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    2024年数论研讨会

    2024-12-06   至   2024-12-09


    2024-12-06 签到
    2024-12-07 上午会场 主持人: 欧阳毅           厦门大学海韵园行政楼C503
    09:00-09:50 郗平
    西安交通大学    		 Amplifications in analytic number theory 摘要
    10:00-10:50 赵立璐
    中国科学技术大学    		 Survey on mean value theorems of the smooth Weyl sums 摘要
    10:50-11:10                      茶歇
    11:10-12:00 黄治中
    中国科学院    		 Ternary quadratic forms and the circle method 摘要
    下午会场 主持人: 王标           厦门大学海韵园行政楼C503
    14:30-15:20 吴涵
    中国科学技术大学    		 On a Generalization of Motohashi's formula 摘要
    15:30-16:20 吴小胜
    合肥工业大学    		 Non-vanishing of Dirichlet $L$-functions at the central point 摘要
    16:20-16:40                      茶歇
    16:40-17:30 范洋宇
    北京理工大学    		 Local harmonic analysis and Euler systems 摘要
    2024-12-08 上午会场 主持人: 翟帅           厦门大学海韵园行政楼C503
    09:00-09:50 李加宁
    山东大学    		 Ideal class groups in radical towers and a stable lemma 摘要
    10:00-10:50 郭宁
    哈尔滨工业大学    		 Geometric Presentation and applications 摘要
    10:50-11:10                      茶歇
    11:10-12:00 杨丽萍
    成都理工大学    		 Unit roots of the unit root $L$-functions 摘要
    下午会场 自由讨论
    2024-12-09 离会

    摘要

    郗平 - 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.

    赵立璐 - 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.

    黄治中 - 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.

    吴涵 - 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.

    吴小胜 - 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.

    范洋宇 - 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.

    李加宁 - 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.)

    郭宁 - 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.

    杨丽萍 - 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.