Workshop on Number Theory and Integrable Systems

Date : June 2 (Mon) - 4 (Wed), 2025

Venue : Room 301, Building B, Graduate School of Science, Kobe University (Access, Campus Map)

Organizers : Yasuhiro Ishitsuka (Kyushu), Tetsushi Ito (Kyoto), Tatsuya Oshita (Gumma), Takashi Taniguchi (Kobe), Yukihiro Uchida (Tokyo Metropolitan)

Program (5/1 updated)

6/2 (Mon)

• 13:30 - 14:30 : Yukihiro Uchida (Tokyo Metropolitan University)

  • Title: Periods modulo p of integer sequences associated with division polynomials of genus 2 curves

  • Abstract: An elliptic divisibility sequence is an integer sequence associated with division polynomials of an elliptic curve. Ward proved the periodicity of the reduction modulo p of an elliptic divisibility sequence for all but finitely many primes p. In this talk, we study an integer sequence associated with Cantor's division polynomials of a genus 2 curve having an integral point. We show that the reduction modulo p of such a sequence is periodic for all but finitely many primes p, and describe the relation between the period of the reduction modulo p of the sequence and the order of the integral point on the reduction modulo p in the Jacobian variety explicitly. This generalizes Ward's results on elliptic divisibility sequences. This talk is based on joint work with Yasuhiro Ishitsuka (Kyushu), Tetsushi Ito (Kyoto), Tatsuya Oshita (Gumma), and Takashi Taniguchi (Kobe).

• 14:45 - 15:45 : Yasuaki Gyoda (Nagoya University)

  • Title: Markov Numbers and Their Cluster Matrix Formulation

  • Abstract: A Markov number is an integer that appears as a positive integer solution to the Markov equation $x^2 + y^2 + z^2 = 3xyz$. It is known that the set of positive integer solutions to this equation possesses a combinatorial structure called a cluster structure. In recent years, there has been growing interest in investigating the properties of these integers from the perspective of cluster algebras. 
          In this talk, we focus on a class of matrices introduced by Cohn in 1955, now known as Cohn matrices. These matrices are elements of $SL(2, \mathbb{Z})$ whose $(1,2)$ entry is a Markov number, and they exhibit a combinatorial structure analogous to the cluster structure of Markov numbers. 
          We explain how various properties of Cohn matrices can be understood within the framework of cluster algebra theory by "clusterizing" these matrices. This talk is based on joint work with Esther Banaian of University of California, Riverside.

• 16:15 - 17:15 : John A. G. Roberts (UNSW Sydney, Australia/The University of Tokyo)

  • Title: Signatures of integrability and non-integrability for birational maps over finite fields

  • Abstract: We consider various aspects of the dynamics of birational maps in two or more dimensions, reduced over finite fields. The maps may possess algebraic properties such as the possession of integrals or symmetries and the broad aim is to investigate the signature over finite fields of these dynamical structures (usually studied over the real or complex numbers).  We show how the finite field dynamics can be modelled by various heuristics that rely on assuming the map acts randomly subject to the constraints imposed by its algebraic properties.
          The talk reviews work over many years with Franco Vivaldi (Queen Mary, London) and my ex-students Danesh Jogia and Tim Siu:

    • J A G Roberts & F Vivaldi: Phys. Rev. Lett. 90 (2003) 034102; Nonlinearity 18 (2005) 2171; Nonlinearity 22 (2009) 1965

    • Danesh Jogia, Algebraic aspects of integrability and reversibility in maps, UNSW PhD thesis 2008
      https://doi.org/10.26190/unsworks/17873

    • Tim Siu, Order, randomness and orbit distributions for dynamics of birational maps over finite fields, UNSW PhD thesis 2019
      https://doi.org/10.26190/unsworks/21440

6/3 (Tue)

• 09:30 - 10:30 : Genki Shibukawa (Kitami Institute of Technology)

  • Title: Some topics in Number Theory and Integrable Systems

  • Abstract: In this talk, I would like to talk about the following topics in Number Theory and Integrable Systems.

    • (1) Cluster algebras and the inverse Galois problem

    • (2) A source identity of the Macdonald operators and a reciprocity law of the Dedekind-Carlitz polynomials

    • (3) Some exponential sums and some specializations of q-hypergeometric functions.

