第 18 回広島仙台整数論集会アブストラクト
2019 年 7 ⽉ 9 ⽇（⽕）〜 7 ⽉ 12 ⽇（⾦）
広島⼤学理学部（東広島キャンパス）B 棟 B707 号室

7 ⽉ 9 ⽇（⽕）
10:45 – 11:45

佐藤信夫（九州⼤学）/ Nobuo Sato (Kyushu University)

A multivariable generalization of Zagier’s identity for ζ(2, .., 2, 3, 2, .., 2) and Zhao’s 2-1 type formula
Zhao showed a beautiful mysterious identity which relates the block structure of multiple zeta star values
and the depth structure of Euler sums. Zagier proved a formula expressing block degree one Hoﬀman basis
ζ(2, . . . , 2, 3, 2, . . . , 2) by a Q-linear combination of ζ(2l + 1)π 2k . Zagier’s formula can be reformulated andgeneralized into a simple iterated integral identity which has striking similarity to Zhao’s formula. In mytalk, I’ll introduce a new type of diﬀerential forms and generalize those identities to multivariable iterated
integral identities. This is a joint research work with Minoru Hirose.

12:00 – 13:00

広瀬 稔（九州⼤学）/ Minoru Hirose (Kyushu University)

Alternating multiple zeta values and their bases
Alternating multiple zeta values are signed analogues of multiple zeta values sometimes called Euler sums,
mod two multiple zeta values, etc.. In his work on motivic multiple L-values, Deligne gave an explicit basis
for the space of motivic alternating multiple zeta values. His proof makes use of the theory of mixed Tate
motives, which does not provide a way to compute the expression (coeﬃcients) of a given alternating multiple
zeta values by his basis. The explicit reduction of alternating multiple zeta values into linear combinationsof Deligne’s basis (or any other explicit basis) is an open problem. In this talk, I report some recent progresson this problem made in a collaboration with Nobuo Sato at Kyushu University.

14:30 – 15:30

⻑町 ⼀平（東京⼤学）/ Ippei Nagamachi (University of Tokyo)

On the Hom version of the Grothendieck Conjecture for morphisms from regular varieties to hyperbolic polycurves ofdimension 2Mochizuki proved the Hom version of the Grothendieck Conjecture for hyperbolic curves, that is, a conjecture that the mapopenMordom
K (Y, X) → HomGK (π1 (Y, ∗), π1 (X, ∗))/Innis bijective for any sub-p-adic field K, any hyperbolic curve X over K, and any smooth variety Y over K. Inthe case where X is a hyperbolic polycurve of dimension 2, Hoshi proved the injectivity of the map and thatany element ϕ ∈ Homopen
GK (π1 (Y, ∗), π1 (X, ∗))/Inn with topologically finitely generated kernel is in the imageof the map. In this talk, we give a “group theoretic” condition for ϕ such that ϕ satisfies this condition ifand only if ϕ is in the image of the map. We also show that any open outer homomorphism is induced by amorphism of the varieties under the assumption that the Grothendieck Section Conjecture holds.

15:45 – 16:45

笠浦⼀海（東京⼤学）/ Kazumi Kasaura (University of Tokyo)

On extension of overconvergent log isocrystals on log smooth varieties
By works of Kedlaya and Shiho, it is known that, for a smooth variety X over a field of positive characteristic
and its simple normal crossing divisor Z, an overconvergent isocrystal on the compliment of Z satisfying a()
certain monodromy condition can be extended to a convergent log isocrystal on X, MZ , where MZ is thelog structure associated to Z. In this talk, we explain a generalization of this result: for a log smooth variety()
X, M satisfying some conditions, an overconvergent log isocrystal on the trivial locus of a direct summand()of M satisfying a certain monodromy condition can be extended to a convergent log isocrystal on X, M .
17:00 – 18:00

⾼⽥ 芽味（九州産業⼤学） / Megumi Takata (Kyushu Sangyo University)p 進体 F の過収束性と F 解析性との関係について
On a relation of overconvergence and F -analyticity on p-adic Galois representations of a p-adic field F
Let p be a prime number. There are a property called “overconvergence” for p-adic Galois representationsof a p-adic field F . If a given p-adic Galois representation of F is overconvergent, then we can study it byusing the theory of (φ, Γ)-modules over the univariable Robba ring. The case F = Qp is nice, since any p-adic
representation is overconvergent, which is proved by Cherbonnier-Colmez. Unfortunately, if F ̸= Qp , thenthere is a p-adic Galois representation of F which is not overconvergent. Berger has proved that anotherproperty called “F -analyticity” yields overconvergence. We note that F -analyticity is very strong, as one cansee that the p-adic cyclotomic character is not F -analytic if F ̸= Qp . Hence it seems to be a natural questionthat how strict F -analyticity is in comparison to overconvergence. In this talk, I discuss a result that, inmany cases, an overconvergent Galois representation is F -analytic up to a twist by a character. This resultemphasizes the necessity of the theory of (φ, Γ)-modules over the multivariable Robba ring, by which we
expect to study all p-adic Galois representations.
7 ⽉ 10 ⽇（⽔）
9:30 – 10:30

