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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