第 19 回仙台広島整数論集会アブストラクト
2020 年 9 月 8 日（火）〜9 月 11 日（金）Zoom によるオンライン開催

9 月 8 日（火）10:00 – 11:00

松坂俊輝（名古屋大学） / Toshiki Matsusaka (Nagoya University)

Rademacher 記号の双曲類似
Hyperbolic analogue of the Rademacher symbol
One of the most famous results of Dedekind is the transformation law of log η(z). In 1956,
Rademacher introduced an SL2 (Z)-conjugacy class invariant (integer-valued) function Ψ(γ) by
modifying Dedekind’s result. We call it the Rademacher symbol. The symbol appears in several
ﬁelds of mathematics. Inspired by Ghys’ work on modular knots, Duke-Imamoḡlu-Tóth (2017)
constructed a hyperbolic analogue of the symbol Ψ(γ).
In this talk, we give a new expression of their hyperbolic Rademacher symbol in terms of thegeodesic integral of the hyperbolic Eisenstein series, and show some properties.

11:15 – 12:15

植木潤（東京電機大学） / Jun Ueki (Tokyo Denki University)

ツイスト結び目の非自明な L 不変量と 1 の 3n − 1 乗根

Non-trivial L-invariants of twist knots and (3n − 1)-th roots of unity
We study irreducible SL2 representations of the fundamental group of the exterior of the twist
knot J(2, 2n) with n ∈ Z. Our main tools are the character variety, the (acyclic) Reidemeister
torsion τ , and the Chebyshev polynomials. For the universal deformation over a CDVR O of aresidual representation, the (algebraic) L-invariant is deﬁned as a certain torsion invariant. Our
results are as follows.
Over any ﬁeld, every non-acyclic representation corresponds to 3n − 1-th root of unity. Nonacyclic representations are topologically characterized (factoring through a certain exceptional
representation +α). Every non-acyclic residual representation admits a universal deformation
over SL2 O[[x − α]]. The L-invariant is nontrivial iﬀ the residual representation is non-acyclic.
For a certain polynomial kn (x) ∈ Z[x] relating to 3n − 1-th roots of unity, we have L = (kn )2 up
to multiplication by units.
These results would shed new light on the analogy between Hida–Mazur’s Galois deformation theory and Thurston’s hyperbolic deformation theory. This talk is based on the jointwork(s) with Ryoto Tange (Kogakuin U.), Anh T. Tran (U.Texas at Dallas), and Leo Benard(GAU, Göttingen).

13:30 – 14:30

前田洋太（京都大学）/ Yota Maeda (Kyoto University)

ユニタリ型志村多様体上の特異点と小平次元への応用

On the singularities of unitary Shimura varieties and their applications to the Kodaira dimension
The Kodaira dimension of Shimura varieties has been studied by many peolple. Kondo
showed that certain orthogonal Shimura varieties related with the moduli spaces of K3 surfaces
have non-negative Kodaira dimension by using the Borcherds lift. Recently, Gritsenko-HulekSankaran proved there exist toroidal compactiﬁcations of orthogonal Shimura varieties whichhave canonical singularities and no branch divisors in the boundary components. As anapplication, they obtained better results on the Kodaira dimension than Kondo’s results. In
this talk, we consider similar problems for unitary Shimura varieties. We construct toroidal
compactiﬁcations of some unitary Shimura varieties satisfying the above properties. Finally, we
give an application to the Kodaira dimension of unitary Shimura varieties in terms of reﬂective
modular forms.

14:45 – 15:45

高松 哲平（東京大学）/ Teppei Takamatsu (University of Tokyo)

On the ﬁniteness of twisted forms of hyperkahler varieties
For a ﬁnite ﬁeld extension L/K and a variety X over K, let TwL/K be a set of isomorphismclasses of varieties Y over K which are isomorphic to X after a base change to L. Such a variety
Y is called a twisted form of X, and the diversity of twisted forms is one of the diﬃculties of thestudy of varieties over number ﬁelds. In this talk, we consider the ﬁniteness of TwL/K for K3
surfaces of characteristic away from 2, and hyperkähler varieties of characteristic 0. This work
is a generalization of Cattaneo–Fu’s work on real forms of hyperkähler varieties. We also give
an application to the ﬁniteness of derived equivalent hyperkähler varieties.

