, Onishi obtained an elliptic generalization of these congruences, which
gives congruences between the order of the Tate–Shafarevich group of certain elliptic curves with complex√
multiplication by the Gaussian integers Z[ −1] and Mordell–Weil rank 0, and a coefficient of power series√
expansion of an elliptic function associated to Z[ −1]. In this paper, we provide Onishi’s type congruences√

for the Eisenstein integers Z[ −1+2 −3 ].

14:00 – 14:50

任鑫（関西大学）/ Ren Xin (Kansai University)

On q-deformation of some real numbers
The q-real numbers was introduced by S. Morier-Genoud and V. Ovsienko in 2019. In this talk, we
focus on q-metallic numbers and q-rational sequences which converge to q-metallic numbers. We found arecurrence formula of the sequence that emerged from the coefficient of q-rational numbers and q-metallic
numbers. We consider the radiuses of convergence of them when we assume that q is a complex number.
We give an estimation of radiuses of convergence of them, and we solve on conjecture of the lower boundexpected which is introduced by L. Leclere, S. Morier-Genoud, V. Ovsienko, and A. Veselov for the metallic
numbers and its convergence rational sequence.
15:00 – 15:50

長町

一平（京都大学数理解析研究所） / Ippei Nagamachi (RIMS, Kyoto University)

Minimal log regular models of hyperbolic curves over discrete valuation rings
Deligne and Mumford proved that a proper hyperbolic curve over a discrete valuation ring R has stablereduction if and only if the Jacobian variety of the curve has stable reduction in the case where the residue
field k of R is algebraically closed. In the proof, the theory of minimal regular models played an importantrole. In this talk, we establish a theory of minimal log regular models of curves. As a key tool for this theory,
we introduce a notion of “log blow-down” and give a scheme-theoretic characterization of 2-dimensionallog regular local log schemes. Moreover, as an application of this theory, we prove the above equivalence
without the assumption on k.