The 8th East Asia Number Theory Conference
Department of Mathematical Sciences
KAIST, Daejeon, Korea
August 25-30, 2019
Venue: Room 1501 at E6 (Natural Sciences Building) of KAIST Main (Daejeon) Campus (PDF). For the location at google map, Click Here.
Titles and Abstracts
Sunday, August 25, 2019
Monday, August 26, 2019
- Registration (8:30-)
- Wansu Kim (09:20-10:10)
Title: Geometry of Newton Stratification
Abstract: In this talk, I'd like to motivate and explain the geometry of the Newton strata in Hodge-type Shimura varieties obtain by me and Paul Hamacher. We'll focus on a few motivating examples of the moduli of polarised abelian varieties or K3 surfaces.
- Keisuke ARAI (10:30-11:20)
Title: Non-existence of rational points on Shimura curves and its function field analogue
Abstract: The moduli space of QM-abelian surfaces (i.e., two-dimensional abelian varieties with an action of a maximal order of a quaternion algebra) is called a Shimura curve. I will talk about non-existence of rational points on Shimura curves as well as Hasse principle violations. The moduli space of D-elliptic sheaves (or Drinfeld-Stuhler modules) is a function field analogue of a Shimura curve. I will also talk about analogous results for function fields. The latter is a joint work with Satoshi Kondo and Mihran Papikian.
- Jilong TONG (11:40-12:30)
Title: Crystalline comparison theorem in p-adic Hodge theory
Abstract: Crystalline comparison theorems in p-adic Hodge theory relate the p-adic etale cohomology with the crystalline cohomology of varieties with good reduction defined over p-adic fields. In this talk, I shall discuss some results in this direction base on my joint works with Fucheng Tan and Yichao Tian.
- WonTae HWANG (14:00-14:50)
Title: Automorphism groups of simple polarized abelian varieties of (odd) prime dimension over finite fields
Abstract: Let X be an abelian variety of dimension g over a field k. In gen- eral, the group Aut_k(X) of automorphisms of X over k is not finite. But if we fix a polarization L on X, then the group Aut_k(X,L) of automorphisms of the polarized abelian variety (X, L) over k is known to be finite. Then it is natural to ask which finite groups can be realized as the full automorphism group of a polarized abelian variety over some field k.
In this talk, we give a classification of such finite groups for the case when k is a finite field, g >=3 is a prime number, and X is simple. Somewhat surprising thing is the fact that all such groups are cyclic. If time permits, we briefly introduce a similar classification for the case when g = 2.
- Yongqiang ZHAO (15:10-16:00)
Title: A Galois theoretic perspective of scrollar syzygy theory
Abstract: The theory of scrollar syzygy resolution of algebraic curves was introduced by Schreyer in his work for Green's conjecture. In this talk, we will give a Galois theoretic perspective of this theory. We will explain the number theory motivations for this project and will discuss, if time permits, its number fields analogues. This is a joint project with Wouter Castryck.
- Chia-Liang SUN (16:20-17:10)
Title: A new Mordell-Lang type problem motivated from a switch of roles in the unit equations
Abstract: We propose a new Mordell-Lang type problem which can be viewed as a dual to the problem generalized from that of unit equations. In positive characteristic, we will prove the following result. Let $K$ be a function field over a finite field $k$ of characteristic $p$, and $M$ be a finitely generated $k[\phi]$-submodule of K with infinitely many elements, where $\phi$ is the $p$-Frobenius map. Under the assumption that there is some natural number $n_0$ such that $M \cap K^{p^{n_0}}\subset \phi ( M )$ , we fully characterize those positive-dimensional subgroups $G$ of a split algebraic torus naturally embedded into the affine space of the same dimension with the property that the set of points on $G$ with coordinates in $M$ is equal to the orbit under $\phi$ of the set of points on a proper subvariety of $G$ with coordinates in $M$.
