Jul 25 (Thu), 2024, 16:00--17:00, Room 1424

Title: On the infinite product expansions of meromorphic modular forms

Speaker: Gyucheol Shin (Sungkyunkwan University)

Abstract: Many modular forms are usually expressed using Fourier expansions, but in some cases, such as the modular discriminant, it is more usefulto express them using infinite product expansions. The most important result, initiated by Borcherds, associated with the infinite product expansion of modular forms, is that there exists a meromorphic modular form of integral weight for some character of SL(Z) with integer Fourier coefficients such that its exponents in the infinite product expansion are equal to the Fourier coefficients of a modular form of weight 1/2 on Γ0(4) satisfying Kohnen plus condition. In this talk, we introduce an operator acting on the exponents of the infinite product expansion of meromorphic modular forms and investigate its properties. This is joint work with Chang Heon Kim.

March 7 (Thu), 2024, 16:00--17:00, Room 8101

Title: Representation numbers of Bell-type quadratic forms

Speaker: Yeongwook Kwon (UNIST)

Abstract: In 1834, Jacobi proved his four-square theorem. The sum of four squares can be viewed as an example of the so-called Bell-type quadratic forms, and the representation number of Bell-type quadratic forms were studied by several authors. However, most preceding results only dealt with Bell-type forms of class number 1. In this talk, we derive a closed formula for the representation numbers of each Bell-type quadratic form of class number less than or equal to 2. This is joint work with Chang Heon Kim, Kyoungmin Kim and Soonhak Kwon.

May 30 (Thu), 2024, 16:00--17:00, Room 1424

Title: Growth of torsion groups of elliptic curves over number fields, rational isogenies, and number fields without rationally defined CM

Speaker: Hansol Kim (KAIST)

Abstract: We find an equivalent condition that a number field $K$ has the following property: There is a prime $p_{K}$ depending only on $K$ such that if $d$ is a positive integer whose minimal prime divisor is greater than $p_{K}$, then for any extension $L/K$ of degree $d$ and any elliptic curve $E/K$, we have $E\left(L\right)_{\operatorname{tors}} = E\left(K\right)_{\operatorname{tors}}$. For the purpose, we study the relations among torsion groups of elliptic curves over number fields, rational isogenies, and number fields without rationally defined CM. As a collorary of our result, we prove that any quadratic number field which is not an imaginary number field whose class number is not $1$ has the above property. This is a joint work with Bo-Hae Im.