Low dimensional topology and number theory XVC502 Lecture Rm, 5F West 1st Bd, Kyushu University (Ito Campus)

5th March, 2024 ∼ 8th March, 2024
ProgramMarch 5th (Tues)
10:00 – 10:25
Ryoto Tange（Waseda University)
Survey on Iwasawa theory for representations of knot groups
10:35 – 11:00
Mishty Ray (University of Calgary)
Introduction to Geometry of local Arthur packets and Vogan’s conjecture
11:20 – 12:10
Sohei Tateno（Nagoya University）
On Kida’s formula for Zdp -covers of graphs
13:40 – 14:30
Mahiro Atsuta（Tsuda University）
Euler system for Tate motives over a totally real ﬁeld
14:50 – 15:40
Anwesh Ray（Chennai Mathematical Institute)
Topological Iwasawa invariants and arithmetic statistics
March 6th (Wednes)
9:40 – 10:30
Xin Ren （Kansai University）
q-deformed rational numbers, q-deformed Farey sum and a triangulated category of A2 quiver
10:50 – 11:40
Mai Katada (Kyushu University)
The ﬁrst homology of the IA-automorphism groups of free groups with coeﬃcients in spaces of Jacobi diagrams
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13:20 – 14:10
Benjamin Collas（RIMS, Kyoto University)
Oda’s prediction, a motivating conjecture
14:40 – 15:30Séverin Philip (RIMS, Kyoto University)
The maximal degeneration method for special loci
15:50 – 16:40
Yasuhiro Wakabayashi（Osaka University）
Topological quantum ﬁeld theory for diﬀerential equations in characteristic pN
March 7th (Thurs)
9:40 – 10:30
Nadav Gropper (University of Pennsylvania/University of Haifa ）TQFTs for pro-p Poincaré duality groups
10:50 – 11:15
Yuki Ishida（Kyushu University）
The density of Borromean primes
11:25 – 11:50
Yuqi Deng (Kyushu University)
Arithmetic Dijkgraaf–Witten invariants for real quadratic ﬁelds, quadratic
residue graphs, and density formulas
13:20 – 14:10
Toshiki Matsusaka (Kyushu University)
Introduction to quantum modular forms and discontinuity property of Habiro
series
14:40 – 15:30
Hisatoshi Kodani (IMI, Kyushu University)
Quantum master equation and Hodge correlators
15:50 – 16:40
Soichiro Uemura (Kavli IPMU, The University of Tokyo)2

A proof of the Teichmüller TQFT volume conjecture for the knot 7 3
March 8th（Fri)10:00 – 10:50
Jun Ueki（Ochanomizu University）
Arithmetic topology and the proﬁnite rigidity of the multivariable Alexanderpolynomials of links
11:10 – 12:00
Hiraku Nozawa (Ritsumeikan University)
Harmonic measures and rigidity for the Fuchsian actions of surface groups onthe circle3

Abstract• Ryoto Tange (Waseda University)

Survey on Iwasawa theory for representations of knot groups
In the spirit of arithmetic topology, I will survey a knot theoretic analogue ofIwasawa theory for Galois representations. I especially focus on the L-function
associated to the twisted knot module for the universal deformation of a representation of a knot group, which may be seen as an analogue of the algebraic
p-adic L-function associated to the Selmer module for the universal deformation of a Galois representation. This is joint work with Takahiro Kitayama,
Masanori Morishita, Yuji Terashima, Leo Benard, Anh Tran, and Jun Ueki.
• Mishty Ray (University of Calgary)

Introduction to Geometry of local Arthur packets and Vogan’s conjecture Local
Arthur packets, or A-packets, are sets of representations of p-adic groups that
help us realize an important classes of automorphic forms. Vogan’s geometricperspective on the Langlands correspondence establishes a bijection betweenequivalence classes of smooth irreducible representations of G(F ) and simple
equivariant perverse sheaves on a moduli space of Langlands parameters. Thisgives us the notion of an ABV-packet, which is also a set of smooth irreducible
representations of G(F ), but now attached to any Langlands parameter. Conjecturally, ABV-packets generalize A-packets; we call this Vogan’s conjecture.
In recent work joint with Clifton Cunningham, we prove this conjecture forp-adic GLn . In this talk, we will explore the geometry of the moduli space ofLanglands parameters for GLn via examples. We will summarize basic results
towards Vogan’s conjecture and discuss other interesting problems in this area.
• Sohei Tateno (Nagoya University)

