Titles and Abstracts

Kenichi Bannai (Keio)

Title:

On the Equivariant Polylogarithm and the Special Values of Hecke L-functions 

for totally real fields

Abstract: In this talk, we give the construction of the equivariant polylogarithm class 

for the algebraic torus associated to totally real fields. We will then give its relation to the Shintani generating class, a class in equivariant coherent cohomology which generates the special values of Lerch zeta functions. We will then state a conjecture concerning the plectic version of this construction and its implication to the Beilinson conjecture in this case.  Some observations in the p-adic case will also be discussed.  This is a joint work with Hohto Bekki,

Kei Hagihara, Tetsuya Ohshita, Kazuki Yamada, and Shuji Yamamoto.

Sanoli Gun (Chennai)

Title: Product of prime ideals in Ray class groups

Abstract: In this talk, we will discuss a sharp explicit Brun-Titchmarsh theorem for ray classes and an equally explicit improved Brun-Titchmarsh theorem for large subgroups of ray class groups. These tools are used to represent ideal classes of ray class groups by product of prime ideals of small size. This is an ongoing joint work with Jyothsnaa Sivaraman and Olivier Ramare.

Ming-Lun Hsieh (Acad.Sinica, Taiwan)

Title: On the first derivatives of the Katz p-adic L-functions for CM fields

Abstract: Buyukboduk and Sakamoto in 2019 proposed a precise conjectural formula relating the leading coefficient at s=0 of the cyclotomic Katz p-adic L-functions associated with ray class characters of a CM field K to suitable L-invariants/regulators of K. 

They were able to prove this formula in most cases when K is an imaginary quadratic field thanks to the existence of the Euler system of elliptic units/Rubin-Stark elements. 

In this talk, we will present a formula relating the first derivative of the cyclotomic Katz p-adic L-functions attached to ring class characters of general CM fields to the product of L-invariant and the value of some improved Katz p-adic L-function at s=0. 

In particular, we show that these Katz p-adic L-functions have a simple trivial zero if and only if their cyclotomic L-invariants are non-zero.

Our method uses the congruence of Hilbert CM forms and does reply on the existence of the conjectural Rubin-Stark elements. 

This is a joint work with Adel Betina based on a previous joint work with Masataka Chida in the case of imaginary quadratic fields.

Bingrong Huang (Shandong)

Title: On the Rankin-Selberg problem

Abstract: In this talk, I will introduce a method to solve the Rankin-Selberg problem on the second moment of Fourier coefficients of a GL(2) Hecke modular or Maass cusp form. This improves the classical result of Rankin and Selberg (in 1939/1940). If time permit, some other results will also be presented.

Mahesh Kakde (Bangalore)

Title: On the Brumer-Stark conjecture

Abstract : The first part of the talk will focus on statement of the Brumer-Stark conjecture and its refinements. The second half of the talk will focus on giving a brief sketch of the proof away from p=2. This is a joint work with Samit Dasgupta.

Kwang- Seob Kim (Chosun)

Title: Class numbers of large degree nonabelian number fields

Abstract: If a number field has a large degree and discriminant, the computation of the class number becomes quite difficult, especially without the assumption of GRH. In this article, we will unconditionally show that a certain nonabelian number field of degree 120 has class number one. This field is the unique A5×C2 extension of the rationals that is ramified only at 653 with ramification index 2. It is the largest degree number field unconditionally proven to have class number 1. The proof uses the algorithm of Guàrdia, Montes and Nart to calculate an integral basis and then finds integral elements of small prime power norm to establish an upper bound for the class number; further algebraic arguments prove the class number is 1. It is possible to apply these techniques to other nonabelian number fields as well.

Masanori Morishita (Kyushu)

Title: On arithmetic Dijkgraaf-Witten theory

Abstract: I will talk about an arithmetic analog for number rings of Dijkgraaf-Witten's (2+1)-dimensional topological quantum field theory, and related topics, based on arithmetic topology and Minhyong Kim's arithmetic Chern-Simons theory.

Trung Hieu Ngo (Hanoi U.Sci.tech.)

Title: Quadratic congruences and Weyl sums

Abstract: Given a sequence of real numbers arising from an arithmetic context, how are their fractional parts distributed on the unit interval? There are many beautiful instances of this question in number theory.

In a classical context, we may consider a quadratic congruence, varying the moduli and forming a sequence of roots-to-moduli ratios. In 1963, Christopher Hooley showed that if the moduli vary over the natural numbers, then the sequence is uniformly distributed modulo one. The equidistribution modulo one when the moduli take values in the prime numbers is a much more challenging problem. It was first established for negative-discriminant quadratics by Duke-Friedlander-Iwaniec in 1995 and then for positive-discriminant quadratics by Toth in 2012. In both cases, a crucial ingredient is a sufficiently strong estimate for exponential sums of Weyl type associated with the congruence roots.

