The 9th East Asia Number Theory Conference
Titles and Abstracts

JST (UTC+9)
10:30 - 11:20
11:40 - 12:30
15:00 - 15:50
16:10 - 17:00

23 (Mon)M. Kurihara
M. Yu
P. Xi
R. Sugiyama

24 (Tue)
D. WeiT. Takamatsu
J. GuoJ. Lee25 (Wed)H. YooR. Zhang
Y. Okumura
Y.-L. Kuan

26 (Thu)
S. Lim
P.-J. Wong
M. Oi
S.-Y. Chen

27 (Fri)
W. KuoD. KimJ. Fang
L. Fu

Monday August 23, 2021
10:30 - 11:20
Masato Kurihara (Keio University)
Title: Some analytic quantities in the arithmetic of elliptic curves
Abstract: For a rational elliptic curve, we introduce and discuss some quantities deﬁned
analytically, giving information on the Mordell-Weil rank and on Selmer groups, in a form
diﬀerent from the conjecture of Birch and Swinnerton-Dyer. We explain that these analytic
quantities control the maps of reduction mod primes on the Mordell Weil group and Selmergroups.

11:40 - 12:30
Myungjun Yu (KIAS)
Title: Bounds for 2-Selmer ranks in terms of a modiﬁed ideal class group
Abstract: Let E be an elliptic curve over the rational number ﬁeld Q. Selmer Groups and
Ideal Class Groups are important and widely-studied objects in number theory. Brumer andKramer studied relations between these two objects in their paper in 1977. They actuallyfound an upper bound for the 2-Selmer rank of E in terms of the ideal class group of acertain cubic ﬁeld extension of Q. As an application, they determined the Mordell-Weilranks of (most) elliptic curves of prime conductor assuming the BSD conjecture. In this talk,we will talk about a generalization of Brumer-Kramer’s work to the case of elliptic curves
over an arbitrary number ﬁeld. We will give both upper and lower bounds for the 2-Selmer
rank in terms of a (modiﬁed) ideal class group, and the bounds turn out to be sharp in manycases. This is joint work with Hwajong Yoo.

15:00 - 15:50
Ping Xi (Xi’an Jiaotong University)
Title: Lang–Trotter conjecture for CM elliptic curves
Abstract: For any elliptic curve E over Q and any non-zero integer r, the Lang–Trotter
conjecture has predicted the asymptotic behaviours of the number of good primes p ⩽ x,
denoted by πE,r (x), such that the Frobenius trace of E at p is equal to the given integer
r. Quite recently, we are able to prove an estimate for πE,r (x) which conﬁrms the upperbound part of the conjecture for CM elliptic curves. Moreover, intimate connections of thisconjecture and Hardy–Littlewood conjecture can also be established to characterize the shapeof the Lang–Trotter constant in πE,r (x). This is based on the joint work with Daqing Wan(in progress).

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16:10 - 17:00
Rin Sugiayama (Japan Women’s University)
Title: Tensor structures of modulus sheaves with transfers
Abstract: Towards to a generalization of Voevodsky’s category of motives, Kahn-SaitoYamazaki and Kahn-Miyazaki-Saito-Yamazaki have introduced modulus sheaves with transfers, its cube-invariance and reciprocity sheaves with transfers. The tensor product of homotopy invariant sheaves with transfers is generalized to reciprocity sheaves via the theory ofcube-invariant modulus sheaves. I will talk about some properties of the tensor structures.
In particular, I will show new motivic presentations of the absolute Kähler diﬀerentials andthe ﬁrst inﬁnitesimal neighborhood of the diagonal. This is a joint work with Kay Rüllingand Takao Yamazaki.

Tuesday August 23, 2021
10:30 - 11:20
Dasheng Wei (Academy of Mathematical and System Science)
Title: Rational points on ﬁbration with few non-split ﬁbers
Abstract: Let f : X → P1 be a dominant map whose generic ﬁbre is rationally connected.
Assume that the Brauer-Manin obstruction controls Hasse principle and weak approximation
for rational points on all (or most) smooth ﬁbers. A natural question is whether the same
holds for the whole space X. With some assumptions, we will try to answer this questionand give some applications. This is a joint work with Harpaz and Wittenberg.

