Bill Allombert on Thu, 23 Jun 2016 18:49:16 +0200

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```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).

There is libpari function ZpXQ_log that you can use to compute log(1+u):

install(ZpXQ_log,GGGL)
p=7; e=10; T=polcyclo(p); /*define Q_p[X]/(T) to prec e*/
a=1+p*random(p^e*x^(poldegree(T)-1)); /* Some random element =1 mod p */
b=ZpXQ_log(a,T,p,e); /* the logarithm */
A=a*(1+O(p^e))*Mod(1,T);
B=b*(1+O(p^e))*Mod(1,T);
expB = liftpol(sum(i=0,2*e,B^i/i!));
valuation(A-expB,p)

This function is quite fast, unfortunately I wrote it with the assumption
that T was irreducible mod p (and not ony in Q_p), so I am not sure it
what happens for Q_p(zeta_p) (though it seems to work in the example
above).

Cheers,
Bill.

```

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