16:00 – 17:00

山田一紀（慶應義塾大学） / Kazuki Yamada (Keio University)

Hyodo–Kato theory with syntomic coeﬃcients
Hyodo–Kato theory in the original sense is a theory of comparison between two p-adic cohomologies, the de Rham cohomology and the log crystalline cohomology. In this talk, we study
coeﬃcients in Hyodo–Kato theory under the rigid analytic reinterpretation given by our previous work with Veronika Ertl. We ﬁrst introduce a notion of log overconvergent F-isocrystals
as coeﬃcients of the Hyodo–Kato cohomology, and give a rigidity theorem for them. Then wedeﬁne a certain notion of coeﬃcients of the p-adic Hodge cohomology, and study their fundamental properties, e.g. the independence from a choice of uniformizer, the relation with ﬁltered(φ, N )-modules, and the admissibility.

9 月 9 日（水）10:00 – 11:00

清水陵嗣（京都大学）/ Ryoji Shimizu (Kyoto University)

制限分岐版ノイキルヒ・内田の定理について

The Neukirch-Uchida theorem with restricted ramiﬁcation
The Neukirch-Uchida theorem is one of the most important results in anabelian geometry.
In this talk, I will discuss the following generalization of the Neukirch-Uchida theorem undersome assumptions: “For i = 1, 2, let Ki be a number ﬁeld and Si a set of primes of Ki . We
write Ki,Si /Ki for the maximal extension of Ki unramiﬁed outside Si and GKi ,Si for its Galois
group. If GK1 ,S1 and GK2 ,S2 are isomorphic, then K1 and K2 are isomorphic.” The assumptions
include: the Dirichlet density of Si is not zero for i = 1, 2; Ki is Galois over Q for i = 1, 2; K1 is
totally imaginary; and so on.

11:15 – 12:15

山口永悟（京都大学）/ Naganori Yamaguchi (Kyoto University)

The m-step solvable Grothendieck conjecture for genus 0 curves over ﬁnitely generated ﬁelds
In anabelian geometry, the Grothendieck conjecture is one of the most important problem. For
this conjecture, many previous results were obtained so far (Tamagawa, Mochizuki, et al). Asa variant, we can also consider the m-step solvable Grothendieck conjecture. Brieﬂy, this is theproblem of reconstructing the isomorphism class of a curve from the maximal geometricallym-step solvable quotient of the arithmetic fundamental group of the curve. For this variant,previous results were obtained by Nakamura and Mochizuki. In this talk, I will outline theproof of my result on the m-step solvable Grothendieck conjecture for curves of genus 0 overﬁelds ﬁnitely generated over the prime ﬁeld.

13:30 – 14:30

田中秀宜（東洋大学）/ Hidenori Tanaka (Toyo University)

On the Euler product expressions of the absolute tensor products of the Dirichlet L-functions
In 1992 Kurokawa deﬁned the absolute tensor product (Z1 ⊗ · · ·⊗ Zr )(s) of some zeta functions
F1F1Zj (s) (j = 1, · · · , r) as a function which had zeros or poles only at s = ρ1 + · · · ρr where ρj ∈ Cwith Zj (ρj ) = 0 or ∞ and predicted it to have the expression by the Euler product over r-tuples
of primes if Zj (s) was represented by the Euler product over primes for each j ∈ {1, · · · , r}.
The validity of Kurokawa’s prediction has been conﬁrmed in some cases, for example, the casesof the Hasse zeta functions of ﬁnite ﬁelds by Koyama and Kurokawa for r = 2, by Akatsuka forr = 3 and by Kurokawa and Wakayama for general r. Also, the case of the Riemann zeta function
for r = 2 was ﬁrst proved by Koyama and Kurokawa, and then by Akatsuka in a diﬀerent way.
In this talk, I will explain the method to construct the Euler product expression of the absolutetensor product on the Dirichlet L-functions L(s, χj ) (j = 1, · · · , r) which is a generalization of
Akatsuka’s method for the Riemann zeta function and show that Kurokawa’s prediction is valid
in the case of the Dirichlet L-functions for r = 2.

14:45 – 15:45

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

Rikuna’s generic cyclic polynomial and the monogenity
Let K/k be an extension of number ﬁelds and OK , Ok be the rings of integers of K, k, respectively. It is said that an element α of K forms a power integral basis for K/k if OK = Ok [α]. Insuch a case, we say K/k is monogenic. For an odd prime l, we put k = Q(ω) with ω = ζ + ζ −1
where ζ is a primitive l-th root of unity. We consider a cyclic extension Ks /k of degree l deﬁned
by Rikuna’s generic cyclic polynomial, parameterized by s ∈ Ok . In this talk, We introduce
a suﬃcient condition for the monogenity of Ks /k and a proof that there exist inﬁnitely many
monogenic cyclic extensions Ks /k, although monogenic cyclic extensions of degree l ≥ 5 over
Q are rather rare.

