For the record, a version of my generic code (with a bit of help from
craigcitro) has been merged into 3.0.1alpha1, as ticket #3031. Anyone
now interested in tackling #793 should let me know.
Kiran
On Apr 15, 1:56 pm, Kiran Kedlaya <[EMAIL PROTECTED]> wrote:
> There is an old ticket #793 about implementing a zeta_function method
> for hyperelliptic curves. Such a method would have to have a default
> behavior in case none of the special-purpose methods we have already
> implemented are appropriate.
>
> So I thought I'd try writing a generic method for schemes over finite
> fields. Unfortunately, it's limited right now to schemes over prime
> fields because it requires at least a random coercion from F_q to
> F_{q^n}, which we don't yet support for q composite. (Maybe soon...)
>
> It's also a bit stupid because it counts rational points over F_q by
> actually constructing the list of them. Better would be to construct a
> scheme consisting of these points and then compute the length of this
> over the field. For example, if your scheme were the affine scheme
> V(I) for I an ideal in F_q[x_1, ..., x_n], you could form the ideal J
> = I + (x_1^q - x_1, ..., x_n^q-x_n) and then call
> J.vector_space_dimension().
>
> But in any case, see
> http://math.mit.edu/~kedlaya/Zeta_functions.sws
> for a notebook containing what I have so far. Comments and
> improvements welcome...
>
> Kiran
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