We plan on giving a general introduction to this circle of equidistribution questions.

We will review the classical results of Hooley, Duke-Friedlander-Iwaniec, and Toth. We will discuss our recent improvements on bounds of congruence Weyl sums for positive-discriminant quadratics. Time permitting, we will highlight interesting applications of congruence Weyl sums.

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Duy Tan Nguyen (Hanoi U. Sci. Tech.)

Title: Relations in the maximal pro-p quotients of absolute Galois groups

Abstract: An open question in Galois theory is to describe the absolute Galois groups of fields among profinite groups. A description of the maximal pro-p quotients of absolute Galois groups of general fields for a given prime number p is already a challenging problem.

For a field F, we denote G_F the absolute Galois group of F and G_F(p) its maximal pro-p quotient.

I. R. Shafarevich essentially showed that G_F(p) is a free pro-p group if F is a local field which does not contain a primitive p-th root of unity. In 1954, Y. Kawada showed that if F is a local field containing a primitive p-th root of unity, then G_F(p) admits a presentation

1 \to R \to S \to G_F(p) \to 1,

where S is a free pro-p group and R is a normal subgroup of  S generated by a single relation r. By the works of Demushkin, Serre and Labute, one now knows the explicit shape of such a relation r.

In this talk, I will discuss some interesting restrictions on the structure of Galois groups of maximal p-extensions of fields containing a primitive pth root of unity. Namely, we show that certain shapes of relations (which though look very similar to the relation in the case of local field as above), cannot be appeared as a relation for a presentation of  G_F(p) for any F containing a primitive pth root of unity.

(Joint work with J\'an Min\'a\v{c} and Michael Rogelstad)

Hae-Sang Sun (UNIST, Korea)

Title: Dynamics of continued fractions and distribution of modular inverses

Abstract: Let R(n,x) be the number of modular inverses modulo n that are less than x. It is well-known (e.g. Heath-Brown) that R(p,p^{3/4+\epsilon})\gg p^{1/2+\epsilon} for primes p. An open problem is to improve the exponent 3/4. In the talk, I will introduce how to study the average version of the problem in terms of the dynamics of continued fractions. This is research in progress.

Masatoshi Suzuki (Tokyo Inst.Tech.)

Title: On canonical systems related to roots of polynomials

Abstract: A canonical system is a first-order system of ordinary differential equations given by a function taking values in positive semidefinite real symmetric matrices, called its Hamiltonian. Since a polynomial whose roots are all inside the unit circle gives a spectral datum of a canonical system, the inverse problem of finding the Hamiltonian of a canonical system that recovers that datum can be considered. The goal of this talk is to state the explicit answer to this inverse problem. That reveals a relation with the classical SchurCohn test, which is a way to determine the distribution of the roots of a polynomial with respect to the unit circle by the number of sign changes in the determinants of the minors of a particular matrix consisting of the coefficients.

Fang- Ting Tu (Louisiana State U.)

Title: Hypergeometric functions over finite fields and a Whipple formula

Abstract:Hypergeometric functions over finite fields have nice character sums ｒrepresentations  and have applications to counting points of certain varieties over finite fields, which provide information on the corresponding Galois representations.  In this talk, I will give an introduction to hypergeometric functions over finite fields  and some

recent developments including the recent  joint work

<https://arxiv.org/pdf/2103.08858.pdf> with Wen-Ching Winnie Li and Ling Long.  In this joint project, we consider the hypergeometric data corresponding to a formula due to Whipple which relates certain hypergeometric values _7F_6(1) and _4F_3(1). When the hypergeometric data are primitive and defined over \mathbb Q, we explain a special structure of the corresponding Galois representations behind Whipple's formula leading to a decomposition that can be described by the Fourier coefficients of Hecke eigenforms. In this talk,  I will use an example to demonstrate our approach and relate the hypergeometric values to certain periods of modular forms.

Fangyang Tian (Zhejiang)

Title: Period Relations of Standard L-functions of Symplectic Type

Abstract: A classical result of Euler says that the value of the Riemann-Zeta function at a positive even integer 2k is a rational multiple of \pi^{2k}. In the 1970s, a successive pioneering work of G. Shimura revealed the relation of different critical values of L-function that are attached to modular forms of\mathrm{GL}_2. This type of result, conjectured by D. Blasius for general linear groups, is called period relation of a certain automorphic L-function, which is closely related to a celebrated conjecture of P. Deligne.

  In this talk, I will discuss my work joint with Dihua Jiang and Binyong Sun on the period relation for the twisted standard L-function L(s, \Pi\otimes\chi), where \Pi is an irreducible cuspidal automorphic representation of GL_{2n}(\mathbb{A}) which is regular algebraic and of symplectic type.  Along this talk, I will also discuss the key ingredient of this project - the existence of uniform cohomological test vector, which provides the most precise information on the archimedean local integrals.

