Dec 19 (Thu), 2024, 15:00--16:00, Room 1424

Title: Goldbach counting and Zeta Zeros

Speaker: Ade Irma Suriajaya (Kyushu University)

Abstract: The Goldbach conjecture asks if we can always write an even number greater than or equal to 4, as a sum of two prime numbers. This will imply that all integers at least 6 can be written as a sum of three primes. The latter for odd integers ≥ 7 is a weaker conjecture and has recently been proven by Harald Helfgott. The even number case remains unsolved. We introduce the conjecture in a quantitative form due to G.H. Hardy and J.E. Littlewood in 1919 (with details published in 1923). A weaker version of this conjecture implies the non-existence of ``exceptional zeros” of certain L-functions. This is joint work with John B. Friedlander, Daniel A. Goldston and Henryk Iwaniec. We also introduce the notion of non-vanishing regions of the Riemann zeta function and its connection to the prime number theorem. The analogue for Dirichlet L-functions also holds except for a possible real zero which we call an ``exceptional zero”. Counting the number of Goldbach representations itself is difficult, and taking its average tells us a bit more information. In fact, the asymptotic formula for the average number of Goldbach representations is very closely related to the quantitative form of the Prime Number Theorem (PNT) and the error is determined by a non-vanishing region of the Riemann zeta-function. This is obtained in the student project with Keith Billington and Maddie Cheng from San Jose State University, together with Jordan Schettler.

Dec 19 (Thu), 2024, 16:00--17:00, Room 1424

Title: Structural stability of nonarchimedean Julia sets

Speaker: Junghun Lee (Chonnam National University)

Abstract: In this talk, we mainly consider the nonarchimedean Julia set, which is the chaotic locus of dynamics, and its structural stability. More precisely, this talk will present three parts of structural stability for the nonarchimedean Julia set. In the first part, which is based on the author’s own work, we will see a nonarchimedean analogue of the celebrated result by Mañe-Sad-Sullivan in complex dynamics. These results implies hyperbolic dynamics is structurally stable so it is natural to ask if hyperbolic dynamics is only dynamics with structural stability. This question is directly related to the Fatou conjecture in complex settings. We will obtain a negative answer of this Fatou conjecture in nonarchimedean settings in the second part, which is based on the joint work with Prof. Robert Benedetto. Namely, it will turn out that it is possible to extend our structural stability theorem with much weaker assumptions than so-called hyperbolicity. One natural question is where the weaker assumption came from unlike complex dynamics. We will see another reasoning of this question in our third part, which is based on the joint work with Prof. Tomoki Kawahira, by considering the stability of the Julia point.

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.

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.

Oct 31 (Thu), 2024, 16:00--17:00, Room 1424

Title: On the genericity of global theta lifts for classical groups

Speaker: Jaeho Haan (Catholic Kwandong University)

Abstract: The fundamental questions in the global theta correspondence concern the non-vanishing and cuspidality of global theta lifts. After 30 years of efforts by Rallis and many others, these questions have been resolved. The next natural question is preserving the genericity of global theta lifts. We address this in the style of Rallis's tower property concerning cuspidality. We plan to tackle this problem for all classical groups. For the orthogonal and symplectic groups, see https://arxiv.org/abs/2410.02625. This is ongoing work with Sanghoon Kwon.

Nov 21 (Thu), 2024, 17:00--18:00, Online

Title: Restriction norms and symplectic Petersson's formula

Speaker: Gilles Felber (Alfr ́ed R ́enyi Institute of Mathematics)

Abstract: Restriction norms are a measure of equidistribution of functions, such as automorphic forms. By restricting the function to a subspace of lower dimension, one detects different fluctuations from conjectures like Quantum Unique Ergodicity. I will present a result on average for Siegel modular forms of degree 2. The proof is based on a generalization of Petersson's trace formula by Kitaoka. I will also speak about ongoing work on a generalization of Kitaoka's work to higher degrees.

Dec 5 (Thu), 2024, 16:00--17:00, Room 1424

Title: Arithmetic geometry of character varieties

Speaker: GyeongHyeon Nam (Ajou University)

Abstract: Character varieties are constructed from homomorphisms of the fundamental group of a space into an algebraic group G. In particular, when the fundamental group arises from a Riemann surface, these varieties are deeply connected to important areas of mathematics, such as the Yang-Mills equations, the P=W conjecture, and mirror symmetry. The work of Hausel, Letellier, and Rodríguez-Villegas has sparked extensive research on character varieties through point-counting methods, particularly for G = GL_n. In this talk, we will explore how these ideas extend to arbitrary reductive groups, beyond GL_n. This is based on joint work with Masoud Kamgarpour, Bailey Whitbread, and Stefano Giannini. If time permits, we will see the additive analogue of this story.