Smart elliptic curve rational point search given RegSha

[Iftikhar Burhanuddin]

Hi folks,

Let E be a global minimal model of an elliptic curve over QQ, with a
2-torsion point which generates the torsion subgroup, and with
Mordell-Weil rank 1 (under BSD). Let RegSha be equal to
Regulator(E)*#Sha(E) which has been computed using the BSD formula.

(Assume computing R the non-torsion rational point generator of the
Mordell-Weil group E(QQ) using descent by 2-isogeny is time
consuming.)

Does the RegSha information make it "easier" to compute R? If so
kindly supply search strategy.

Regards,
Ifti

[William Stein]

This seems more like a http://mathoverflow.net/ question that a
sage-nt mailing list question.

-- William

>
> Regards,
> Ifti
>
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--
William Stein
Professor of Mathematics
University of Washington
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[Iftikhar Burhanuddin]

http://mathoverflow.net/questions/87511/smart-elliptic-curve-rational-point-search-given-regsha

[John Cremona]

On 4 February 2012 05:52, William Stein <wst...@gmail.com> wrote:
> On Fri, Feb 3, 2012 at 6:46 PM, Iftikhar Burhanuddin
> <iftikhar.b...@gmail.com> wrote:
>> Hi folks,
>>
>> Let E be a global minimal model of an elliptic curve over QQ, with a
>> 2-torsion point which generates the torsion subgroup, and with
>> Mordell-Weil rank 1 (under BSD). Let RegSha be equal to
>> Regulator(E)*#Sha(E) which has been computed using the BSD formula.
>>
>> (Assume computing R the non-torsion rational point generator of the
>> Mordell-Weil group E(QQ) using descent by 2-isogeny is time
>> consuming.)
>>
>> Does the RegSha information make it "easier" to compute R? If so
>> kindly supply search strategy.
>
> This seems more like a http://mathoverflow.net/ question that a
> sage-nt mailing list question.
>

On the other hand *I* am more likely to read it here (I made my very
first ever post on MO yesterday!).

Silverman wrote a paper about this, and the answer is basically "yes".
Ask again if you cannot find it. The idea is that you know the
height of R, hence you know that the real component of the height is
one of a finite (usually) small set of possibilities, hence via
numerical solving you can find the x-coordinate approximately (and up
to a finite set of possibilities).

John