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§3. Definition of Supersingular Elliptic Curves 259

2g
m m
N m = q + 1 − α .
i
i=1
Then zeta function is given by
P(q −s )
ζ C (s) = ,
(1 − q −s )(1 − q 1−s )
where P(T ) = (1 − α 1 T )...(1 − α 2g T ). Further ζ C satisfies a functional equation
g(2s−1)
q · ζ C (s) = ζ C (1 − s)
which is equivalent to the assertion that the α j pair off such that after reordering
¯ α 2 j−1 = α 2 j and so α 2 j−1 α 2 j = q.
The above theorem is due to Weil [1948]. For other discussions see Monsky
[1970] and Bombieri [1973]. The article by Katz [1976] puts this result in the per-
spective of general smooth projective varieties and outlines how the above form of
the zeta function for curves has a natural general extension first conjectured A. Weil
and finally completely proven by P. Deligne.
Observe that the zeta function for a smooth projective curve is a rational function
of q −s . Its poles are at s = 0and s = 1, and its zeros are on the line Re(s) = 1/2.
√
This last assertion is equivalent to |α j |= q and is called the Riemann hypothesis
for algebraic curves over a finite field. It is the most difficult statement in the above
theorem to demonstrate.
For elliptic curves there were three ways to obtain the quadratic polynomial f E
which is the nontrivial factor in the zeta function. First, it is a quadratic equation for
−1 0
π with constant term 1 satisfied in End (E) over Q. Second, it is the characteristic
E
polynomial of the inverse Frobenius of E acting on any Tate module V (E), where
is unequal to the characteristic p. Third, we can calculate N 1 directly to obtain
Tr(π E ). In the case of general smooth curves C the nontrivial factor P(T ) of (2.5)
in the zeta function can be obtained from the theory of correspondences on C which
are certain divisors on the surface C × C. In this way Grothendieck reproved the
basic results for ζ C (s) obtained earlier by A. Weil. Originally Weil looked at the Ja-
cobian J(C), an abelian variety of dimension g, and the corresponding Tate module
V (J(C)). The charcteristic polynomial of π −1 on V (J(C)) with constant term 1 is
C
the factor P(T ) of (2.5). In a more general context this Tate module is the first -adic
cohomology group and for higher-dimensional smooth varieties all -adic cohomol-
ogy groups together with the action of Frobenius must be bought into the analysis of
the zeta function of the variety.

§3. Definition of Supersingular Elliptic Curves

In line with our general program of developing as much of the theory of elliptic
curves as possible via the theory of cubic equations, we use the following definition
of supersingular for elliptic curves.