Karim Belabas on Fri, 07 Aug 2015 09:37:15 +0200

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 Re: elllocalred vs. ellglobalred

```* John Cremona [2015-08-06 16:43]:
> On 6 August 2015 at 14:11, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr
> > wrote:
>
> > On Thu, Aug 06, 2015 at 01:13:30PM +0100, John Cremona wrote:
> > > For E and elliptic curve over Q,  the 3rd component of elllocalred(E,p)
> > is
> > > the [u,r,s,t]-transformation required to take E to a local minimal model
> > at
> > > p.  The 5th component of ellglobalred(E) is supposed to be a list all the
> > > elllocalred(E,p) for all bad primes;  but in the output of
> > > ellglobalred(E)[5] all the 3rd components are 0!
> >
> > The documentation says:
> >
> > * L is a vector,  whose i-th entry contains the local data at the i-th
> > prime
> >   divisor of N,  i.e.  L[i] = elllocalred(E,F[i,1]),  where the local
> >   coordinate change has been deleted, and replaced by a 0.
> >
> >
> In my version it is different:
>
> ?ellglobalred
> ellglobalred(E): E being an elliptic curve, returns [N,[u,r,s,t],c, faN,L],
> where N is the conductor of E, [u,r,s,t] leads to the standard
> model for E, c is the product of the local Tamagawa numbers c_p, faN is
> factor(N) and L[i] is elllocalred(E, faN[i,1]).

Well, Bill quoted the extended help (??ellglobalred), which has
been correctly updated, whereas the basic help (?ellglobalred) has not. :-(

I'm fixing that.

> so this must have changed.

Indeed, about 2 years ago : in commits 35191d80, e98a75c6, c280f9e6.

I "discovered" the (global) Laska/Kraus/Connell algorithm at that time
[ for which I later found a better presentation in your book ! ] and used it
to rewrite ellminimalmodel so that it skipped Tate's local algorithm
(elllocalred, which gives more information, of course).

As a result, the operation became quite inexpensive, and allowed to fix
in a cheaper way the (very annoying, albeit documented) bug that many functions
gave wrong results on non minimal inputs. So I rewrote all local tests in
terms of Kraus's criterion.

As a side effect of these minimality-related rewrites, I decided to delete the
(redundant) information from the ellglobalred output. And forgot to update the
basic help.

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 69 50
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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