|Bill Allombert on Thu, 23 Jun 2016 18:49:16 +0200|
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|Re: p-adic logarithm|
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.