Silver workshop VI
Date: Aug. 7-9, 2023
Venue: OIST Lab.4, F01 and Zoom
Organized by Noriko Yui (Queen’s Univ.), Kyoji Saito (RIMS), Shinobu Hikami (OIST)Silver Workshop VI (2023, Aug.) (talk 50 min and discussion 10 min.)7th(Mon)10:00-11:00

11:20-12:20

12:20-14:00

14:00-15:00

15:20-16:20

Kyoji Saito (zoom)

J. Sekiguchi(Lunch)T. Suwa (zoom)

Yoshihisa Saito(zoom)
8th(Tue)

M. Mase (zoom)

K. Watanabe(zoom)(Lunch)Seoyoung Kim(zoom)
9th(Wed)N. YuiMinkyu Kim

Tea time (chat) andStrings*(Lunch)

A. Petrou

S. Hikami* La Primavera (A. Vivaldi) by Hokulani orchestra (OIST member).

Abstract:
Seoyoung Kim
Title: From the Birch and Swinnerton-Dyer conjecture to Nagao’s Conjecture
Abstract: In 1965, Birch and Swinnerton-Dyer formulated a conjecture on the Mordell-Weil rank r of elliptic
curves which also implies the convergence of the Nagao-Mestre sum. We show that if the Nagao-Mestre
sum converges, then the limit equals −r+1/2, and study the connections to the Riemann hypothesis for E.
We also relate this to Nagao’s conjecture for elliptic curves. Furthermore, we discuss a generalization of theabove results for the Selberg classes and hence (conjecturally) for larger classes of L-functions.

Yui, Noriko (Queen’s University)
Title: Calabi–Yau threefolds of Borcea–Voisin type and Arithmetic Mirror Symmetry
Abstract: We consider Calabi–Yau threefolds of Borcea–Voisin type defined over the field of rationals. Sucha Calabi–Yau threefold is constructed as the quotient of the product of a K3 surface and an elliptic curve bya specific involution.
For appropriate choices of K3 surfaces, we can establish the modularity/automorphy of the L-series of theCalabi–Yau motive.

We also consider mirror pairs of Calabi–Yau threefolds of Borcea–Voisin type, anddiscuss arithmetic mirror symmetry and its variants. Finite path integral model and toric code based onhomological algebra.

Minkyu KIM
Finite path integral is a mathematical methodology to construct TQFT’s (topological quantum field theories)
from finite gauge theory. In three dimensions, it is generalized to state sum models closely related with finiteHopf algebra gauge theory. Toric code originated from topological quantum computation provides a local
and combinatorial description of state sum models. This talk concerns a refinement of finite path integral
models by replacing finite (Hopf algebra) gauge theory with homological algebra based on bicommutative
Hopf algebras. We introduce and study a relative version of Haar integrals for Hopf algebras to formulate
finite path integral in our framework. It leads to a construction of a homotopy-theoretic version of projective
TQFT’s from Mayer-Vietoris functors valued in the category of bicommutative Hopf algebras. The
projectiveness gives rise to an obstruction problem, which is solved for some Mayer-Vietoris functors suchas the dimension reduction and bounded homology theories. Furthermore, we relate toric code to chain
complexes of bicommutative Hopf algebras.

Jiro Sekiguchi
Simple singularity of type E7 and the complex reflection group No.34
In this paper, we shall treat Arnold’s problem for the complex reflection group called Mitchell group G
whose number is 34 in the list of Shephard and Todd [2]. First of all, we recall the formulation of the problem.
In the book [1], p.20, it is written that“1974-5 Find applications of the (Shephard-Todd) complex reflection groups to singularity theory.”
We introduce a family F of surfaces Sτ : fτ (x, y, z) = 0 of C3 depending on parameters τ = (t1, t2, t3, t4, t5,t7). where
33 234 22 fτ(x,y,z)=y +x y+t7x+t5x +t3x +t1x +y(t4x+t2x )−z . (1)We point out some of basic properties of F:(a) The surface (1) has a simple singularity of type E7, if t1 = ··· = t5 = t7 = 0. (b) If the weights attached tox,y,z,tj (j = 1,2,3,4,5,7) are 2,3,9/2,j(j = 1, 2, 3, 4, 5, 7), respectively, then fτ is weighted homogeneous.
(c) If t7 = 0, the origin is an isolated singular point of the surface (1).
(d) There is an irreducible polynomial ∆ of τ such that Sτ is smooth if and only if t7 ·∆̸=0.
Since it is known that the degrees of basic invariants of G are 6, 12, 18, 24, 30, 42, the set of weights of t1, ·· · , t5, t7 and that of degrees of basic invariants are parallel. It can be shown that if t1, · · · , t5, t7 are
regarded as basic invariant of G, then ∆ is the discriminant of G.
This is regarded as an answer to Arnold’s Problem for G. If I have a time, I am going to explain the reason
why I recognized the existence of the family above which is not the versal family of E7 singularity and
Arnold’s Problem for other complex reflection groups.

References[1] V.I. Arnold: Arnold’s Problems, Edited by V.I. Arnold. Springer-Verlag. 2005.
[2] G. C. Shephard and A. J. Todd: Finite unitary reflection groups. Canadian J. Math. (1954),
6, 274-304.

Tatsuo Suwa (Hokkaido University)
Title: Relative Bott-Chern cohomology
Abstract: The Bott-Chern cohomology of a complex manifold refines both the de Rham and Dolbeault
cohomologies. The relative versions of the latters have many applications, including a recent one to the Sato
hyperfunction theory. In this talk, a relative Bott-Chern cohomology theory will be presented together withapplications and examples (based on a joint work with M. Corrˆea).

