Number Theory Seminars 2023

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March 23 (Thu), 2023, 16:00--17:00, Room 1424

Title: Regular ternary octagonal forms

Speaker: Mingyu Kim (Sung Kyun Kwan University)

Abstract: Let $P_8(x)=3x^2-2x$. A polynomial of the form $a_1P_8(x_1)+a_2P_8(x_2)+\cdots+a_kP_8(x_k)$ ($a_i\in \mathbb{N}$) is called a $k$-ary octagonal form. An octagonal form is called regular if it represents every positive integer which is locally represented. In this talk, we show that there are at most 15 regular ternary octagonal forms and prove the regularity of 12 forms.

April 20 (Thu), 2023, 16:00--17:00, Room 1424

Title: Diversity in Rationally Parameterized Fields

Speaker: Benjamin Klahn (Graz University of Technology)

Abstract: Let $F(x,y) \in \mathbb{Q}[x,y]$ be an irreducible polynomial of degree $d \geq 2$ in $x$. Hilbert's Irreducibility Theorem (HIT) states that for the vast majority of integers $n$ the polynomial $F(x,n) \in \mathbb{Q}[x]$ is irreducible, i.e. $[\mathbb{Q}(\theta_{n}):\mathbb{Q}]=d$ for any root $\theta_{n}$ of $F(x,n)$. However, HIT does not answer the following questions:
(1) Given an integer $N$, what is the degree of $\mathbb{Q}(\theta_{1},\theta_{2}, \dots, \theta_{N})$?
(2) How many distinct fields are there among $\mathbb{Q}(\theta_{j})$, $1 \leq j \leq N$?
These questions were first studied by Dvornicich and Zannier, who showed that there is a positive constant $c$ such that $\mathbb{Q}(\theta_{1},\dots, \theta_{N}) \geq e^{cN/\log N}$, and consquently that there are at least $c' N/\log N$ many distinct fields among $\mathbb{Q}(\theta_{j})$ with $j \leq N$.

We will consider the larger set of fields $\mathbb{Q}(\theta_{r})$ where $r \in \mathbb{Q}$ varies over rational numbers of height $H(r) \leq N$. Under some assumptions on $F$ we will obtain a lower bound on the number of distinct fields among $\mathbb{Q}(\theta_{r})$, $H(r) \leq N$.

May 18 (Thu), 2023, 16:00--17:00, Room 1424

Title: The Hasse principle for homogeneous polynomials with random coefficients over thin sets

Speaker: Kiseok Yeon (Purdue University)

Abstract: In this talk, we introduce a framework via the circle method in order to confirm the Hasse principle for random projective hypersurfaces over $\mathbb{Q}$. First, we give a motivation for developing this framework by providing the overall history of the problems of confirming the Hasse principle for projective hypersurfaces over $\mathbb{Q}$. Next, we provide a sketch of the proof of our main result and show a part of the estimates used in the proof.

May 25 (Thu), 2023, 16:00--17:00, Room 1424

Title: Transcendence in automorphic forms and $L$-functions

Speaker: Henry Kim (University of Toronto)

Abstract: We study transcendence of values of automorphic forms and automorphic $L$-functions. In particular, we discuss the transcendence of Petersson norm and the Koecher-Maass series of the Ikeda type lift of the exceptional group of type $E_7$.

June 8 (Thu), 2023, 16:00--17:00, Room 1424

Title: Zeta functions of Lie rings and their associate varieties

Speaker: Seungjai Lee (Seoul National University)

Abstract: Let $L$ be a finite-dimensional $\mathbb{Z}$-Lie algebra (also called a Lie ring), and for a prime $p$ let $L_p:=L\otimes \mathbb{Z}_p$, where $\mathbb{Z}_p$ is the usual ring of $p$-adic integers. We call $\zeta_{L_{p}}^{\triangleleft}(s):=\sum_{H\triangleleft L_{p}}|L_p:H|^{-s}$ the local ideal zeta functions of $L$, enumerating ideals of finite index in $L_p$.

In this talk, we show how these local factors $\zeta_{L_{p}}^{\triangleleft}(s)$ can be expressed in terms of certain $p$-adic integrals, allowing us to associate $L$ with certain algebraic varieties intrinsically encoded in its structure. Then we discuss how, under certain conditions, one can control and study these varieties.

June 29 (Thu), 2023, 16:00--17:00, Room 1424

Title: Galois structure of units in totally real $p$-rational field

Speaker: Dong Hyeok Lim (Ewha Institute of Mathematical Sciences)

Abstract: The theory of factor-equivalence of integral lattices gives a far-reaching relationship between the Galois module structure of units of a number field and its arithmetic. For a number field $K$ that is Galois over $\mathbb{Q}$ or an imaginary quadratic field, we prove a necessary and sufficient condition on the quotients of class numbers of subfields of $K$, for the quotient $E_{K}$ of the group of units of $K$ by the subgroup of roots of unity to be factor equivalent to the standard cyclic Galois module. We study the factor equivalence class of units in totally real $p$-rational fields by using their strong arithmetic properties. If time permits, we also discuss the Galois module structure of units in cyclic $p$-extensions.

