Computing the zeta function of curves over finite fields.

Question

Given a nonsingular curve $X/\mathbb{F}_q$, how can one efficiently compute its zeta function $Z(X/\mathbb{F},T)$?

My current strategy is to determine the genus $g$ of $X$, then count the points $N_1,...N_g$, then use that these determine $Z(X/\mathbb{F}_q,T)$.

This is doable by hand for small primes/genus, but quickly gets out of hand, and I am wondering if there are more intelligent ways to get $Z(X/\mathbb{F}_q,T)$.

For instance, how could one efficiently find $Z(X/\mathbb{F}_q,T)$ by hand where $X$ is the nonsingular curve associated to $y^2+y+x^5+x^3+1$ over $\mathbb{F_2}$?

Presumably computer algebra packages can solve this problem, but I'm not really familiar with any, so any reference would be much appreciated.

Answer 1

On magma http://magma.maths.usyd.edu.au/calc/

F<b> := FiniteField(5^2);
 A<x,y> := AffineSpace(F,2);
 f := y^2+2*x^3+x-b-1;
 C := ProjectiveClosure(Curve(A,f));
 IsSingular(C); // SingularPoints(C);
 g:= Genus(C);
 for n in [1..2*g] do
   Cn := BaseChange(C,FiniteField((5^2)^n));
   cn := #Points(Cn);
   {n,cn};
 end for;

Then (for non-singular projective curve) you want to identify the $\alpha_j$ such that $$c_n = q^{n}+1-\sum_{j=1}^{2g} \alpha_j^n$$

Let $$\zeta(T) = \frac{\prod_{j=1}^{2g}(1-\alpha_j T)}{(1-T)(1-q T)}= \exp(\sum_{n=1}^\infty \frac{c_n}{n} T^n), \qquad F(T) = \sum_{n=1}^{2g} \frac{c_n}{n} T^n $$

Then $$\zeta(T) = \exp(F(T)+O(T^{2g+1})) =\exp( F(T))(1+O(T^{2g+1}))$$

$$\prod_{j=1}^{2g}(1-\alpha_j T) = (1-T)(1-qT)\exp( F(T))(1+O(T^{2g+1}))$$ Let $$(1-T)(1-qT) \exp(F(T)) = \sum_{l=0}^{2g}b_l T^l +O(T^{2g+1})$$ then $$ \prod_{j=1}^{2g}(1-\alpha_j T) = \sum_{l=0}^{2g}b_l T^l$$