Kevin Buzzard on Sat, 25 Jun 2016 16:34:17 +0200 |
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Re: p-adic logarithm |
I think that one efficient way to compute Iwasawa's p-adic log (i.e. normalised so that log(p)=0) is to take your random element x of your random p-adic field, raise it to some power n until its valuation is an integer multiple of the valuation of p, divide by an appropriate power of p to get a unit u=x^n/p^m, raise the unit to an appropriate power (e.g. q-1) until it's a 1-unit, raise this 1-unit to an appropriate power (a power of p) until it's congruent to 1 modulo a sufficiently large power of p for the power series for log to converge, and then plug it into the power series. Once we've computed ell:=log(x^N/p^M) we just divide by N to get log(x). As long as you can compute valuations on your p-adic field this should be both efficient and fairly simple to code. I guess the disadvantage of this method is that if N is a multiple of a large power of p (which can happen if the p-adic field is badly ramified) then one might lose a bit if p-adic precision, but it makes up for this in speed. Kevin On 25 June 2016 at 13:21, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr> wrote: > On Thu, Jun 23, 2016 at 06:49:07PM +0200, Bill Allombert wrote: >> On Thu, Jun 23, 2016 at 02:48:00PM +0000, LECOUTURIER Emmanuel wrote: >> > Hello, >> > Is there any function which allows us to compute directly Iwasawa >> > p-adic logarithm in finite extensions of Q_p ? (like Q_p(zeta_p)) >> >> GP itself only handles Q_p, however you can factor x as >> x = t*(1+u) where t is of torsion and u is in the convergence domain of >> log(1+X) and then use the power series for log (or for atanh). > > Writing this gives me an idea: instead of computing t, if n is the order of > the torsion subgroup, then x^n = (1+u)^n = 1+v and > log(x^n)= log(x)/n > In some case this can be simpler than actually computing t. > > Cheers, > Bill. >