If *A* is any finite dimensional
-algebra, then
*a* *a*^{q}
gives a ring homomorphism from *A* to itself. Moreover if *A* is
local, then the only elements of *A* that are fixed under this
homomorphism are elements of
. From this one can show that the
intersection of the diagonal with the graph
of
the Frobenius in *X*×*X* is precisely
*X*(); the points of *X*
over
.

Now for any regular scheme *X* over
, the Frobenius *F* is a
flat morphism and thus gives an endomorphism of *K*_{0}(*X*). The latter
group thus acquires some ``structure'' in addition to being an abelian
group. In the case when *X* is a curve (or more specifically a
hyperelliptic curve) this has additional consequences. As we remarked
above *K*_{0}(*X*) is decomposed as the free group on
_{a} plus the
free group on [] (which can be any
point of *X*) and
the group
Pic^{0}(*X*). Moreover, there is a group scheme *J* so that
Pic^{0}(*X*) = *J*(). Thus, one way to determine the order of the group
Pic^{0}(*X*) is to determine the fixed points for the action of
Frobenius on this group scheme. Now, let be a prime that is
invertible the field
. On can show that the points of order
in *J*(*E*) for a large enough field extension *E* of
form
a vector space of rank 2*g* over
/ (here *g* is the
genus of the curve *X*). Moreover, there is a polynomial *P*(*T*) of
degree 2*g* with *integer coefficients* that is satisfied by the
automorphism of this vector space that is given by the Frobenius
endomorphism; the important point is that this polynomial is * independent* of . Another important fact is that this
polynomial has roots that are complex numbers of absolute value
*q*^{1/2}. Finally, given *P*(*T*) one can determine the number of
elements in *J*(*E*) for any finite extension of
. These results
were proved by Weil and were generalised to other varieties in the
form of the ``Weil conjectures'' which were proved by Grothendieck,
Deligne and others.

This approach was used by Schoof to calculate the order of
Pic^{0}(*T*)
in the case *T* is an elliptic curve (or a hyperelliptic curve of
genus 1). In this case *P*(*T*) is a quadratic polynomial of the form
*T*^{2} + *aT* + *q*; moreover, *J* = *T* in this case. One can write polynomials
*f*_{}(*x*) that are satisfied by the *x* co-ordinates of points of
order *l*. Thus we can use the action of the Frobenius on this
polynomial to determine *a* modulo for a number of primes
. The additional inequality
| *a*| *q*^{1/2}, can then we used
to determine *a*. One could attempt to generalise this to other
hyperelliptic curves. One must write down the equations that define
the -torsion in the Jacobian *J*. From the action of the
Frobenius on this we can write down the coefficients of *P*(*T*) modulo
. The inequalities resulting from the knowledge of the absolute
value of the complex roots can then be used to bound the number of
for which this needs to be done in order to determine the
coefficients uniquely.