The Elven Amulet does not work.
At least, it doesn't work for me. I went to the Jeweller, I got the Necklace from him, but it did not appear in my inventory, and propably the quest should change when accepting the necklace, but that didn't happen.
So I get back to the Elven Beauty and all I can do is ask for more specific information on the whereabouts of this necklace.

I haven't tried selling the Jeweller the necklace and buying it again afterwards, but I don't think that does the trick. Well, one more unsolved quest for me.