On Kida’s formula for Zdp -covers of graphs
Riemann–Hurwitz formula for Zp -extensions over number ﬁelds is called Kida’s
formula, which describes the relations among Iwasawa λ-invariants, extension
degrees, and ramiﬁcation indices. Recently, Ray and Vallières have established
Kida’s formula for Zp -covers of graphs. In this talk, by using generalized Iwasawa µ and λ-invariants introduced by Cuoco and Monsky, we give Kida’s
formula for Zdp -covers of graphs. We also provide several examples. This is a
joint work with Kosuke Mizuno.
4• Mahiro Atsuta (Tsuda University)

Euler system for Tate motives over a totally real ﬁeld
In recently work, Burns-Sano constructed a higher rank Euler system for Zp (1)over a totally real ﬁeld that interpolates the special values of the Dedekind zetafunction. In this talk, we discuss a reﬁnement of their work. More precisely, we
introduce the existence of an Euler system which interpolates more values ofL-functions. This is a joint work with Naoto Dainobu and Takenori Kataoka.
• Anwesh Ray (Chennai Math. Inst.)

Topological Iwasawa invariants and arithmetic statistics
There are interesting analogies between number theory and topology. Suchanalogies were systematically observed by Mazur who observed parallels between the arithmetic of primes in number rings and the geometry of knots
embedded in 3 manifolds. With respect to this analogy, we may formulate
certain questions in arithmetic statistics and then consider their topologicalanalogues. Iwasawa theory is the study of the asymptotic growth of classnumbers in certain inﬁnite abelian towers of number ﬁelds. This asymptotic
growth is determined by Iwasawa invariants associated to the number ﬁeld.
Given a knot in a 3 sphere, a fundamental polynomial invariant is the Alexander polynomial; which is regarded as the analogue of the Iwasawa polynomialfrom number theory. We consider certain natural families of links embeddedin the 3-sphere for which we prove results about the distribution of Iwasawa
invariants. In greater detail, we consider distribution questions for the familyof 2-bridge links. In time permits we shall also talk about the family of links
associated to certain braids on a ﬁxed number of strings.
• Xin Ren (Kansai University)

q-deformed rational numbers, q-deformed Farey sum and a triangulated category of A2 quiver
Let q be a formal parameter. The left and right q-deformed rational numbers
were introduced by Bapat, Becker and Licata via regular continued fractions,and the right q-deformed rational number is exactly q-deformed rational number considered by Morier-Genoud and Ovsienko [2], when q is a formal parameter. They gave a homological interpretation for left and right q-deformed
rational numbers [1; Theorems 4.7 and 4.8].
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In this talk, we give a formula for computing the q-deformed Farey sumof the left q-deformed rational numbers based on the negative continued fractions. We use this formula to give a combinatorial proof of the relationshipbetween the left q-deformed rational number and the Jones polynomial of thecorresponding rational knot which was proved by Bapat, Becker and Licata
using a homological technique. We combine the homological interpretation ofthe left and right q-deformed rational numbers and the q-deformed Farey sum,and give a homological interpretation of the q-deformed Farey sum.
References[1] Bapat, A., Becker, L., Licata, A. M.: q-deformed rational numbers and the2-Calabi–Yau category of type A2 , Forum Math. Sigma 11 (2023), Paper No.
e47, 41 pp.
[2] Morier-Genoud, S., Ovsienko, V.: q-deformed rationals and q-continued
fractions, Forum Math. Sigma 8 (2020), Paper No. e13, 55 pp.
[3] Ren, X.: On q-deformed Farey sum and a homological interpretation ofq-deformed real quadratic irrational numbers, arXiv:2210.06056, 2022.
• Mai Katada (Kyushu Universsity)

The ﬁrst homology of the IA-automorphism groups of free groups with coeﬃcients in spaces of Jacobi diagrams
The Kontsevich invariant for bottom tangles takes values in the spaces of Jacobi diagrams on oriented arcs. We have an action of the automorphism groupof the free group on the space of Jacobi diagrams. We study the ﬁrst homologyof the IA-automorphism group, which is a normal subgroup of the automorphism group of the free group, with coeﬃcients in the space of Jacobi diagramsof degree 2.

