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Special values of L-functions

L-函数的特殊值

会议编号:  M240101

时间:  2024-01-02 ~ 2024-01-06

浏览次数:  2826

组织者:   Emmanuel Lecouturier, 万昕, Ashay Burungale

会议介绍

    会议摘要(Abstract)

    The topic of this conference is the arithmetic of special values of L-functions. The Birch and Swinnerton-Dyer (BSD) conjecture relates the leading term of the L-function attached to an abelian variety at s=1 to arithmetic invariants of that abelian variety. A vast generalization is the Bloch--Kato conjecture, which applies to motives, e.g. Artin motives, which is the setting for the Stark conjectures. There are also p-adic versions of these conjectures, e.g. the Mazur--Tate--Teitelbaum conjecture which is an analogue of the BSD conjecture and the Gross--Stark conjecture which is an analogue of the Stark conjecture. There even are in some cases some "refined" conjectures modulo p, e.g. the Mazur--Tate conjecture.

    Many important results related to these conjectures have been proved in recent years. For instance, we have results in many cases toward the BSD conjecture for elliptic curves over Q when the order of vanishing of the L-function at s=1 is at most one. Among the tools involved to study these conjectures are: p-adic L-functions, Euler systems, Iwasawa theory, Eisenstein congruences, special value formulae. The conference aims to gather experts who have been recently working on these topics. 

    这个会议的主题是L-函数特殊值的算术。BSD猜想将阿贝尔簇的L-函数在s=1处的首项系数与其算术不变量联系在一起。Bloch-Kato猜想是它对于一般motive的大幅推广(包括Artin motive的Stark猜想)。此外它们还有p进情形的类比,例如Mazur-Tate-Teitelbaum猜想(类比于BSD猜想)以及Gross-Stark猜想(类比于Stark猜想)。在某些情形还有一些精细化的猜想(在模p情形),例如Mazur-Tate猜想。

    近年来关于这些猜想有很多重要结果,例如对于秩不大于1情形的BSD公式。这些研究中涉及到的工具有:p进L-函数,欧拉系,岩泽理论,爱森斯坦同余,特殊值公式等。这次会议的目的是让这些领域的专家们集中起来。


    举办意义(Description of the aim)

    There are several aims for this workshop.

    - Keeping up to date with the recent advances on the various cases of the Bloch—Kato conjecture. There have been several recent notable achievements including the work of Burungale—Tian, Fouquet—Wan, Dasgupta—Kakde,Liu-Tian-Xiao-Zhang-Zhu, etc.

    -   Bringing together experts on the BSD conjecture (or other similar conjectures on L-values) so that they can exchange their new ideas and collaborate. For instance, F. Castella, G. Grossi and C. Skinner have been working on cases of the BSD conjecture in rank zero and one in the residually Eisenstein case. Other people like E. Lecouturier, J. Wang or Y. Dong have also studied this situation under different assumptions and using different ideas. It would be good understand intersections between these works and how to go further.

    -   Focusing on more refined special values conjectures with different coefficients (eg. modulo p), in particular from the point of view of the Equivariant Tamagawa Number Conjecture (a conjecture of Kato). These conjectures are not so well-studied and it would be interesting to learn from experts like D. Burns, M. Flach or P. Wake. Another aspect would be to interact with experts on p-adic conjectures and try to study whether some of these p-adic conjectures could have refined analogues (in the spirit of the Mazur—Tate conjecture).

    这次会议有几个目标:

    -交流近年来关于Bloch-Kato猜想在各种情形的最新进展,例如Burungale-Tian, Fouquet-Wan, Dasgupta-Kakde, Liu-Tian-Xiao-Zhang-Zhu等。

    -让研究BSD猜想的专家们集中起来交流他们的新想法和合作,例如Castella-Grossi-Skinner对于模p爱森斯坦情形的BSD猜想,以及Lecouturier,Wang,Yan等对于该问题基于不同想法的研究。人们需要理解这些研究之间的关系以及如何才能走得更远。

    -聚焦于一般系数情形的更精细的猜想(例如模p情形),例如Kato的等变玉河数猜想。这些猜想目前还有待进一步研究(例如从Burns, Flach,Wake等学者)。另外一个问题是与p进理论的学者们互动交流,尝试看这些p-进猜想是否有精细化的类比(类似Mazur-Tate-Teitelbaum猜想)


组织者

Emmanuel Lecouturier, YMSC
Xin Wan, MCM/AMSS
Ashay Burungale, University of Texas in Austin

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