Title and Abstracts

Junhwa Choi (Korea Institute for Advanced Study)

Title: Quadratic twists of an elliptic curve
Abstract: In this talk, we prove the 2-part of Birch and Swinnerton-Dyer conjecture for an explicit infinite family of rank 0 quadratic twists of the modular elliptic curve $X_0(14)$, using an explicit form of the Waldspurger formula. This is joint work with Yongxiong Li.

Hikaru Hirano (Kyushu University)

Title : On arithmetic Dijkgraaf–Witten theory
Abstract : We present some basic constructions and properties in arithmetic Chern–Simons theory, which was initiated by Minhyong Kim, with finite gauge group along the line of topological quantum field theory, TQFT for short, in the sense of Atiyah. We construct an arithmetic analogue of Dijkgraaf–Witten TQFT in a certain special situation, namely, we construct arithmetic analogues, for a finite set S of finite primes of a number field k, of the prequantization bundles, the Chern–Simons 1-cocycle, the Chern–Simons functional, the quantum Hilbert space (space of conformal blocks) and the Dijkgraaf–Witten partition function. We show some basic and functorial properties of those arithmetic analogues and gluing formulas for arithmetic Chern–Simons invariants and arithmetic Dijkgraaf–Witten partition functions. This is joint work with Junhyeong Kim and Masanori Morishita.

Yasuhiro Ishitsuka (Kyushu University)

Title : Local-global properties on flexes and bitangents
Abstract : We discuss local-global properties on some substructures of algebraic varieties defined over number fields. In particular, we introduce results including those of flexes on plane cubics, bitangents of plane quartics. These are joint works with Tetsushi Ito, Tatsuya Ohshita, Takashi Taniguchi, and Yukihiro Uchida.

Keunyoung Jeong (Ulsan National Institute of Science and Technology)

Title: On the rank of the Jacobian of some hyperelliptic curves
Abstract: Let $J_A$ be the Jacobian of hyperelliptic curves $y^2 = x^5 + A$. Stoll and Yang show that the analytic and algebraic rank of $J_1$ is zero. More precisely, they give an explicit expression of the central value of $L$-function of $J_1$ based on the work of Yang connects the central $L$-value and the theta series, and the work of Stoll computes various arithmetic invariants of $J_1$. Together with Stoll’s other work computes the Selmer group of $J_A$ for some $A$, they obtain the main theorem. In the talk, we will talk about a generalization of this result. On the algebraic side, we compute Selmer groups which are not computed in Stoll’s work using Schaefer’s method. On the analytic side, we generalize the work of Stoll—Yang and give an expression of the central $L$-values for some $A$. Each result gives a sufficient condition that makes algebraic/analytic rank zero. Unfortunately, these conditions are incompatible, but we give an example that satisfies the Birch—Swinnerton-Dyer conjecture. This is a joint work with Junyeong Park and Donggeon Yhee.

Jangwon Ju (University of Ulsan)

Title : Ternary quadratic forms representing the same integers
Abstract : In 1997, Kaplansky conjectured that if two positive definite ternary quadratic forms with integer coefficients have perfectly identical integral representations, then they are isometric, both regular, or included either of two families of ternary quadratic forms. In this talk, we prove the existence of pairs of ternary quadratic forms representing the same integers which are not contained in Kaplansky's list.

Hisatoshi Kodani (Tohoku University)

Title : Arithmetic Orr invariants for absolute Galois groups
Abstract : In the late 1980s, K.E. Orr introduced an invariant, called Orr invariant, of a link in 3-sphere by homotopy theoretic way and answered several questions asked by J. Milnor. For example, he determined the dimension of Milnor invariants of a link using his invariant. Following an analogy between braid groups and absolute Galois groups initiated by Y. Ihara, we construct an arithmetic analogue of Orr invariant and show its properties such as the dimension of certain space on which arithmetic Orr invariant lies. We will also discuss relations between arithmetic Orr invariants and some obstruction classes introduced by J. Ellenberg in the context of Grothendieck's section conjecture. This talk is based on a joint work with Y. Terashima.

Dayoon Park (Ulsan National Institute of Science and Technology)

Title: An applicable representation of quadratic form to show representation of $m$-gonal form
Abstract

Biplab Paul (Kyushu University)

Title : On analogues of Ramanujan conjecture for Siegel modular forms
Abstract : The conjectures of Ramanujan played a crucial role in developing the theory of elliptic modular forms. Siegel modular forms are one of the important generalizations of elliptic modular forms. In this lecture, we shall discuss various analogues of Ramanujan conjecture in the context of Siegel modular forms and explain some of our recent results in this context. The lecture contents are based on two joint works: one is with Gun and Sengupta; another one is with Kumar.

Ryoko Tomiyasu (Kyushu University)

Title : Application of geometry of numbers to problems in the real world
Abstract : In mathematical crystallography, there are various problems of discrete mathematics. Firstly, I will introduce some examples where geometry of numbers and number theory were useful. In the latter half of my talk, I will present the case of the Kaplansky conjecture on the representation of ternary quadratic forms. The existence of counterexamples was conjectured in Tomiyasu (2020), and proved by J. Ju (2020). I would like to introduce some remaining problems, including one related to simultaneous representations of pairs of quadratic forms.

Hwajong Yoo (Seoul National University)

Title: The rational torsion subgroup of J_0(N)
Abstract: For any positive integer N, we propose a conjecture on the rational torsion subgroup of J_0(N). Also, we prove this conjecture up to finitely many primes. More precisely, we prove that the prime-to-2n parts of the rational torsion subgroup of J_0(N) and the rational cuspidal divisor class group of X_0(N) coincide, where m is the largest perfect square dividing 3N.

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