|Bill Allombert on Thu, 22 Jul 2010 15:20:07 +0200|
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|Re: Dirichlet L-Functions|
On Mon, Jul 12, 2010 at 02:07:22PM -0500, Ariel Pacetti wrote: > > A stupid question (from my ignorance): I heard a talk recently by > Guardia-Montes-Nart about a package they implemented in magma to > compute factorization of ideals and many other number field > invariants without computing explicitely the ring of integers (which > I presume Pari does, right?). They posted the algorithm in arxiv > > http://arxiv.org/abs/1005.4596 > > where they can work with quite big number fields (I suggested them > to implement L-series computations, and answered will do), do you > (experts in Pari) think it is worth it to have something similar > implemented in GP? (I saw it has nothing to do with the standard way > to write down ideals nor integers which might be a LOT of work to > do, and so far I only can think in computing L-functions and ideal > factorizations as applications). > > I am just sharing my question with you all.... Last time I check, PARI integral basis algorithm (Zassenhaus/Ford/Pauli/Roblot round4) was faster than Magma/KANT algorithm. For example: nfbasis(x^100-x^75+x^50+2^500,1); take about 3 seconds. So I would say this is not a priority for us now (we have more pressing issues to adress first like the handling of fundamental units). I do not know you would use it to compute L-series: this only helps for the ramified primes, and for large discriminants, the expected convergence of the series is too slow to be carried out on a computer (even assuming the GRH). Cheers, Bill.