Sep 26 (Thu), 2024, 16:00--17:00, Room 8406

Title: Orbital integrals and ideal class monoids for a Bass order

Speaker: Jungtaek Hong (POSTECH)

Abstract: A Bass order is an order of a number field whose fractional ideals are generated by two elements. Majority of number fields contain infinitely many Bass orders. For example, any order of a number field which contains the maximal order of a subfield with degree 2 or whose discriminant is 4th-power-free in $\mathbb{Z}$, is a Bass order. In this talk, we will propose a closed formula for the number of fractional ideals of a Bass order $R$, up to its invertible ideals, using the conductor of $R$. We will also explain explicit enumeration of all orders containing $R$. Our method is based on local-global argument and exhaustion argument, by using orbital integrals for $\mathfrak{gl}_n$ as a mass formula. This is joint work with Sungmun Cho and Yuchan Lee.

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.

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.

Aug 8 (Thu), 2024, 14:00--15:00, Room 1424

Title: Monodromy and Irreducibility of type A_1 automorphic Galois representations

Speaker: Wonwoong Lee (University of Hong Kong)

Abstract: As part of the Langlands conjecture, it is predicted that every $\ell$-adic Galois representation attached to an algebraic cuspidal automorphic representation of $\mathrm{GL}_n$ over a number field is irreducible. In this talk, we will prove that a type $A_1$ Galois representation attached to a regular algebraic (polarized) cuspidal automorphic representation of $\mathrm{GL}_n$ over a totally real field $K$ is irreducible for all $\ell$, subject to some mild conditions. We will also prove that the attached Galois representation is residually irreducible for almost all $\ell$. Moreover, if $K=\mathbb Q$, we will prove that the attached Galois representation can be constructed from two-dimensional modular Galois representations up to twist. This is a joint work with Chun-Yin Hui.

Sep 12 (Thu), 2024, 16:00--17:00, Room 8406

Title: Generation of Hecke fields by the square of absolute values of modular $L$-values with cyclotomic twists

Speaker: Junhwi Min (UNIST)

Abstract: Let $f$ be a non-CM elliptic newform, which does not have a quadratic inner twist. Let $p$ be an odd prime and $\chi$ a $p$-power conductor Dirichlet character. We show that the compositum $\mathbb{Q}_{f}(\chi)$ of the Hecke fields associated to $f$ and $\chi$ is generated by the square of the absolute value of the corresponding central $L$-value $L^{alg}(1/2, f \otimes \chi)$ over $\mathbb{Q}(\mu_p)$, as $\chi$ varies over Dirichlet characters of $p$-power conductor and order. The proof is based on the recent resolution of unipotent mixing conjecture due to Blomer and Michel.