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