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 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 and here, which use rigid syntomic cohomology.

3. Besser's overview talk at the conference in Loen (notes available here) was a real joy.

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 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 and here, which use rigid syntomic cohomology.

3. Besser's overview talk at the conference in Loen (notes available here) was a real joy.