• Benjamin Collas (RIMS, Kyoto University)

Oda’s prediction, a motivating conjecture
The action of the absolute Galois group of rational numbers on the (proell) fundamental group of moduli spaces of curves of genus g with m marked
points indicates general arithmetic properties for curves. One of those is given
by Oda’s prediction (OP; MPI Bonn 1993), which states the independence ing and m of the stable number ﬁeld associated with these actions.
Oda’s prediction has since acted as a motivating conjecture in arithmetic
geometry – in terms of Lie algebra, divisorial arithmetic, or combinatorial an6

abelian geometry (resp. Matsumoto and Takao, Ihara-Nakamura, and HoshiMochizuki), but also in low dimensional topology (Johnson homomorphismand Morita’s obstruction; Morita, Nakamura, Hain, Takao et al.)Our goal will be to provide a panorama of techniques and results that will
also serve as a motivating introduction to Philip’s talk on the application ofstack arithmetic to Oda’s prediction.
• Séverin Philip (RIMS, Kyoto University)

The maximal degeneration method for special loci
Collas and Maugeais classiﬁed the irreducible components of the cyclic special
loci, a moduli space for curves admitting an action from a cyclic group. We
will introduce an analogue of Oda’s problem for this context and then explain
the relations between the ﬁelds involved in this problem and the one from theclassical Oda’s problem. From this point we will see how an adaptation of themethod of maximal degeneration from Ihara and Nakamura can be used tosolve the problem.
• Yasuhiro Wakabayashi (Osaka University)

Topological quantum ﬁeld theory for diﬀerential equations in characteristic pN
We will discuss the enumerative geometry of linear diﬀerential equations (ormore generally, opers) on algebraic curves in prime-power characteristic pN .
One of the central themes is to know how many such equations have a fullset of solutions. In the case of N = 1, this counting problem has been largely
resolved by eﬀectively using connections with the Gromov-Witten theory ofrelative Grassmannians, the conformal ﬁeld theory of aﬃne Lie algebras, andcombinatorics of rational polytopes, etc. Our recent work is an attempt toextend this theory to larger N .
• Nadav Gropper (University of Pennsylvania/University of Haifa ）TQFTs for pro-p Poincaré duality groups
In the talk, I will discuss the Turner-Turaev formalism for unoriented Topological Quantum Field Theory (TQFT). Building upon this formalism, I will introduce an analogous version for (d + 1)-dimensional TQFT for pro-p Poincaré
duality groups. In the case of d = 1, this enables us to study cobordisms
and TQFTs for both the maximal pro-p quotient of absolute Galois groups of7

p-adic ﬁelds and pro-p completions of fundamental groups of surfaces. This
generalisation gives a framework for arithmetic TQFTs and strengthens theanalogies within arithmetic topology, which relates p-adic ﬁelds to surfaces(oriented mod pr ). I will also provide an outline of the classiﬁcation of such
TQFTs for d = 1, in terms of Frobenius algebras with some extra structure.
The talk is based on joint work with Oren Ben-Bassat.
• Yuki Ishida (Kyushu University)

The density of Borromean primes
In this talk, I will show that the density of Borromean primes is 1/128 among
all of the triples of primes. Borromean primes are analogues of Borromeanrings in topology. This analogy depends on the relationship between numbertheory and 3-dimensional topology in arithmetic topology”, originated byB. Mazur, M. Morishita, M. Kaparanov and A. Reznikov. The idea of density of primes lies in arithmetic statistics”, which has been developed fromPrime Number Theorem. Our study is in the intersection of these areas. I will
overview these two areas and brieﬂy explain my main theorem and numericalresults.
• Yuqui Deng (Kyushu University)

