Re: Artin's method for class field computation

[Karim Belabas <Karim.Belabas@math.u-bordeaux.fr>]

* lucas legrand [2021-04-30 11:16]:
> Looking at 'kummer.c', I (think I) understand that your current
> bnrclassfield implementation break the looked for extension into
> intermediate prime degree extensions and then apply rnfkummer to each
> of them, following closely Hecke's method in Cohen Vol. 2.

Yes, in fact incorporating the key idea in Artin's (in fact Fieker's) method: 

1) at the end of what Cohen calls Hecke's method, there is an expensive
exhaustive enumeration of all extensions K(zeta_p)(u^(1/p)) for some
S-units u of K(zeta_p) in a finite (large!) set.

Besides being expensive, the book version is actually subtly
wrong: as described the algorithms do not work when K(zeta_p) != K:
it does work when the (abelian) splitting field of X^p - u over
K(zeta_p) lifts an abelian extension of K, which is not always the case.

N.B. They are also very expensive for two unrelated reasons:
- the first is trivial and easy to deal with: those algorithms assume
  known the fundamental units (and generally S-units and "virtual units"
  from p-Selmer groups) as algebraic numbers, which can be huge and even
  prevent the computation, whereas everything can be done with compact
  representations (products of tiny S-units with huge exponents). Think
  of first expanding 2^(10^10) when you only need various congruence
  checks and determination of signs ...

- the second is also easy but technically painful: the algorithms work with
  the full ray class group Cl_f which introduces discrete logarithm
  problems in the residue fields attached to the prime divisors of f,
  which is a disaster if a prime of large non-smooth norm divides f.
  Everything can be made to work directly in quotients of exponent p
  (with lots of technical modifications in previous algorithms in the
  book).

2) instead of 1), rnfkummer uses the splitting of primes to immediately
pinpoint a single suitable u (X^p - u factors completely in the residue
field mod \wp iff \wp lies in the congruence subgroup), more or less the
basis of what Cohen calls Artin's method.

> Maybe I'm mistaken, but I can't find anything in sources related to
> Artin's method which uses Artin reciprocity map and can deal with
> prime power degree extensions at once. Is there reasons Artin's method
> is not implemented ?

This is correct. With the modification described above, there is no real
advantage in supporting directly prime power degree extensions. In fact
it's a little less efficient than a succession of prime degree extensions. :-)
(One needs to compute the invariants of K(zeta_{p^r}) and start with
a "large" multiple of the conductor; instead of L_{r-1}(zeta_p) where
L_{r-1} is the class field attached to a quotient of exponent p^{r-1}
of Cl_f/H and exact conductors using the explicit formulas specific to
prime degrees.)

In fact what the code *should* (but currently doesn't) do is to compute
the invariants of prime degrees extensions using the invariants of their
maximal subfield, which is actually possible and helpful in a variety of
cases. Aurel Page is working of this and the theoretical part is finished,
but the code is not yet ready for production.

Cheers,

    K.B.
--
Karim Belabas, IMB (UMR 5251)  Tel: (+33) (0)5 40 00 26 17
Universite de Bordeaux         Fax: (+33) (0)5 40 00 21 23
351, cours de la Liberation    http://www.math.u-bordeaux.fr/~kbelabas/
F-33405 Talence (France)       http://pari.math.u-bordeaux.fr/  [PARI/GP]
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