Re: New GP function ellrank (2-descent)

[Bill Allombert <Bill.Allombert@math.u-bordeaux.fr>]

On Tue, Feb 16, 2021 at 12:04:39PM +0000, John Cremona wrote:
> Bill,
> 
> Can you add some documentation for the parameter 'effort'?  (That name
> sounds very Magma-like, by the way).   I assume that it is related to
> a bound on a search for rational points on 2-covers (quartics), but is
> it linear or exponential?

It is a bound on the number of Selmer classes to inspect, a bound of the number
of quartic models to try for each Selmer classes, and a bound (* 10000) on
the naive height of the points for hyperellratpoints.

For example you can try:
? \g2
? E=ellinit("2170b1");
? setrand(2);ellrank(E,10)

The selmer rank is 1, so there is a single Selmer classes, however
we can generate different quartics by taking different representatives,
for example

Y^2 = 5812437140379431936*x^4+1347241111842961644544*x^3+117156424752666733221120*x^2+4530122383197205883049984*x+65719549423500847171260416

Y^2 = 221275160067200000*x^4-475741594144480000*x^3+756678069244798800*x^2-644795816435820800*x+377052872754508800

This second one has a small rational point:
[48513/7192,14958243512261070/808201]
while the point on the first is much (~10000 time) harder to find: 
[-4298555/68258,83178143745675062784/1164788641]

It is more efficient to try more quartics than to use a large bound for
hyperellratpoints (as an analogy, MPQS is more efficient than QS).

I came up with this trick by experimenting with Denis program.
If you have a geometrical interpretation, I would be happy to hear it.

> Also what does the output of ellrankinint() contain?   (If you say
> that it is very technical and I should not worry about it, I will feel
> like a Mathematica user!)   In particular, does it contain the
> quartics which represent elements of (or generators of) the 2-Selmer
> group?

No, it only contains the bnfinit of the 2-division polynomial.
This way you can also use it for twists.

Cheers,
Bill.