9.5 Frobenius Endomorphism

Since all our varieties are defined over a finite field $\mathbb {F}$, there is a special endomorphism to consider. Let q be the number of elements of the field, then for any element of the field a = a^q. Thus for any polynomial f (t₁,..., t_r) with coefficients in $\mathbb {F}$ we have

f (t₁,..., t_r)^q = f (t₁^q,..., t_r^q)

Now consider the endomorphism of $\mathbb {P}$^k given by (X₀ : ^... : X_k) $\mapsto$ (X₀^q : ^... : X_k^q). If X = V(F₁,..., F_p;G₁,..., G_q) is a subscheme of $\mathbb {P}$^k and the polynomials F_i and G_j have coefficients in $\mathbb {F}$, then this endomorphisms sends X to itself. This endomorphism of X is called the Frobenius Endomorphism F : X$\to$X.

If A is any finite dimensional $\mathbb {F}$-algebra, then a $\mapsto$ 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 $\mathbb {F}$. From this one can show that the intersection of the diagonal $\Delta_{X}^{}$ with the graph $\Gamma_{F}^{}$ of the Frobenius in X×X is precisely X($\mathbb {F}$); the points of X over $\mathbb {F}$.

Now for any regular scheme X over $\mathbb {F}$, the Frobenius F is a flat morphism and thus gives an endomorphism of K₀(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<sub>0</sub>(X) is decomposed as the free group on $\mathbb {G}$<sub>a</sub> plus the free group on [$\infty$] (which can be any $\mathbb {F}$ point of X) and the group Pic<sup>0</sup>(X). Moreover, there is a group scheme J so that Pic<sup>0</sup>(X) = J($\mathbb {F}$). Thus, one way to determine the order of the group Pic<sup>0</sup>(X) is to determine the fixed points for the action of Frobenius on this group scheme. Now, let $\ell$ be a prime that is invertible the field $\mathbb {F}$. On can show that the points of order $\ell$ in J(E) for a large enough field extension E of $\mathbb {F}$ form a vector space of rank 2g over $\mathbb {Z}$/$\ell$$\mathbb {Z}$ (here g is the genus of the curve X). Moreover, there is a polynomial P(T) of degree 2g 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 $\ell$. Another important fact is that this polynomial has roots that are complex numbers of absolute value q<sup>1/2</sup>. Finally, given P(T) one can determine the number of elements in J(E) for any finite extension of $\mathbb {F}$. 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 Pic⁰(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 T² + aT + q; moreover, J = T in this case. One can write polynomials f_$\ell$(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 $\ell$ for a number of primes $\ell$. The additional inequality | a| $\leq$ 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 $\ell$-torsion in the Jacobian J. From the action of the Frobenius on this we can write down the coefficients of P(T) modulo $\ell$. The inequalities resulting from the knowledge of the absolute value of the complex roots can then be used to bound the number of $\ell$ for which this needs to be done in order to determine the coefficients uniquely.