(I hesitate to post that question here, but I received on answer on FB:) Does anyone know how the current status of work on them is? And how the possible generalizations etc. which one thinks currently on, look like?
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2$\begingroup$ I hesitate to post that question here, but I received on answer on FB??? I don't understand this sentence. $\endgroup$– Marc PalmCommented Apr 6, 2013 at 15:45
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5$\begingroup$ I understand this sentence as « I have received no answer on Facebook ». I would downvote the question if I was sure I am not mistaken. $\endgroup$– Jonathan ChicheCommented Apr 6, 2013 at 15:58
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1$\begingroup$ @Marc and Jonathan: Yes, I had posted the question on facebook, but had so far received no answer (probably because I failed to find some recent survey or article and therefore an answer would be too simple; the question came up for me from random bedside yesterday reading some old article on these conjectures, i.e. the typical sort of things for social network sites. It is then too simple for MO, but the weight of curiosity ...). $\endgroup$– Thomas RiepeCommented Apr 6, 2013 at 16:07
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2$\begingroup$ @Jonathan: I would upvote your comment if you explain it (isn't MO just another such site, specialized on mathematics?). $\endgroup$– Thomas RiepeCommented Apr 6, 2013 at 18:30
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2 Answers
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The survey of Nekovar tells you what was known about the Beilinson conjectures in the early 90s. Other surveys/introductions from that time include Scholl-Deninger, Soulé, Ramakrishnan (in Contemporary mathematics 83), and the volume edited by Rapoport, Schappacher and Schneider (introduction here, all articles here). Since then, not a lot has happened I think. There is a fairly recent article of Otsubo with some results for Fermat curves (published version, arXiv version). There is a nice reformulation of the conjectures in terms of Arakelov motivic cohomology, by Jakob Scholbach (see articles on his webpage). You might also want to check out some of the articles of Rob de Jeu and his coauthors. In addition, there have been various attempts at new descriptions of the Beilinson regulator, most recently by Bunke and Tamme (their work will by the way be the topic of a summer school in Freiburg in July), but this does not in itself imply any progress on the Beilinson conjectures themselves.
If you are interested in recent progress on special values in general, there are other areas where more exciting things are happening. Search for work on the Birch and Swinnerton-Dyer conjecture, the Equivariant Tamagawa number conjecture, and Weil-etale cohomology (starting point for Weil-etale: webpage of Baptiste Morin).
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$\begingroup$ By the way, one of the causes of my curiosity is a similar impression as you express: Having noticed the conceptual work on BSD etc. - i.e. "the other side of the special-values-questions" -, I failed to notice similar things the "Beilinson side". $\endgroup$ Commented Apr 6, 2013 at 18:21
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In addition to Andreas's excellent answer, we should also mention the Tamagawa number conjecture of Bloch and Kato, which predicts the undetermined rational factor arising in Beilison's conjectural description of the $L$-value. The Bloch-Kato conjecture was later reformulated and generalized by Fontaine and Perrin-Riou to the case of motives with coefficients in an arbitrary number field. Here are some references :
Bloch, Kato, L-functions and Tamagawa numbers of motives.
Fontaine, Perrin-Riou, Autour des conjectures de Bloch et Kato: cohomologie galoisienne et valeurs de fonctions L.
Colmez, Fonctions L p-adiques.
Kings, The Bloch-Kato conjecture on special values of L-functions. A survey of known results.
Flach, The equivariant Tamagawa number conjecture : A survey.
Gealy, On the Tamagawa Number Conjecture for Motives Attached to Modular Forms.
Bellaïche, An introduction to the conjecture of Bloch and Kato.