Yingnan Wang (Shenzhen)

Title: ON THE EXCEPTIONAL SET OF THE GENERALIZED RAMANUJAN CONJECTURE 

Abstract: For any prime p and Hecke-Maass form φ on GL(n)(n > 2), αφ,1(p), . . . , αφ,n(p) denote the corresponding Satake parameters of φ at p. The Generalized Ramanujan Conjecture asserts that |αφ,i(p)| = 1, i = 1, . . . , n. In this talk we will survey the recent results and developments centered on this problem. This is a joint work with Yuk-Kam Lau and Ming Ho Ng. 

Ping Xi (Xian Jiaotong U.)

Title: Arithmetic exponent pairs: individual and averaged

Abstract: The theory of classical exponent pairs for analytic exponential sums have been extensively developed since 1920’s thanks to van der Corput, Philipps, Bombieri, Iwaniec, Bourgain, et al, motivated by various applications to analytic number theory. Recently, we were able to develop the so-called arithmetic exponent pairs for general Frobenius trace functions composited by $\ell$-adic sheaves over finite fields. Moreover, an averaged version of arithmetic exponent pairs may also be developed for some special trace functions to produce sharper estimates than individual ones. Some applications will also be mentioned if time permits.

Jingwei Xiao (IAS)

Title: An RTF approach to Unitary Friedberg-Jacquet periods

Abstract: Let G be a reductive group over a number field F and H a subgroup. Automorphic periods study the integrals of cuspidal automorphic forms on G over H(F)\H(A_F). They are often related to special values of certain L functions. In this talk, I will explain my work in progress with Wei Zhang that study (G,H)=(U(2n), U(n)☓U(n)) using a relative trace formula comparison. We prove the required fundamental lemma using a limit of the Jacquet-Rallis fundamental lemma.

Daxin Xu (Chinese Acad. Sci.)

Title: Bessel F-isocrystals for reductive groups

Abstract: I will first review the relationship between the Kloosterman sums and the classical Bessel differential equation. Recently, there are two generalizations of this story (corresponding to GL_2-case) for arbitrary reductive groups using ideas from the geometric Langlands program, due to Frenkel-Gross and Heinloth-Ngô-Yun. I will discuss my joint work with Xinwen Zhu where we unify previous two constructions from the p-adic aspect and identify the exponential sums associated to different groups as conjectured by Heinloth-Ngô-Yun. If time permits, I will also talk about my recent joint work with Masoud Kamgarpour and Lingfei Yi on the generalization of the above story to hypergeometric sheaves.

Myungjun Yu (KIAS)

Title: The rational cuspidal subgroup of J_0(p^2M)

Abstract: For a positive integer N, let C_N(Q) be the rational cuspidal subgroup of J_0(N) and let C(N) be the rational cuspidal divisor class group of X_0(N), which are both subgroups of the rational torsion subgroup of J_0(N). We prove that two groups C_N(Q) and C(N) are equal when N=p^2M for any prime p and any squarefree integer M. To achieve this, we show that all modular units on X_0(N) can be written as products of certain functions, which are constructed from generalized Dedekind eta functions. Also, we determine the necessary and sufficient conditions for such products to be modular units on X_0(N) under a mild assumption.

Akihiko Yukie (Kyoto)

Title: On the GIT stratification of prehomogeneous vector spaces

Abstract: The notion of GIT stratification was established in 1980's by Kempf, Ness and Kirwan, The rationality question including the non-split case was answered by the Tajima and the speaker. It is possible to use computer to determine the parametrizing set of the GIT stratification when the group is not too big. We applied this method to some important prehomogeneous vector spaces and determined all orbits rationally over any perfect field when the group is a product of general linear groups.

Yichao Zhang (Harbin)

Title: Fourier coefficients of double Eisenstein series and their analytic continuation

Abstract: Based on a previous work with Choie and Kohnen, we compute the Fourier coefficients of a twisted double Eisenstein series, and then analytically continue them to the critical strip. As an application,  we obtain a rationality result of Kohnen-Zagier type on the Petersson inner product of Cohen kernels. This is a joint work with Yuanyi You.

Yitang Zhang (Santa Barbara)

Title:Discrete mean formulas and the Landau-Siegel Zero

Abstract: As a special and (probably much) weaker form of the Generalized Riemann Hypothesis, the Landau-Siegel zero problem has its own interest and various applications in number theory. In this talk we introduce a new approach that first gives a relationship between the existence of the Landau-Siegel zero and the distribution of zeros of a family of Lfunctions. The problem is further reduced to showing that certain discrete sums over the zeros are not always non-negative.