奥村喜晶（東京⼯業⼤学） / Yoshiaki Okumura (Tokyo Institute of Technology)

Parametrization of Q-virtual Drinfeld modules
Elliptic curves over Q are classical and important objects in number theory. As a generalization, it is natural
to consider elliptic curves over Q̄ which are isogenous to all GQ -conjugates, which are called Q-curves. It is
known that various properties of elliptic curves over Q can be extended to Q-curves. For example, Elkies
proved that all isogeny classes of non-CM Q-curves are corresponds to Q-rational points of some modularcurves.
Let Q = Fq (T ) be a rational function field over a finite field Fq . In the arithmetic of function fields, Drinfeldmodules (of rank two) over Q play the role that elliptic curves over Q do in the arithmetic of number fields.
In this talk, we introduce the notion of Q-virtual Drinfeld modules as a function field analogue of Q-curves,and to parametrize (up to isogeny) them by Q-rational points of the Drinfeld modular curves X∗ (n).

10:45 – 11:45

関川隆太郎（東京理科⼤学） / Ryutaro Sekigawa (Tokyo University of Science)

Relative power integral bases in ray class fields of an imaginary quadratic number field
Let L/K be an extension of number fields and let OL and OK be the rings of integers of L and K, respectively. It is said that OL has a OK -power integral basis when there is α ∈ OL such that OL = OK [α]. Let K
be an imaginary quadratic number field and H be the Hilbelt class field of K. Schertz proved that ray class√fields of K have a OH -P.I.B. with some exceptions. Let K = Q( −10) and p ⊂ K be the prime ideal over 2.
In this talk, I explain the way to construct a P.I.B. for Kp4 /H that is one of the exceptions and introduce its
application. The main tools of this way are fundamental units by PARI/GP.
12:00 – 13:00

⽥嶋 和明（仙台⾼専・東北⼤学） /Kazuaki Tajima (Sendai National Colledge of Technology, Tohoku University)
On the GIT stratification of prehomogeneous vector spaces
We have established a combinatorial method to determine a stratification (GIT stratification) based on geometric invariant thoery. We applied this method to certain prehomogeneous vector spaces using computor
computations and determinedtheir GIT stratifications. The result has been known if the ground field is C.
This method enables us to determine the stratification rationally over any perfect field. This is joint workwith Akihiko Yukie.

14:30 – 15:30

福永 健吾（⼤阪⼤学）/ Kengo Fukunaga (Osaka University)p-adic triple product L-function attached to p-adic families of modular forms
In his preprint “Hida families and p-adic triple product L-functions”, Ming-Lun Hsieh constructed threevariable p-adic triple product L-functions attached to triples (F, G, H) of primitive Hida families and heproved the interpolation formulas.
In the unbalanced case, I generalized his result and constructed a three-variable p-adic triple productL-function attached to a primitive Hida family F and general p-adic families of modular forms G, H. For
example, we can take G and H to be Colman families or CM-families. I will explain the construction ofthis p-adic triple product L-function.
15:45 – 16:45

伊藤 望（京都⼤学）/ Nozomi Ito (Kyoto University)

On Miyawaki lifts with respect to Hermitan Maass lifts
Miyawaki lifts are liftings constructed by using the pullbacks of Ikeda lifts. They have been studied intensively in recent years, but their properties, especially non-vanishing, have not been suﬃciently understood.
In this talk, we study Miyawaki lifts of automorphic characters of U(1) with respect to Hermitian Maasslifts, which is Ikeda lifts for U(2, 2).

7 ⽉ 11 ⽇（⽊）
9:30 – 10:30

⽯本宙（京都⼤学）/ Hiroshi Ishimoto (Kyoto University)

Local intertwining relation for metaplectic groups
In 2012, Gan and Savin established the local Langlands correspondence (LLC) for metaplectic groups. Theytransferred LLC for odd special orthogonal groups, via the theta correspondence. LLC should satisfy a lot ofnatural properties, and local intertwining relation (LIR) is one of them. We can also transfer LIR by applying
the the mixed model, which was introduced by Gan and Ichino, to our case.
10:45 – 11:45

⼭本 祐輝（東京⼤学）/ Yuki Yamamoto (University of Tokyo)

GLN の内部形式の超尖点表現の type について
On the types for supercuspidal representations of inner forms of GLN
Let G be the multiplicative group of a central simple algebra over a non-Archimedean local field. Whenwe consider smooth representations of G, the theory of types is useful. A type is an irreducible smooth
representation of some compact open subgroup in G, which can classify irreducible representations of G
in a certain sense. I will explain types and discuss the existence and uniqueness of types. In particular, Iwill show that for an irreducible supercuspidal representation π, [G, π]G -types defined over some maximal
compact subgroup in G are unique up to G-conjugation under some unramifiedness assumption on a simple
stratum for π.
12:00 – 13:00

松村 英樹（慶應義塾⼤学） / Hideki Matsumura (Keio University)

