• 제목: Some refined applications of zeta elements

• 초록: We discuss the notion of zeta elements and their (refined) arithmetic applications. The concept of zeta elements is introduced by K. Kato, and zeta elements are regarded as the Galois-cohomological realization of zeta and L-functions in various settings. In particular, the collection of zeta elements is usually expected to form an Euler system and to recover the special values of the corresponding zeta and L-functions via the explicit reciprocity law. We will focus on explaining how to extract more arithmetic information from zeta elements in certain explicit situations.

• 제목: Dynamical irreducibility of rational functions modulo primes

• 초록: A key problem in arithmetic dynamics is the question when and "how often" a polynomial map (or rational function) over a number field can be expected to be dynamically irreducible, meaning that all of its iterates are irreducible. This question is relevant both from the global and the local viewpoint. Several previous works have given rise to the expectation that "usually", for a given polynomial f, the set of primes modulo which f remains dynamically irreducible (also called "stable primes") is a "small" set, but proven results were available only in very special cases. In this talk, I will exhibit and apply a new group-theoretical approach which shows for the first time that, indeed, for "most" polynomials of a given degree, the set of stable primes is of density 0. I will also discuss conjectures on the precise shape of polynomials which violate this assertion.

• 제목: The automorphism group of the torsion subgroup of an elliptic curve over a field of characteristic p≥5 

• 초록: For a field 𝕂⊇𝔽𝑝^{alg} of characteristic p≥5, the elliptic curve Es,t​:y2=x3+sx+t defined over the function field 𝕂(s,t) of two variables s and t, a non-negative positive integer e, and a positive integer N which is not divisible by p, we prove that the automorphism group of the normal extension 𝕂(s,t)(𝑬s,t [p^{e}𝑵])/𝕂(s,t) is isomorphic to (Z/p^{e}Z)^{×}×SL2(Z/𝑵Z). This work is a joint work with Bo-Hae Im. 

• 제목: A central limit theorem related to continued fraction expansion of two real numbers with small difference

• 초록: Lochs proved a theorem on the number of continued fraction digits of a real number x that can be determined from its first nth decimal expansion, which increases as the length of the cylinder set containing x decreases. In 1998 C.Faivre provided a proof of a central limit theorem related to this quantity. Over the years this theorem has been refined using other two number theoretic fibered maps. In this presentation, we outline the proof a version of central limit theorem concerning the number of continued fraction digits of a real number x that can be determined from small difference with x. For this, we also prove a Lochs type theorem.

• 제목: Uniform Diophantine approximation on the Hecke group H₄

• 초록: Dirichlet's uniform approximation theorem is a fundamental result of Diophantine approximation that provides an optimal rate of approximation. In this talk, we investigate uniform Diophantine approximation properties on the Hecke group H₄ using the Rosen continued fractions. For a given real number, the best approximations are the convergents of both the Rosen continued fraction and the dual Rosen continued fraction of the number. We examine analogues of Dirichlet's uniform approximation theorem and Legendre's theorem with optimal constants.

• 제목: Orbital integrals and ideal class monoids for a Bass order

• 초록: 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 Z , is a Bass order.

In this talk, I 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. I 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 gln as a mass formula. This is joint work with Sungmun Cho and Yuchan Lee.

• 제목: Arithmetic Fundamental Lemmas and Higher Derivatives of L-functions

• 초록: The Fundamental Lemma (FL), initially developed within the framework of the Langlands program, has since found applications beyond its original context, connecting number theory, automorphic representations, harmonic analysis, and geometry. In this talk, I will outline the history and significance of the FL and its arithmetic generalization, the Arithmetic Fundamental Lemma (AFL), which arises from the Relative Trace Formula in the study of the Gross–Zagier formula. These lemmas serve as a bridge between intricate integrals and their combinatorial or geometric interpretations, reflecting their foundational role in mathematics.

I will present a recent collaborative project with Andreas Mihatsch and Wei Zhang on higher derivatives of L-functions, with particular emphasis on the connection between the Guo–Jacquet Fundamental Lemma and intersection theory. After discussing the technical challenges in proving the Guo–Jacquet FL, I will delve into the study of higher derivatives of orbital integrals in this context.

Our work interprets these derivatives as intersection numbers on local Shtukas, drawing inspiration from Yun and Zhang’s work on the higher Gross–Zagier formula. Additionally, I will touch on the geometrization of the Guo–Jacquet FL for function fields, highlighting its link to Hitchin fibrations and offering insights into the relationship between geometry and analysis. This approach illustrates the evolving connections and potential for further exploration in the field.

• 제목: Rank 2 strongly divisible modules over Qpf

• 초록: Strongly divisible modules plays a key role in determining the mod-p reduction of semi-stable Galois representations, yet their construction is often challenging. In this talk, we present a method that reduces constructing strongly divisible modules for 2-dimensional semi-stable representations of Gal(Qp/Qpf) to solving systems of linear equations and inequalities. We then describe the mod-p reduction of these representations explicitly in terms of the solutions to these systems. We expect that our method exhausts all the necessary strongly divisible modules of rank 2 for general f. Our approach recovers previously known results for f=1 and is illustrated with an example for f=2. This is a joint work with Chol Park.

• 제목: The Sparsity of Character Tables for Finite Reductive Groups

• 초록: Character values of finite reductive groups are fundamental objects in representation theory, but computing them in full generality is a challenging problem. In this talk, I will present an arithmetic statistical perspective on this issue, demonstrating that as the semisimple rank of reductive groups increases, almost all character values tend to be zero. If time permits, we will also explore a version of this problem in the context of Lie algebras. This is joint work with Anna Puskás.