July 20 (Thu), 2023, 16:00--17:00, Room 1424

Title: When is a polarized abelian variety determined by its $p$-divisible group?

Speaker: Chia-Fu Yu (Academia Sinica)

Abstract: If two principally polarized abelian varieties are isomorphic, then their associated $p$-divisible groups are isomorphic. However, the converse is not true in general. We shall describe precisely when a principally polarized abelian variety is determined by its $p$-divisible group. We also explain some ingredients of the proof, as well as the connection with results of Chai and Oort on central leaves. This talk is based on the joint papers with Tomoyoshi Ibukiyama, Valentijn Karemaker and Fuetaro Yobuko.

July 27 (Thu), 2023, 16:00--17:00, Room 1424

Title: A stratification on Lagrangian varieties and supersingular arithmetic invariants

Speaker: Chia-Fu Yu (Academia Sinica)

Abstract: Let $(V_0, \psi)$ be a non-degenerate symplectic space over a finite field $k_0$ and $k$ an algebraically closed field containing $k_0$. For any maximal isotropic $k$-subspace $W$ we define the endomorphism algebra of $W$ with respect to $k_0$-structure of $V_0$. This induces a stratification on the Lagrangian variety $L(V_0)$. In this talk we shall report the joint work with V.-Karemaker in progress for this stratification for $\text{dim} V_0=4$ and its connection with supersingular arithmetic invariants.

August 14 (Mon), 2023, 16:00--17:00, Room 1424

Title: Orbital integrals for unitary Lie algebras

Speaker: Taeyeoup Kang (POSTECH)

Abstract: Orbital integral is an object occurring in a geometric side of the trace formula, which is a powerful tool in Langlands program. A traditional method to study orbital integral is through Bruhat-Tits building or Shalika germs. Recently, Cho and Lee proposed a new method to study orbital integrals, called smoothening. Using this idea, we will introduce our ongoing work about stable orbital integrals for unitary Lie algebras. This is a joint work with Sungmun Cho and Yuchan Lee.

November 23 (Thu), 2023, 16:00--17:00, Room 1424

Title: Primitively $2$-universal senary integral quadratic forms

Speaker: Jongheun Yoon (Seoul National University)

Abstract: For a positive integer $n$, a positive definite integral quadratic form is called primitively $n$-universal if it primitively represents all positive definite integral quadratic forms of rank $n$. Ju, Kim, Kim, Kim, and Oh proved that there are exactly $107$ primitively $1$-universal quaternary quadratic forms up to isometry. In this talk, we prove that the minimal rank of a primitively $2$-universal integral quadratic form is six, and we prove that there are exactly $201$ primitively $2$-universal senary integral quadratic forms up to isometry. This is joint work with Prof. Byeong-Kweon Oh.

December 8 (Fri), 2023, 10:00--11:30, Room 8101

Title: Application of Tadic's structure formula

Speaker: Yeansu Kim (Chonnam National University)

Abstract: Tadic's structure formula gives explicit information about Jacquet modules of induced representations, which has a numerous application in Langlands program. In this talk, we briefly explain Tadic's structure formula and its applications.

December 8 (Fri), 2023, 11:30--13:00, Room 8101

Title: Rankin-Selberg integrals and symmetric square $L$-factors

Speaker: Yeongseong Jo (Ewha Womans University)

Abstract: Automorphic L-functions are generalizations of Riemann zeta functions. Their analytic properties have profound implications in dynamical systems, including, but not limited to, the prime number theorem. A problem of recurring interest, with wide-ranging applications, is to relate the special value of $L$-function with period of integrals. In this context, an answer to the fundamental question boils down to understanding poles and zeros of a family of certain "Rankin-Selberg" type integrals that represent various $L$-factors. We will review some earlier results on exterior and symmetric square $L$-factors. The proof involves the Bernstein-Zelevinsky derivatives and exceptional/regular poles, building upon the seminal work of Cogdell and Piatetski-Shapiro. If time permits, we will also discuss the strategy for generalizing the results from $\text{GL}(2)$ to higher rank groups, as well as the Rankin-Selberg method for classical groups.

December 14 (Thu), 2023, 16:00--17:00, Room 1424

Title: Non-vanishing mod $p$ of Hecke $L$-values and abelian modular symbols over real quadratic fields

Speaker: Jaesung Kwon (Seoul National University)

Abstract: Non-vanishing mod $p$ of $L$-values of $\ell$-adic families has been widely studied by many researchers. One of the famous results is Washington's celebrated theorem published in 1978. In 2007, Sun reproved this result by studying the distribution of abelian modular symbols on the cylinder over the rational number field. In this talk, I will introduce a notion of cylinder over real quadratic fields and abelian modular symbols on it. Also, I will suggest a generalization of Washington 1978. This work is an ongoing research and joint with Hae-Sang Sun.