You may not know this, but we can see the search terms people use to find our blog. Yesterday, four people came to our blog using the search string “p-adic 2 pi i”. Presumably, people want to know what the -adic analogue of is.
There isn’t one, and there is a good reason why. I assume that we can all agree that the most important property of is that it is the period of the complex exponential function. Unfortunately, there are no continuous periodic functions on except for the locally constant functions. The reason is very simple. Suppose that was periodic with some nonzero period . Then we would have for every integer . But, in the -adic toplogy, the integers are dense in any neighbourhood of the identity. So would take the value infinnitely often near , and would thus be constant in a neighbourhood of . The same argument gives that is constant in a neighborhood of any . (And the same arguement applies if you take to be complex valued.)
There is an interesting -analgoue of , related to the Carlitz exponential. But that is a more complicated, and more interesting, subject.
Judging from our searches, what people want to know today is when Ed Witten’s seminar meets. I’m afraid I don’t know that!
Update: This post has drawn some comments which are far smarter than what I wrote. There is a way to make -adic sense of , although I don’t understand what it is yet. Come and see our very smart commenters try to explain it to me!
Update: The conversation seems to have stopped for the moment, but I am still trying to understand these period rings, with help from Jay. If I get it, I’ll be sure to post an explanation here.