Tuesday, August 27, 2019
- Pin-Chi HUNG (09:20-10:10)
Title: On the growth of Mordell-Weil ranks in $p$-adic Lie extensions
Abstract: Let $A$ be an abelian variety over a number field $F$ with ordinary reduction at every primes above $p$. Under various assumptions, we establish asymptotic upper bounds for the growth of Mordell-Weil rank of $A$ in some $p$-adic Lie extension. This is a joint work with Feng Fai Lim.
- Subong LIM (10:30-11:20)
Title: Pairs of eta-quotients with dual weights
Abstract: Let $D$ be the differential operator defined by $D = \frac{1}{2\pi i}\frac{d}{dz}$. This induces a map $D^{k+1} : M^!_{-k}(\Gamma_0(N)) \to M^!_{k+2}(\Gamma_0(N))$, where $M^!_{k}(\Gamma_0(N))$ is the space of weakly holomorphic modular forms of weight $k$ on $\Gamma_0(N)$. In this talk, we show that the structure of eta-quotients is very rarely preserved under the map $D^{k+1}$ between dual spaces $M^!_{-k}(\Gamma_0(N))$ and $M^!_{k+2}(\Gamma_0(N))$.
- Yoshiyasu OZEKI (11:40-12:30)
Title: Torsion of abelian varieties and Lubin-Tate extensions
Abstract: In this talk, we study some finiteness properties for torsion points of abelian varieties over a p-adic field with values in a finite extension of the Lubin-Tate extension of a p-adic field. The result here is a straightforward generalization of Imai's theorem shown in 1975; his theorem describes the same finiteness result for the case of cyclotomic Z_p-extensions.
- Bingyong XIE (14:00-14:50)
Title : A generalization of Colmez-Greenberg-Stevens formula
Abstract: In their proof of exceptional zero conjecture, Greenberg and Stevens showed a formula for ordinary families of 2-dinmensional Galois representations. Later Colmez generalized this formula to non-ordinary families setting. In this talk, we will give a generalization of this formula to trianguline families of higher dimensional Galois representations.
- Fu-Tsun WEI (15:10-16:00)
Title: Eisenstein series on Goldman-Iwahori spaces in positive characteristic
Abstract: The Goldman-Iwahori space of rank r over a non-archimedean local field F consists of all the norms on the vector space F^r. In this talk, we shall introduce an Eisenstein series on Goldman-Iwahori spaces over local fields with positive characteristic, and present an analogue of the Kronecker limit formula. One application is to express the "Kronecker term"of a partial Dedekind-Weil zeta function of a global function field in terms of an average of certain discriminant quantities along the corresponding "Heegner cycles"on the moduli space of Drinfeld modules.
- Kenji SAKUGAWA (16:20-17:10)
Title: On Jannsen's conjecture for modular forms
Abstract: Let M be a pure motive over Q and let M_p denote its p-adic etale realization for each prime number p. In 1987's paper, Jannsen proposed a conjecture about a range of integers r such that the second Galois cohomology of M(r)_p vanishes for any prime number p. Here, M(r) denotes the rth Tate twist of M. When M is an Artin motive, Jannsen's conjecture had already essentially proven a half by Soule aroud early 1980's. In this talk, we consider the case when M is a motive associated to elliptic modular forms. I will explain my ongoing work about an approach to the conjecture used a weighted completion of the arithmetic fundamental groups of modular curves.
Wednesday, August 28, 2019
- Takashi HARA (09:20-10:10)
Title: On equivariant Iwasawa theory for CM number field
Abstract: I discuss the equivariant version of the Iwasawa main conjecture
for CM number fields over certain noncommutative p-adic Lie extensions.
- Junhwa CHOI (10:30-11:20)
Title: Iwasawa theory and CM elliptic curves
Abstract: In this talk, we will discuss some of the recent progress on the one- and two-variable main conjecture of Iwasawa theory for a finite extension of an imaginary quadratic field. We will also discuss the applications to CM elliptic curves.