• 10:45 - 11:45 : Frank Thorne (Univesity of South Carolina)

  • Title: 6-torsion in Imaginary Quadratic Fields

  • Abstract: I will present work with Peter Koymans, Robert Lemke Oliver, and Efthymios Sofos establishing the average size of the 6-torsion subgroup of the ideal class group of imaginary quadratic fields, along with some related results.
        The proof's ingredients include Sato-Shintani zeta functions and the anatomy of integers, and I will spend a bit of time explaining these ingredients and how they fit together.

• 13:30 - 14:30 : Yasuhiro Ohta (Kobe University)

  • Title: Hankel determinant expressions for nonautonomous Somos-4 sequence

  • Abstract:  The Somos-4 recurrence relation is q-deformed by requiring its nonautonomous form to satisfy the Laurent property.   Several Hankel determinant expressions are obtained for the sequence and the recurrence is reduced to the Jacobi formula for determinants.  The recurrence relation for determinants is similar to the discrete Toda lattice equation in bilinear form.

• 14:45 - 15:45 : Rei Inoue (Chiba University)

  • Title: Quantized 6-vertex model on a torus and tetrahedron equations

  • Abstract:  The 6-vertex model is an integrable statistical model of (1+1) dimensional space time, whose symmetry is governed by the famous Yang-Baxter equation (YBE). The quantized 6-vertex model (q-6v) was introduced by Kuniba-Matsuike-Yoneyama in 2023, by replacing the Boltzmann weight of the 6-vertex model with the elements of the q-Weyl algebra. This gives (2+1)-dimensional lattice models whose symmetry is governed by the tetrahedron equation, a higher dimensional version of the YBE.
          In this talk, we discuss the symmetry of the q-6v model. It turns out that for a class of graphs on a torus (including square lattice) integrability is established by applying various tetrahedron equations.  This model is also related to quantum cluster algebra for a specific quiver.  This talk is based on joint works with Atsuo Kuniba, Xiaoyue Sun, Yuji Terashima and Junya Yagi.

• 16:15 - 17:15 : Andrew Hone (University of Kent)

  • Title: Continued fractions, hyperelliptic curves and higher genus Chebyshev polynomials

  • Abstract: We consider a family of nonlinear maps that are generated from the continued fraction expansion of a function on a hyperelliptic curve of genus g, as originally described by van der Poorten. Using the connection with the classical theory of J-fractions and orthogonal polynomials, we show that in the elliptic case g=1 this provides a straightforward derivation of Hankel determinant formulae for the terms of a general Somos-4 sequence, which were found in particular cases by Chang, Hu, and Xin. We extend these formulae to the higher genus case, and find analogues of (2nd kind) Chebyshev polynomials, which correspond to the case g=0. Moreover, for all g>0 we show that the iteration for the continued fraction expansion is equivalent to a discrete Lax pair with a natural Poisson structure, and the associated nonlinear map is a discrete integrable system, connected with solutions of the infinite Toda lattice. If time permits, we will make some remarks about analogous results for the Stieltjes case (S-fractions), obtained in recent work with Roberts, Vanhaecke and Zullo.

6/4 (Wed)

• 09:00 - 10:00 : Tatsuya Ohshita (Gunma University)

  • Title: Asymptotic behavior of ideal class groups along Galois representations and its topological analogue

  • Abstract: Let T be an integral p-adic representation of the absolute Galois group of  a  number filed.  In this talk, by using the Selmer group of T, we shall describe  the asymptotic behavior of certain quotients of ideal class groups along the along the tower of fields arising from T  (partially based on a joint work with T. Hiranouchi).
          We shall also note the analogous results in the arithmetic topology.

• 10:15 - 11:15 : Nobuo Sato (National Taiwan University) 

  • Title: TBA

  • Abstract: TBA

• 11:30 - 12:30 : Yasuhiro Ishitsuka (Kyushu Univesity)

  • Title: Exponential sums on singular binary forms

  • Abstract:  In 2024, we gave a result on the number of 2-Selmer elements on elliptic curves over Q whose discriminants have at most four prime factors. This is an analogue of results by Takashi Taniguchi (Kobe) and Frank Thorne (University of South Carolina), who estimated the number of cubic and quartic fields under a similar condition.
          In both settings, an explicit computation of exponential sums on singular binary forms plays a central role. In this talk, I will explain the result on exponential sums on binary quartic forms along with some related results.
          This is joint work with Takashi Taniguchi, Frank Thorne, and Stanley Yao Xiao (UNBC).