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Guillaume Hanrot on Thu, 04 Nov 2021 17:32:08 +0100
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Re: Transforming general cubic to standard form
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- To: John Cremona <john.cremona@gmail.com>
- Subject: Re: Transforming general cubic to standard form
- From: Guillaume Hanrot <Guillaume.Hanrot@ens-lyon.fr>
- Date: Thu, 4 Nov 2021 17:32:02 +0100
- Cc: "Grechuk, Bogdan (Dr.)" <bg83@leicester.ac.uk>, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>, "pari-dev@pari.math.u-bordeaux.fr" <pari-dev@pari.math.u-bordeaux.fr>
- Delivery-date: Thu, 04 Nov 2021 17:32:08 +0100
- In-reply-to: <CAD0p0K7UZst0GP0RdnzqQj3pHO8GpyG67XvvLVG8ZEW-UUi6CQ@mail.gmail.com>
- Mail-followup-to: Guillaume Hanrot <Guillaume.Hanrot@ens-lyon.fr>, John Cremona <john.cremona@gmail.com>, "Grechuk, Bogdan (Dr.)" <bg83@leicester.ac.uk>, Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>, "pari-dev@pari.math.u-bordeaux.fr" <pari-dev@pari.math.u-bordeaux.fr>
- Organisation: ENS Lyon
- References: <DB7PR06MB496685293708862D8C86C10C9E8A9@DB7PR06MB4966.eurprd06.prod.outlook.com> <CAD0p0K7y7nmmJpNUB28gi28jXmkPPu9bq3pVxGM7NUhtPc2_qw@mail.gmail.com> <YYAzwpjPPCdZQij3@seventeen> <DB7PR06MB4966CAFBA73245F8D5DE74719E8D9@DB7PR06MB4966.eurprd06.prod.outlook.com> <CAD0p0K7UZst0GP0RdnzqQj3pHO8GpyG67XvvLVG8ZEW-UUi6CQ@mail.gmail.com>
Dear colleagues,
> line at infinity. So I assume that you have an affince plane cubic,
> not in Weierstrass form, and want to find its integral points. That
> is a perfectly good question, but I do not know an answer to it.
> There are only finitely many (by Siegel's Theorem
> https://en.wikipedia.org/wiki/Siegel%27s_theorem_on_integral_points).
If my (old) memories are good, on this precise question I would suggest
consulting :
R. Stroeker, B.M.M de Weger, Solving Elliptic Diophantine Equations:
The General Cubic Case, Acta Arith. 87 (4) (1998).
Sincerely yours,
Guillaume Hanrot