John Cremona on Wed, 15 Feb 2012 16:24:28 +0100

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 Re: Complex AGM

• To: pari-dev@pari.math.u-bordeaux.fr
• Subject: Re: Complex AGM
• From: John Cremona <john.cremona@gmail.com>
• Date: Wed, 15 Feb 2012 15:24:21 +0000
• Delivery-date: Wed, 15 Feb 2012 16:24:48 +0100
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• References: <20120215150953.GA11263@yellowpig>

```For a definition of what "optimal" means and why it matters for
elliptic curve period computations, see http://arxiv.org/abs/1011.0914

I am pretty sure that taking the principal square root will never give
a sequence converging to zero.  Using the optimal branch always gives
the largest limit (and hence the smallest periods), though there is al
ittle more to the question than that.

John

On 15 February 2012 15:09, Bill Allombert
<Bill.Allombert@math.u-bordeaux1.fr> wrote:
> Hello PARI developers,
>
> I noticed the complex AGM function in PARI do not always return the so-called 'optimal' AGM.
> However this is documented:
>
> ??agm
> ...
> In the case of complex or negative numbers, the principal square root is always chosen.
> ...
> So I am not sure whether this should be fixed.
> However, do we have a proof of that PARI agm never diverges to zero ?
>
> Example of bad case are:
> agm(1,-.00001)
> which gives -0.0403257332 - 0.00327376459*I.
> but the optimal AGM is
> 0.114955789 + 0.0279973724*I
>
> Cheers,
> Bill.
>

```

• References:
• Complex AGM
• From: Bill Allombert <Bill.Allombert@math.u-bordeaux1.fr>