11:40 - 12:30
Teppei Takamatsu (The University of Tokyo)
Title: Minimal model program for semi-stable threefolds in mixed characteristic
Abstract: The minimal model program, which is a higher-dimensional analog of the classiﬁcation method of surfaces, is a tool to ﬁnd a ”simplest” variety in each birational equivalenceclass. The minimal model program is also studied for more general schemes not necessarily
deﬁned over a ﬁeld, and play an important role in studies of reductions of varieties. Kawamata showed that the minimal model program holds for strictly semi-stable schemes overan excellent Dedekind scheme of relative dimension two whose each residue characteristic
is neither 2 nor 3. In this talk, I will introduce a generalization of the result of Kawamata
without any assumption on the residue characteristic. This talk is based on a joint work withShou Yoshikawa.

15:00 - 15:50
Ji Guo (Academia Sinica)
Title: A Truncated Second Main Theorem for Algebraic Tori with Moving Targets andApplications
Abstract: In this talk, we will establish a second main theorem for algebraic tori with moving
targets with truncation to level 1. Then we will apply it to a special case of Green-GriﬃthLang conjecture and the integrability of the ring of exponential polynomials in the ring ofentire functions. We will also discuss the connections with Vojta’s conjecture and linear
recurrences over number ﬁelds. This is a joint work with Chia-Liang Sun and Julie Wang.

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16:10 -17:00
Jungin Lee (KIAS)
Title: Counting algebraic tori over Q by Artin conductor
Abstract: Counting number ﬁelds by discriminant is one of the most important topics inarithmetic statistics. In this talk, we discuss its natural generalization: counting algebraic
tori over Q of given dimension by Artin conductor. We propose analogues of Linnik’s andMalle’s conjecture for tori over Q and provide several evidences for them. After that, wesummarize our results on counting two and three-dimensional tori over Q.

Wednesday August 23, 2021
10:30 - 11:20
Hwajong Yoo (Seoul National University)
Title: The rational torsion subgroup of J0 (N )Abstract: In this talk, we review the work of Mazur and Ohta about the rational torsion
subgroup of J0 (N ), and discuss a natural generalization of their work.
11:40 - 12:30
Runlin Zhang (Beijing International Center for Mathematical Research, Peking University)
Title: Counting integral points on aﬃne homogeneous varieties and homogeneous dynamics
Abstract: Given an aﬃne variety over the integers, assuming there are inﬁnitely many solutions, it is natural to ask what the asymptotic count of the solutions is. It has been observedthat when the variety is homogeneous, this problem is naturally related to a problem fromhomogeneous dynamics. I will report on progress on this problem and what remains to bedone.
15:00 - 15:50
Yoshiaki Okumura (Toyo University)
Title: On congruence of Galois representations attached to A-motives
Abstract: The notion of A-motives is a higher-dimensional generalization of Drinfeld modules,
and p-adic representations attached to them are one of the most important objects in function
ﬁeld arithmetic. In this talk, we give a criterion for two congruent p-adic representations
arising from A-motives deﬁned over a global function ﬁeld to be isomorphic up to semisimpliﬁcation when they are restricted to a decomposition group, in terms of Hodge-Pinktheory. This is a function ﬁeld analogue of the criterion for ℓ-adic representations given by
Ozeki and Taguchi.
16:10 - 17:00
Yen-Liang Kuan (National Dong Hwa University)
Title: Linear Relations among Triple Multiple Zeta Values
Abstract: Multiple zeta values in the positive characteristic were initially studied by Thakur,
who deﬁned them as analogues of classical multiple zeta values of Euler. In this talk, wewill give necessary conditions for linear relations among triple multiple zeta values of sameweights.

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Thursday August 23, 2021
10:30 - 11:20
Subong Lim (Sungkyunkwan University)
Title: Independence between coeﬃcients of two modular forms
Abstract: A modular form is determined by its Fourier coeﬃcients. A natural question
is to ﬁnd relations between two modular forms when a connection between their Fouriercoeﬃcients is given. Suppose that f and g are nonzero weakly holomorphic modular forms
on SL(2, Z). We prove that if the number of ratios of Fourier coeﬃcients of f and gare ﬁnite,then f is a constant multiple of g.

