The cyclotomic character

Stream: maths

[Kevin Buzzard]

Constructing the p-adic cyclotomic character of the absolute Galois group of a field would on the other hand be a really neat project (given that, you can make a nontrivial 2-d rep). It would definitely go in mathlib (assuming it's not there -- you could check). This would be a nice simple side project.

[Kevin Buzzard]

I'm sure that we'll need the theorem that the determinant of Deligne's rho is chi * cyclo^{k-1} at some point.

[Kevin Buzzard]

Sorry, those were supposed to be DMs to someone, I was just being incompetent. Perhaps a coherent public thing to say would be: given a field K and a prime number p, we should construct the cyclotomic character from Aut_K(K-bar) or Aut_K(K-sep) to Z_p^* and prove it's continuous.

[Filippo A. E. Nuccio]

I think it would also be crucial to prove that the construction is "universal" in the sense that if you consider an extension $L/K$ then the restriction of $\chi_K$ is $\chi_L$. On the other hand I do not know what to do about $\mathrm{char}(K)$: if it is $p$, the character is not really interesting, yet it could be defined...