Re: p-adic logarithm

[Kevin Buzzard <kevin.m.buzzard@gmail.com>]

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.
>