- Zhefeng XU (11:40-12:30)
Title: D.H. Lehmer problem and its generlizations
Abstract: Let $q\geq2$ be an integer, for any $a$ and $\overline{a}$ in the least positive reduced residue class modulo $q$, $\overline{a}$ satisfies $a\overline{a}\equiv 1\pmod q$, let $r(q)$ be the number for cases in which $a$ and $\overline{a}$ are of opposite parity. It is $q=p$ a prime that the problem is raised by D. H. Lehmer who asks us to say something nontrivial. In this talk, based on a series of works of Prof. Zhang Wenpeng, Prof. C. Cobeli, Prof. Emre Alkan, Prof. Igor E. Shparlinski and Prof. Jean Bourgain, I will introduce both historical and recent results on D.H. Lehmer problem and give some results and corollaries.
Thursday, August 29, 2019
- Enlin YANG (09:20-10:10)
Title: Twist formula of epsilon factors of constructible etale sheaves
Abstract: We prove a twist formula for the epsilon factor of a constructible sheaf on a projective smooth variety over a finite field. This formula is a modified version of a conjecture by Kato and T. Saito. We also propose a relative version of the twist formula and discuss some applications. This is a joint work with Naoya Umezaki and Yigeng Zhao.
- Yasuhiro WAKABAYASHI (10:30-11:20)
Title: Symplectic geometry of $p$-adic Teichm\"{u}ller uniformization
Abstract: The $p$-adic Teichm\"{u}ller theory established by S. Mochizuki describes the uniformization of $p$-adic hyperbolic curves and their moduli. It may be regarded as an analogue of the Serre-Tate theory of ordinary abelian varieties. In this talk, we would like to give a quick review of a part of the $p$-adic Teich\"{u}ller theory, and explain a $p$-adic version of comparison theorems by S. Kawai, B. Loustau, et al..
- Ming Ho NG (11:40-12:30)
Title: Distribution laws of Hecke eigenvalues
Abstract: The Erdos-Kac central limit theorem asserts that the numbers of prime factors of large integers (suitably normalised) tend to follow the Gaussian distribution. Central limit behaviour is also observed in certain family of automorphic forms. In GL(2), the central limit theorem for Hecke eigenvalues of holomorphic primitive cusp forms was obtained by Nagoshi while in case of Maass cusp forms it was given by Wang and Xiao. In this talk, we will discuss some extensions in GL(N) where N >=3. This is a joint work with Y.-K. Lau and Y. Wang.
- Jangwon JU (14:00-14:50)
Title: Ternary quadratic forms representing same integers
Abstract: Kaplansky conjectured that if two positive definite integral ternary quadratic forms have perfectly identical representations, then they are isometric or, are both regular, or belong to either of two families. In this talk, we prove the existence of pairs of ternary quadratic forms representing same integers which shows that Kaplansky conjecture is not true. This is a joint work with Byeong-Kweon Oh.
- Liang-Chung HSIA (15:10-16:00)
Title: Variations of canonical heights associated to a one-parameter families of H\'enon maps
Abstract: Let ${\mathbf H} = \{H_{t}\}$ be a one-parameter family of H\'enon maps parameterized by points $t\in {\bar K}$, an algebraic closure of the number field $K$ and let ${\mathbf P} = \{P_{t}\}$ be a family of initial points. In this talk, we're interested in the problem of variations of the canonical heights ${\tilde h}_{\mathbf H_{t}}(P_{t})$ associated to the specialized H\'enon map $H_{t}$ and the point $P_{t}$ as $t$ varies over ${\bar K}.$ Let ${\tilde h}_{{\mathbf H}}({\mathbf P})$ be the function field canonical height associated to ${\mathbf H}$ and ${\mathbf P}$. Under a natural condition, we show that the function $h_{\mathbf P}(t):= {\tilde h}_{\mathbf H_{t}}(P_{t})/{\tilde h}_{{\mathbf H}}({\mathbf P})$ (provided that ${\tilde h}_{{\mathbf H}}({\mathbf P}) \ne 0$) on the parameter space is the restriction of the height function associated to a semipositive adelically metrized line bundle on projective line. We'll also discuss some application of this result. This is a joint work with Shu Kawaguchi.
Friday, August 30, 2019
- Morning Session: Discussions.
- Afternoon: Departure