Infinitely many hyperelliptic curves with exactly two rational points
In arithmetic geometry, it is one of central problems to determine the sets of rational points on algebraic
varieties. In this talk, we construct some families of infinitely many hyperelliptic curves with exactly two
rational points. The key tools of the proof are the 2-descent argument, the Lutz-Nagell type theorem for hyperelliptic curves (Grant’s theorem) and the Richelot isogeny. This is based on a joint work with Yoshinosuke
Hirakawa (arXiv: 1904.00215v2).
14:30 – 15:30

厳 冬（⼤阪⼤学）/ Dong Yan (Osaka University)

剰余可約な肥⽥変形における岩澤不変量の変動について

Variation of the two-variable Iwasawa invariant in residually reducible Hida deformation
The two-variable Iwasawa theory for Hida deformation is studied by Ochiai, Skinner-Urban and OchiaiShimomoto. Especially, they proved the two-variable Iwasawa main conjecture partially when the residual
representation is irreducible.
In this talk, I study the two-variable Iwasawa theory when the residual representation is reducible. Oneof subtle points in this case is that the Selmer group and its characteristic ideal depend on the choice oflattices of the two-variable Galois representation associated to Hida family. I show that there is a one-to-one
correspondence between the set of all characteristic ideals and the set of all factors of the reflexive closureof Eisenstein ideal. Combining this result with a recent result of Bellaïche-Pollack, we give some examples
where the two-variable Iwasawa main conjecture for residually reducible Hida deformation is true.

15:45 – 16:45

村上和明（慶應義塾⼥⼦⾼等学校） / Kazuaki Murakami (Keio Girls Senior High School)

On the cyclicity of the unramified Iwasawa modules of the maximal multiple Zp -extensions over imaginary quadraticfields
For an odd prime number p, we study the number of generators of the unramified Iwasawa modules ofthe maximal multiple Zp -extensions over Iwasawa algebras. In this talk, for an imaginary quadratic field, wegive an necessary and suﬃcient condition for the Iwasawa module to be cyclic under several assumptions andgive numerical examples. Moreover, as an application, we give an upper bound of λ-invariants for infinitely
many Zp -extensions of an imaginary quadratic field. This is joint work with Takashi Miura, Rei Otsuki, andKeiji Okano.
17:00 – 18:00

⽚岡 武典（慶應義塾⼤学） / Takenori Kataoka (Keio University)

楕円曲線の同変岩澤理論

Equivariant Iwasawa theory for elliptic curves
Iwasawa theory, which began with the study of ideal class groups, has now quite many extensions andrefinements. In this talk, we discuss abelian equivariant Iwasawa theory for elliptic curves over Q. Firstly,
we formulate equivariant main conjectures in this situation. Secondly, under certain assumptions, we prove
one divisibility via Euler system machinery. Finally, as an application, we prove a case of a conjecture of
Mazur-Tate.

7 ⽉ 12 ⽇（⾦）
9:30 – 10:30

⽊村昭太郎（早稲⽥⼤学） / Shotaro Kimura (Waseda University)

Modular diﬀerential equations for skew holomorphic Jacobi forms
In 1998, Kaneko and Zagier introduced modular diﬀerential equations for elliptic modular forms of one
variable. The modular form solutions of these diﬀerential equations is related to the supersingular polynomial and given in terms of hypergeometric polynomials. In 2016, Kiyuna constructed modular diﬀerential
equations and found its solutions for holomorphic Jacobi forms of two variables. I constructed modular differential equations for skew holomorphic Jacobi forms which is not holomorphic of two variables. Theseequations have similarities with the case of elliptic modular forms and diﬀerences from the case of holomorphic Jacobi forms. In this talk, I will begin with reviewing known results and talk about my recent results.
10:45 – 11:45

境優⼀（九州⼤学）/ Yuichi Sakai (Kyushu University)

Modular linear diﬀerential equations in general form
It is well known that modular linear diﬀerential equations(MLDEs) appear as tools in studies related to supersingular elliptic curves and classifications of characters of vertex operator algebras(VOAs). In some cases,
MLDEs give a certain correspondence between modular forms and characters of VOAs. In this talk, we determine the properties of coeﬃcients of MLDEs of any order by using the definition of MLDEs only. Furthermore, we give a general expression of MLDEs under the natural assumption for non-compact (quasi)modularforms. This is a part of a joint work with K. Nagatomo and D. Zagier.

12:00 – 13:00

中屋智瑛（九州⼤学）/ Tomoaki Nakaya (Kyushu University)

On the inductive structure of certain quasimodular forms of depth ≤ 4
The extremal quasimodular forms of weight k and depth r = 1, 2 on SL2 (Z) were studied by Kaneko
and Koike. These forms are defined inductively by using diﬀerential operators, and satisfy certain modularlinear diﬀerential equations(MLDEs). In this talk, we will construct certain quasimodular forms of weight
k and depth ≤ 4 on SL2 (Z) and prove that these forms satisfy corresponding MLDEs of order r + 1.