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What is the (Beilinson) higher regulator of a number field?
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What is the (Beilinson) higher regulator of a number field?
Here is an attempt to answer, but I hope that someone else can give a better explanation.
As Rob H. pointed out in his answer to the previous question, the survey of Nekovar is very nice, and it is also available online here.
About your question: The Beilinson regulator can be defined for number fields but also for varieties over number fields. It is a map from motivic cohomology (or algebraic K-theory) with rational coefficients, to Deligne-Beilinson cohomology, and can be thought of as a kind of Chern character.
The precise definition of the regulator is quite nontrivial, and there are several equivalent ways of defining it. Philosophically, it should be a map between certain Ext groups, induced from a "Hodge realization" of motives. Sorry for not explaining this well - it belongs to your other question about the yoga of motives.
Forgetting about philosophy, there are several ways of actually constructing the regulator. One approach is to use the general framework of characteristic classes developed by Gillet. Nekovar has a "direct" construction in his paper. For another construction in terms of explicit complexes, see the recent thesis of Elisenda Feliu, available on her webpage. Another reference is the Bourbaki talk by Soule, available here.
If you are only interested in number fields, there is another construction of the regulator, named after Borel. An excellent online reference for this, and its relation to the Beilinson regulator, is the book of Burgos, available here. The regulator generalizes the Dirichlet regulator which is covered in most introductory books on algebraic number theory. For a computational approach to Borel's regulator, see recent papers on the arXiv by Choo, Mannan, Sánchez-Garcia and Snaith.
Many, many years ago I happened to be in the room when somebody asked exactly the same question --- "What is the Beilinson regulator?" --- of Marc Levine. Marc went straight to the blackboard and drew the key diagram that makes everything clear. I knew it had to be preserved for posterity, so I ran across the street to buy a disposable camera. I am delighted to be able to share this:
(Click picture for larger version.)