3
$\begingroup$

Let $k$ be a finite field, $X/k$ a smooth curve, $f$ a polynomial of 2 variables which gives an affine model of $X$ and $J$ its Jacobian.

Then how can I compute $J(k)$?

If $X$ is a hyperelliptic curve, then there is an algorithm computing it. (Although I don’t know its theory...)

But what about for non-hyperelliptic curves?

In 12.3.1 of Poonen, Schaefer, Stoll’s “Twists of X(7) and primitive solutions to x^2+y^3=z^7”, the authors give $J(k)$ (and the orders of some particular points) for some non hyperelliptic curves with no arguments. So I think that we can compute them even for non hyperelliptic curves.

It seems that we can compute its order using the Zeta function (since we can compute the explcit action of the Frobenius on the curve). But I can’t find how to compute the groups structures.

And on this page, he says “This has been pursued recently by Andrew Sutherland”. But I can’t find any papers relating it.

And can we compute their torsion part if $k$ is a number field, in particular the rationals?

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ This can be broken down into two questions: ①how to compute $J$, as an algebraic (abelian) variety (i.e., write down equations for it and for its addition law) in function of $X$, and ②given $J$, how to compute its points over a finite field or its torsion points over a number field. It's not entirely clear what interests you. This question asks about part ①, and provides some references. $\endgroup$
    – Gro-Tsen
    Commented May 28, 2021 at 8:27
  • $\begingroup$ Do you want any algorithm or an efficient algorithm? If the former, the only problem is how to perform elementary operations in $J(k)$ (since this is a finite group); this can be done for instance using Riemann-Roch spaces, say using Makdisi's algorithms arxiv.org/abs/math.NT/0105182 $\endgroup$
    – Aurel
    Commented May 28, 2021 at 10:23
  • $\begingroup$ If you want a polynomial time algorithm, this is open: we don't even know how to compute the cardinality of this group in polynomial time (if the genus and the characteristic both vary). $\endgroup$
    – Aurel
    Commented May 28, 2021 at 10:26

0

Reset to default

You must log in to answer this question.