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    2024 Workshop on Number Theory

    2024-12-06   to   2024-12-09


    2024-12-06 Check-in & Registration
    2024-12-07 Morning Session Chair: Yi Ouyang           Conference Room C503
    09:00-09:50 Ping Xi
    Xi'an Jiaotong University    		 Amplifications in analytic number theory Abstract
    10:00-10:50 Lilu Zhao
    University of Science and Technology of China    		 Survey on mean value theorems of the smooth Weyl sums Abstract
    10:50-11:10                      Tea Break
    11:10-12:00 Zhizhong Huang
    Chinese Academy of Sciences    		 Ternary quadratic forms and the circle method Abstract
    Afternoon Session Chair: Biao Wang           Conference Room C503
    14:30-15:20 Han Wu
    University of Science and Technology of China    		 On a Generalization of Motohashi's formula Abstract
    15:30-16:20 Xiaosheng Wu
    Hefei University of Technology    		 Non-vanishing of Dirichlet $L$-functions at the central point Abstract
    16:20-16:40                      Tea Break
    16:40-17:30 Yangyu Fan
    Beijing Institute of Technology    		 Local harmonic analysis and Euler systems Abstract
    2024-12-08 Morning Session Chair: Shuai Zhai           Conference Room C503
    09:00-09:50 Jianing Li
    Shandong University    		 Ideal class groups in radical towers and a stable lemma Abstract
    10:00-10:50 Ning Guo
    Harbin Institute of Technology    		 Geometric Presentation and applications Abstract
    10:50-11:10                      Tea Break
    11:10-12:00 Liping Yang
    Chengdu University of Technology    		 Unit roots of the unit root $L$-functions Abstract
    Afternoon Session Free Discussions
    2024-12-09 Departure

    Abstracts

    Ping Xi - Amplifications in analytic number theory

    The idea of amplifications, dating back at least to van der Corput, Vinogradov, Kloosterman and Selberg, is quite elementary in the sense that one embeds the object of interests into a suitable family, and the individual information can be reflected on average. Such observations turn out to be very powerful in the developments of sieve methods, estimates for oscillatory sums and $L$-function theory. In this talk, we aim to give a short survey on the underlying ideas, and some recent applications to the above three typical objects will also be discussed.

    Lilu Zhao - Survey on mean value theorems of the smooth Weyl sums

    The mean value theorem of the exponential sums has applications to the Waring type problems. Many records in Waring’s type problem are obtained by using smooth numbers. In this talk, we give a survey on mean value theorems of the Weyl sums. We mainly focus on the exponential sums over smooth numbers.

    Zhizhong Huang - Ternary quadratic forms and the circle method

    We report recent results on counting integer solutions to $F(x_1,x_2,x_3)=m$, where $F$ is a non-singular integral ternary quadratic form and m is a non-zero integer, with a view towards quantitative strong approximation. Our approach is based on the $\delta$-variant of the Hardy--Littlewood circle method developed by Heath-Brown.

    Han Wu - On a Generalization of Motohashi's formula

    Spectral reciprocities are equalities between moments of automorphic $L$-functions in different families. They are powerful tools for the study of the moment problem and the subconvexity problem. The first spectral reciprocity formula is Motohashi's formula, which relates the cubic moment of $L$-functions for $\mathrm{GL}_2$ with the fourth moment of $L$-functions for $\mathrm{GL}_1$. The exploitation of this formula (over $\mathbb{Q}$) has led Conrey-Iwaniec (Ann. of Math. 2000) and Petrow-Young (Ann. of Math. 2020, Duke Math. J. 2022) to the uniform Weyl bound for all Dirichlet $L$-functions, a celebrated recent result.

      In this talk, we give an adelic version of a generalization of Motohashi's formula relating $\mathrm{GL}_3 \times \mathrm{GL}_2$ with $\mathrm{GL}_3 \times \mathrm{GL}_1$ and $\mathrm{GL}_1$ moments of $L$-functions, whose study was initiated by Xiaoqing Li (Ann. of Math. 2011). For many types of the $\mathrm{GL}_3$ representation, we describe the local weight transforms via a decomposition of Voronoi's formula in terms of elementary transforms, which generalizes the one given by Miller--Schmid (Ann. of Math. 2006) in a way consistent with the local Langlands correspondence. As application, we announce a local non-archimedean weight estimation on the fourth moment side, generalization of a former joint work with P. Xi.

    Xiaosheng Wu - Non-vanishing of Dirichlet $L$-functions at the central point

    In this talk, we introduce a recent work on non-vanishing of Dirichlet $L$-functions at the central point. It is generally believed that Dirichlet $L$-functions do not vanish at the central point. For a large modulus, it is showed that there are at least $\frac{7}{19}$ of primitive Dirichlet characters, for which the central value $L(\frac12,\chi)$ does not vanish. This talk is based on a joint work with Xinhua Qin.

    Yangyu Fan - Local harmonic analysis and Euler systems

    In this talk, we report our new approach to the horizontal Euler system property of theta elements by the relative Satake isomorphism. This is a joint work with L. Cai and S. Lai.

    Jianing Li - Ideal class groups in radical towers and a stable lemma

    I will talk about the $p$-class groups in radical towers such as $F_n=\mathbb{Q}(\mu_p,\sqrt[p^n]{a})$ when $n$ varies, where $a$ is an integer. Note that $F_2/F_0$ is non-Galois. However, we prove a stable result which assers that if the $p$-class group of $F_0$ and $F_1$ are same, then so is $F_n$, under some usual ramification conditions. (It is previously well known that such a stable result holds for cyclic extensions.)

    Ning Guo - Geometric Presentation and applications

    Various problems concerning cohomology sets and K-theory, including Gersten's injectivity and the Grothendieck-Serre conjecture, have affirmative answers in the equi-characteristic case. A helpful tool for dealing with mixed characteristic cases is geometric presentation theorems. Roughly speaking, these theorems realize the local ring under consideration as a relatively smooth (or étale) object over a regular DVR or a field with a low relative dimension such that problems reduce to the DVR or field cases. In this talk, we will discuss several geometric presentation theorems (due to Gabber-Quillen, Colliot-Thélène, Ojanguren, etc) and see their application to the Grothendieck-Serre conjecture in the mixed characteristic case. In particular, I will introduce the notion of weak elementary fibrations, which traces back to Artin's "bon voisinage" in SGA4. These are individual joint works with Ivan Panin and Fei Liu.

    Liping Yang - Unit roots of the unit root $L$-functions

    Adolphson and Sperber expressed the unique unit root of the toric exponential sums in terms of the $\mathcal{A}$-hypergeometric functions. For the unit root $L$-function of a family of toric exponential sums, Haessig and Sperber conjectured its unit root behaves similarly to the classical case studied by Adolphson and Sperber. Under a lower deformation hypothesis, Haessig and Sperber proved this conjecture is true. We will show that Haessig and Sperber's conjecture is true in general. This is a joint work with Zhang Hao.