11:40 - 12:30
Peng-Jie Wong (NCTS)
Title: Reﬁnements of strong multiplicity one for GL(2)Abstract: Let f1 and f2 be (holomorphic) newforms of same weight and with same nebentypus, and let af1 (n) and af2 (n) denote the normalised Fourier coeﬃcients of f1 and f2 ,respectively. If af1 (p) = af2 (p) for almost all primes p, then it follows from the strong multiplicity one theorem that f1 and f2 are equivalent. Furthermore, a result of Ramakrishnan
1states that if af1 (p)2 = af2 (p)2 outside a set of primes p of density less than 18, then f1 andf2 are twist-equivalent. In this talk, we will discuss some reﬁnements of the strong multiplicity one theorem and Ramakrishnan’s result for general GL(2)-forms. In particular, we will
analyse the density of primes p for which |af1 (p)| ̸= |af2 (p)| for non-twist-equivalent f1 andf2 .

15:00 - 15:50
Masao Oi (Kyoto University)
Title: Comparison of algebraic and geometric constructions of supercuspidal representations
Abstract: Supercuspidal representations are building blocks in representation theory of p-adic
reductive groups. In this talk, I will report a result on comparing two diﬀerent constructionsof supercuspidal representations; one is an algebraic construction due to Yu, and the other one
is a geometric construction due to Chan–Ivanov. I am also going to discuss how interesting
this comparison result is from a viewpoint of the local Langlands correspondence. This isjoint work with Charlotte Chan.

16:10 - 17:00
Shih-Yu Chen (Academia Sinica)
Title: On Deligne’s conjecture for symmetric sixth L-functions of modular forms
Abstract: In this talk, we present our recent result on Deligne’s conjecture for symmetricsixth L-functions of modular forms. We deﬁne automorphic periods associated to globally
generic cuspidal automorphic representations of GSp(4) and show that the algebraicity ofcritical L-values for GSp(4)×GL(2) can be expressed in terms of these periods. In the case of
Kim-Ramakrishnan-Shahidi lifts of GL(2), we establish period relations for the automorphic
periods and powers of Petersson norm of modular forms. The conjecture for symmetric sixth
L-functions then follows from our previous work on the algebraicity of adjoint L-functions
for GSp(4).

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10:30 - 11:20
Wentang Kuo (The University of Waterloo)
Title: On multi-dimensional Hilbert-Kamke’s problem
Abstract: Motivated by Lagrange’s theorem that every natural number is the sum of at mostfour squares, in 1770, Waring asked whether each natural number k has an associated positive
integer s such that every natural number is the sum of at most s natural numbers raised to thepower k. Then, in 1900, Hilbert asked the solubility for a system of Diophantine equationsof Waring type, now called the Hilbert-Kamke system. A multi-dimensional analogue of
those problems is naturally raised by solving the same system in the polynomial rings overintegers. A way to approach those questions is via the Hardy-Littlewood circle method. To
ensure that the major arc from the circle method provides the main contribution, one needs
to show an absolute positive lower bound of the singular series, a product of the densities
of local solutions. This talk will discuss this lower bound problem for multi-dimensional
Hilbert-Kamke’s system by studying its local solubility. The number of variables we need isnear-optimal. It is joint work with Yu-Ru Liu and Xiaomei Zhao.

11:40 - 12:30
Dohyeong Kim (Seoul National University)
Title: Integral unipotent completion and arithmetic applications
Abstract: Unipotent completion is deﬁned over a ﬁeld of characteristic zero. In this talk,we show how to obtain a p-integral generalization of it based on Lazard’s theory. As anapplication one can deﬁne discrete Selmer sets, a notion introduced by Sakugawa, in greater
generality.

15:00 - 15:50
Jiangxue Fang (Capital Normal University)
Title: Composition series for GKZ-systems
Abstract: In the 1980s, Gelfand, Kapranov and Zelevinsky generalized the hypergeometricfunctions in a uniﬁed way. They associated to any matrix A with integer entires a class ofsystems of partial diﬀerential equations or D-modules, which are now called GKZ-systems
or A-hypergeometric D-modules. In this talk, I will review some basic results of the GKZsystems. Especially, I will give a ﬁltration on the GKZ-systems with semisimple subquotientsusing toric varieties.

16:10 - 17:00
Lei Fu (Tsinghua University)
Title: On the p-adic Gelfand-Kapranov-Zelevinsky hypergeometric system
Abstract: Using the Dwork theory, we construct the p-adic GKZ hypergeometric system,
which is a twisted de Rham complex for a rigid analytic space. The underlying D-module
corresponds to the GKZ hypergeometric system of diﬀerential equations. The underlying
Frobenius structure gives an expression for the GKZ hypergeometric exponential sum.

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