16:00 – 17:00

山田智宏（大阪大学）/ Tomohiro Yamada (Osaka University)

On the problem of de Koninck
Let σ(n) and γ(n) denote the sum of divisors and the product of distinct prime divisors ofn, called the radical or the squarefree kernel of n, respectively. De Koninck asked if the only
solutions of σ(n) = (γ(n))2 are n = 1 and 1782. We shall show that if n ̸= 1, 1782 and σ(n) =(γ(n))2 , then there exist odd (not necessarily distinct) primes p, p′ and (not necessarily odd)
primes qi (1 ≤ i ≤ k) with k ≤ 3 such that p, p′ || n, qi2 || n(1 ≤ i ≤ k) and q1 | σ(p2 ), qi+1 |σ(qi2 )(1 ≤ i ≤ k − 1), p′ | σ(qk2 ).

9 月 10 日（木）10:00 – 11:00

伊藤/Sprung Florian（アリゾナ州立大学） / Ito/Sprung Florian (Arizona StateUniversity)
楕円曲線の虚二次体の Z2p 拡大における有理点の Mordell-Weil 階数

The Mordell-Weil rank of elliptic curves in Z2p -extensions of imaginary quadratic ﬁelds
The Mordell-Weil rank remains bounded in cyclotomic Zp -extensions of a number ﬁeld K,
even though there are inﬁnite layers (and therefore the rank has inﬁnite chances to increase).
For K = Q, this is the end of the story because the cyclotomic Zp -extension is the only one.
When looking at the next simplest case – where K is a quadratic imaginary ﬁeld – there areinﬁnitely many Zp -extensions, and it is known that the rank remains bounded in most of them,while for one special extension (the anticyclotomic one), the rank may be unbounded. In joint
work with A. Lei, we give an estimate for the growth of the rank in all the Zp -extensions byconsidering their compositum, the Z2p -extension of K.

11:15 – 12:15

野本 慶一郎（九州大学） / Keiichiro Nomoto (Kyushu University)

On the rank of the elliptic curve y 2 = x3 + px and a recurrence formula
Let p be a prime number and Ep denote the elliptic curve y 2 = x3 + px. It is known that forp which is congruent to 1, 9 modulo 16, the rank of Ep over Q is equal to 0, 2. Under the condition that the Birch and Swinnerton-Dyer conjecture is true, we give a necessary and suﬃcientcondition that the rank is 2 in terms of the constant term of a polynomial that is deﬁned by arecurrence formula.

13:30 – 14:30

金村 佳範（慶應義塾大学） / Yoshinori Kanamura (Keio University)

How to calculate the proportion of everywhere locally soluble diagonal hypersurfaces
Rational points on diagonal hypersurfaces have been studied for a long time in connection
with the Waring problem and the Fermat’s Last Theorem. In this talk, we give a strategy tocalculate the proportion of everywhere locally soluble diagonal hypersurfaces. Here, we consider the family of diagonal hypersurfaces of the projective space for every ﬁxed dimension
and degree. Our strategy is based on the product formula established by Bright, Browning andLoughran. As working examples, we give the proportions of diagonal hypersurfaces with rational points for each dimension under a certain hypothesis on Brauer-Manin obstruction when
they are quadratic and cubic. This is a joint work with Yoshinosuke Hirakawa at Keio University(arXiv:2003.11426).

14:45 – 15:45

中川彬雄（千葉大学）/ Akio Nakagawa (Chiba University)

対角超曲面の Artin L 関数と有限体上の一般超幾何関数

Artin L-functions of diagonal hypersurfaces and generalized hypergeometric functions over ﬁnite ﬁelds

In general, if a ﬁnite group G acts on a variety V then we can deﬁne the Artin L-function ofV associated to a character of G, and the congruence zeta function of V is decomposed to theArtin L-functions of V .
In this talk, we compute the Artin L-function of a diagonal hypersurface Dλ over a ﬁnite ﬁeld
associated to a character of a ﬁnite group acting on Dλ , and under some condition, express it interms of hypergeometric functions and Jacobi sums over the ﬁnite ﬁeld. As an application, weconsider the Dwork hypersurfaces and obtain relations between certain hypergeometric functions over diﬀerent ﬁnite ﬁelds.

