Learning a little Motivic Cohomology

Simply because I find it interesting, I have spent some time studying motivic cohomology from the lectures by Mazza, Voevodsky and Weibel. However, I'm finding it hard to tell if the theory is something I could use for intuition or to prove interesting theorems. I was hoping somebody could give me a few examples if this is the case.

An example I would consider for K-theory could be that Bloch (Ch. 5 in Duke Lectures) proves Roitman's theorem using in part K-theoretic techniques.

$\textbf{Question:}$ For example, I've been told that one can, in a useful way, write down the Abel-Jacobi map using this theory. However I have no example to this effect. Does anyone know of an example where this is the case, or understand maybe why this is predicted to be the case?

$\textbf{Question:}$ There are several conjectures that one can state in complete generality if one uses this theory. However, what one can prove if they learn motivic cohomology? Could one prove (perhaps in an "easier" way) some theorems that an arithmetic algebraic geometer (interpreted however you like) could find interesting? Are there number theoretic things that people study using these tools?

$\textit{Edit}$: I should clarify: Beilinson's conjecture on special values of L-functions of course aims to explain special values of various L-functions via motivic cohomology. Hence, it makes sense that in studying it people use the theory. I'm really interested if there are examples that are not of this form. (In the sense that the theory is used in work on conjectures not written in its language). Examples would be the question about the Abel-Jacobi map above, or in the case of K-theory Bloch's proof mentioned above. Milne's result on the the polynomials $P_{2r}$ appearing in the Weil conjectures (from the paper referenced by Andrew below) is also an example. Somehow I feel this can't be the only result of this kind ...

There are some results that sound extremely interesting: For example, in http://arxiv.org/pdf/1309.4068.pdf Geisser and Schmidt construct a pairing between $\textrm{mod } m$ algebraic singular homology and $\textrm{mod } m$ tame etale cohomology group. This gives a kind of class field theory in a very general setting, and appears to formally resemble the topological situation a lot.

However, it is completely unclear to me if I could ever use such a result as someone who doesn't intend to specialize in this subject.

• Hi Andrew, thanks for your comments and the reference - I wasn't aware of it. I agree, Motivic cohomology has the ability to describe special values of $L$-functions of smooth projective varieties (and more generally) as predicted by Beilinson.