Makiko Mase
Abstract: Introducing coupling duality between weight systems, Ebeling proved there is a "Saito's duality" ofreduced zeta functions associated to simple K3 singularities.
Hinted by this, we discuss some dualities that may explain a relation between the Picard lattice of a family ofK3 surfaces and the Milnor lattice of a simple K3 singularity.

Shinichi Tajima
Title: Holonomic D-modules and non-isolated singularities.
Abstract: First, I give a quick introduction s-parametric annihilators, Bernstein-Sato polynomials and local
cohomology Second, I consider holonomic D-modules associated toa hypersurface with non-isolated
singularities. I give a new method for computing holonomic D-modules and microlocal b-functions. The keysof the method are the use of a Poincare-Birkhoff-Witt algebra and local cohomology. As applications, I discuss
relations between the structure of holonomic D-modules and the vertical monodromy introduced by Siersma
and Le cycles introduced by Massey.

Kyoji Saito joint work with Benoit Guerville
Title: Elliptic Artin braid relations and Zariski-van Kampen relations
Abstract: The fundamental group of the regular orbit space of a finite Weyl group is calculated by E.Brieskorn
by gallarly of chambers. It is called an Artin group and is determined by the Artin braid relations (E.BrieskornK.Saito). It is shown that the Artin braid relations are also given by Zariski-van Kampen method (K.Saito)
The fundamental group of the regular orbit space of an elliptic Weyl group is recently determined as theelliptic Artin group by a system of, what we call, elliptic Artin braid relations in a joint work with Yoshihisa
Saito and Kyoji Saito.
However, it is not known whether the elliptic Artin braid relations are given by Zariski-van Kampen method(it is a conjecture). For some special case of type E_6^{(1,1)}, Zariski-van Kampen relations are calculated
joint with Benoit Guerville. In the present talk, we show that some of the Zariski-van Kampen relations aregenerated by elliptic Artin braid relations.

Shinobu Hikami
Title: Arithmeticity in knot polynomials
Abstract: We discuss the factorization in Harer-Zagier transform (HZ) of Homply polynomial of knot
continuing to the previous talk by A. Petrou. By the multiplication of a factor a^m (a= q^N) to Homply
polynomial or by setting a parameter lambda to be q^m (m is integer), HZ obeys a factorized form where thezeros are roots of unity. This is analogous to the dimensional formula in arithmetic theory such as Igusa cuspform. The link case is also studied.
(Joint work with Andreani Petrou, arXiv:2307.05919).

Yoshihisa Saito
Title: Marked elliptic root systems with non-reduced affine quotients.
Abstract: In the middle of 1980's, motivated by study of singularity theories, K. Saito introduced the notionof "elliptic root systems". Roughly speaking, they are root systems with two null directions. Furthermore, hehad classified elliptic root systems R with one-dimensional subspace of G of two dimensional null directions,
under the assumption that the quotient affine root system R/G is reduced. In our joint work with A. Fialowski
and K. Iohara, we take off the assumption above, and give the classification of the pair (R, G) with noassumption. In addition, we give an overview of the theory of elliptic root systems in this talk, and certain
applications of this theory are also discussed.

IDEAL THEORETIC AND GEOMETRIC PROPERTIES OFINTEGRALLY CLOSED IDEALS IN 2 DIMENSIONAL NORMALSINGULARITIESKEI-ICHI WATANABE

Introduction
This is a joiint work in progress with T. Okuma (Yamagata Univ.) andK. Yoshida (Nihon Univ.).
Let (A, m) be a Noetherian normal local ring and I be an integrally closed idealof A. Take resolution of singularities f : X → Spec(A) so that IOX = OX (−Z). Sothe ideal I is represented by a positive cycle Z on X.
Let Q ⊂ I be a minimal reduction of I, namely, Q = (a, b) is a parameter idealwith IOX = QOX . In this paper, we will study the following invariants;
Deﬁnition 0.1.
nr(I) = min{r ∈ Z≥1 | I r+1 = QI r },r̄(I) = min{r ∈ Z≥1 | I n+1 = QI n for all n ≥ r}.
Also we deﬁne nr(A) (resp. r̄(A)) to be the maximal of all nr(I) (resp. r̄(I) for allI ⊂ A.
These invariants reﬂects the “goodness” of the singularity A. For example,Example 0.2. (1) r̄(A) = 1 if and only if A is a rational singularity.
(2) If A is an elliptic singularity, then r̄(A) = 2.
There are several invariants for the singularity∪r A. Let f : X → Spec(A) is aresolution of singularities of A and we put E := i=1 Ei be irreducible decompositionof E. The invariants below are independent of the resolution X.
Deﬁnition 0.3. (1) pg (A) := h1 (X, OX ).
(2) pa (A) = sup{pa (Z) | Z is a positive cycle supported on E}.
It is well known that pa (A) ≤ pg (A) and if pa (A) = 0, then pg (A) = 0 and in
this case A is rational singularity by deﬁnition. Also, A is called an elliptic
singularity if pa (A) = 1. There are elliptic singularities with arbitrary high pg (A).
Note that pa (A) is a topological invariant of A.
The following is the main result of this talk proved by improved Röhr’s Vanishing
Theorem.
Theorem 0.4. r̄(A) ≤ pa (A) + 1.
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We will discuss by several examples the behavior of nr(I), r̄(I) and r̄(A).
(Kei-ichi Watanabe) Department of Mathematics, College of Humanities and Sciences, Nihon University, Setagaya-ku, Tokyo, 156-8550, Japan and Organization
for the Strategic Coordination of Research and Intellectual Properties, Meiji
University
Email address: watnbkei@gmail.com

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