I'd like to write a better response, but I must be brief.

For now, let me offer some places to read. Long story short, it is  predicted that there's a relationship between special values of $p$-adic $L$-functions and syntomic regulators (which are the analogue of Beilinson's regulators in the $p$-adic world).

1. The beautiful [paper](http://www.springerlink.com/content/vcjlmrmeeh9ktwll/) of Manfred Kolster and Thong Nguyen Quong Do is, I think, a very readable resource.

2. The best results I know in this direction are Besser's papers [here](http://www.math.uiuc.edu/K-theory/0298/) and [here](http://www.math.uiuc.edu/K-theory/0299/), which use rigid syntomic cohomology.

3. Besser's overview talk at the conference in Loen (notes available [here](http://folk.uio.no/rognes/yff/alexandra.html)) was a real joy.