Arithmetic Dijkgraaf–Witten invariants for real quadratic ﬁelds, quadratic
residue graphs, and density formulas
We compute Hirano’s formula for the mod 2 arithmetic Dijkgraaf–Witten invariant (partition function) Zk for the ring of integers of the quadratic ﬁeld√√
k = Q( p1 , . . . , pr ), where pi ’s are distinct prime numbers with pi ≡ 1 (mod4), and give a simple formula for Zk in terms of the graph obtained from
quadratic residues among p1 , . . . , pr . Our result answers the question posedby Ken Ono. We also give a density formula for mod 2 arithmetic Dijkgraaf–Witten invariants.
• Toshiki Matsusaka (Kyushu University)

Introduction to quantum modular forms and discontinuity property of Habiroseries• Hisatoshi Kodani (IMI, Kyushu University)

Quantum master equation and Hodge correlators
8We give a generalization of Goncharov’s Hodge correlator twistor connection.
Our generalized version is a connection 1-form with values in a DG Lie algebraof uni-trivalent graphs which may have loops and satisﬁes some Maurer–Cartanequation. This connection and the Maurer–Cartan equation can be viewed
as an arithmetic analogue of eﬀective action and quantum master equation
respectively in non-acyclic Chern–Simons perturbation theory around trivial
local system. This is joint work with Yuji Terashima.
• Soichiro Uemura (Kavli IPMU, The University of Tokyo)

A proof of the Teichmüller TQFT volume conjecture for the knot 7 3
We will introduce the generalized topological quantum ﬁeld theory (TQFT)
constructed by Andersen and Kashaev. This TQFT is based on the quantum Teichmüller theory and is called the Teichmüller TQFT. The Teichmüller
TQFT is expected to be a mathematical model for the SL(2, C) Chern-Simons
theory. In the Teichmüller TQFT, a partition function is deﬁned for a triangulated 3-manifold with an angle structure. As for the partition function,
Andersen and Kashaev proposed a conjecture analogous to the volume conjecture for the pair of any closed 3-manifold and any embedded hyperbolic knot.
Aribi, Guéritaud and Piguet-Nakazawa proved this conjecture in a reformulated form for the pair of the three-dimensional sphere S 3 and any twist knot.
We proved a theorem similar to the volume conjecture for the pair of S 3 and a
hyperbolic knot 73 , which does not belong to the family of twist knots (arXiv:2307.12848). We will give an outline of the proof.
• Jun Ueki (Ochanomizu University)

Arithmetic topology and the proﬁnite rigidity of the multivariable Alexanderpolynomials of links
Let p be a prime number and d a positive integer. We establish several versions of the Iwasawa-type formula for Zdp -covers of links in rational homology
3-spheres. We also discuss the proﬁnite rigidity of the multivariable Alexanderpolynomials by using the same tools, say, the fundamental two exact sequences.
(based on joint works with Sohei Tateno and Biao Ma.)• Hiraku Nozawa (Ritsumeikan University)

Harmonic measures and rigidity for the Fuchsian actions of surface groups on
the circle

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A measure on a foliated manifold equipped with a leafwise Riemannian metricis harmonic if it is invariant under the leafwise heat ﬂow. In an unﬁnished article, Thurston proposed a construction of an S 1 connection on an S 1 -bundle
over a hyperbolic surface by using a harmonic measure on a foliation transverse to the S 1 -ﬁbers. We will explain that, even if the surface has cusps,
the Gauss-Bonnet formula holds for this S 1 connection and the bounded Euler number due to Burger-Iozzi-Wienhard. As an application, we will give analternative proof of Matsumoto rigidity theorem of Burger-Iozzi-Wienhard for
the Fuchsian actions of hyperbolic surfaces with cusps. This talk is based ona joint work with Masanori Adachi (Shizuoka Univ.) and Yoshifumi Matsuda(Aoyama Gakushuin Univ.).

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