16:00 – 17:00

隈部哲（九州大学）/ Satoshi Kumabe (Kyushu University)

Dwork hypersurfaces of degree six and Greene’s hypergeometric function
Heidi Goodson expressed the number of the rational points on the Dwork hypersurfaces ofdegree four and of odd degree over the ﬁnite ﬁeld by using Greene’s ﬁnite ﬁeld hypergeometricfunction. Furthermore, Goodson asked about an existence of a formula for the Dwork hypersurfaces of even degree. We extend Goodson’s formula to the Dwork hypersurfaces of degree six.
Furthermore, our formula is a higher-dimensional and a ﬁnite ﬁeld analogue of MatsumotoTerasoma-Yamazaki’s formula.

9 月 11 日（金）10:00 – 11:00

浅山 拓哉（東京工業大学）/ Takuya Asayama (Tokyo Institute of Technology)

大きな代数拡大体上の Drinfeld 加群の捩れ点

Torsion points of Drinfeld modules over large algebraic extensions
Fix a ﬁnitely generated ﬁeld K over its prime ﬁeld. For each e-tuple σ = (σ1 , σ2 , . . . , σe ) ∈Gal(K sep /K)e , let K̃(σ) be the ﬁxed ﬁeld by σ in the algebraic closure K̃ of K. Geyer and Jarden
studied the torsion parts of elliptic curves over K̃(σ) and obtained some ﬁniteness or inﬁnitenesstheorems for the torsion subgroups. In this talk, I discuss a Drinfeld module analogue of GeyerJarden’s results and give some partial answers analogous to them.

11:15 – 12:15

石井 竣（京都大学）/ Shun Ichii (Kyoto University)

On a variant of the uniform boundedness conjecture for Drinfeld modules
Drinfeld modules are function ﬁeld analogues of abelian varieties. Based on this analogy,Poonen conjectured the uniform boundedness conjecture for torsion of Drinfeld modules.
In this talk, we prove a result on the uniform boundedness conjecture for 1-dimensionalfamilies of Drinfeld modules over ﬁnitely generated ﬁelds of generic characteristic. This is ananalogue of Cadoret-Tamagawa’s result on the p-primary uniform boundedne ss conjecture for
1-dimensional families of abelian varieties.

13:30 – 14:30

竹内 大智（東京大学）/ Daichi Takeuchi (The University of Tokyo)

Symmetric bilinear forms and Milnor formula for local epsilon factors
Let f : X → A1k be a morphism from a smooth variety to an aﬃne line with an isolated singular point. For such a singularity, we have two invariants. One is a non-degenerate symmetricbilinear form (de-Rham), and the other is the vanishing cycles complex (étale).
In this talk, I will give a formula which expresses the local epsilon factor of the vanishingcycles 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 ofMilnor formula, which compares the dimension of the bilinear form and the total dimension of
the vanishing cycles.
In characteristic 2, we ﬁnd a generalization of Arf invariant, which is an invariant for an ordinary quadratic singularity, to a general isolated singularity.

14:45 – 15:45

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

On congruence of Galois representations attached to A-motives
Let K/Q be a ﬁnite extension of global function ﬁelds over ﬁnite ﬁelds and p a ﬁnite place ofQ. Let GK be the absolute Galois group of K. Now two p-adic GK -representations V and V ′ aresaid to be congruence if associated residue representations are isomorphic in a suitable sense.
Then it is natural to discuss when V and V ′ themselves are isomorphic in a suitable sense. In thistalk we consider this problem for Galois representations coming from mixed A-motives over K,
which are introduced by Anderson as a higher-dimensional analogue of Drinfeld A-modules.
Let v and u be places of K with v ∤ p and u | p. Using the theory of equal-characteristic crystalline representations, for suﬃciently large p, we prove that congruent V and V ′ are isomorphic
on GKv in a suitable sense if they come from mixed A-motives over K with some reduction conditions (with respect to v and u) and bounded Hodge-Pink weights. This is an analogue of thework of Ozeki and Taguchi on congruence of ℓ-adic representations.

16:00 – 17:00

角濱 寛隆（京都大学）/ Kakuhama Hirotaka (Kyoto University)

Formal degree and Theta correspondence: quaternion cases
By the formal degree conjecture, the behavior of formal degree under the theta correspondence (with a certain rank condition) is expected to be described by a special value of standard
γ-factor and some constants. Gan and Ichino has calculated the behavior except for the quaternion cases. In this talk, I extend their work to the quaternion cases. The main ingredient is a
precise analyzation of the local Siegel-Weil formula.