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256 13. Elliptic Curves over Finite Fields

(1.8) Remark. The definition of f E (T ) extends to singular cubic curves C with
singularity c 0 .

1 − T if c 0 is a node with rational tangents,

f C (T ) = 1 + T if c 0 is a node with tangents quadratic over k,

1 if c 0 is a cusp.

Then for C either singular or nonsingular cubic
#C ns (F q ) = qf C (q −1 ).

We end this section by counting the points on a pair of elliptic curves E and E t
over F q . Let g(x) ∈ k[x] be a cubic polynomial, and let t be a nonzero scaler in k.
3
Then denote by g t (x) = t g(x/t) ∈ k[x]. We thank O. Foster for pointing this out
to us.
2
(1.9) Definition. let E be an elliptic curve defined by y = g(x) where g(x) is a
cubic polynomial over a field k. The twist by a nonzero t ∈ k is the elliptic curve E t
2
2
defined by the equation y = g t (x) or t −1 (y/t) = g(x/t). It is isomorphic to curve
y = g(x).
with equation t −1 2
t
In characteristic > 3 by (3.8) the elliptic curves E and E are isomorphic if and
t

only if t is a nonzero square, and further E and E t are isomorphic if and only if

t /t is a square.
(1.10) Proposition. Let E be an elliptic curve over a finite field F q of characteristic
t
p > 3. Then up to isomorphism there is exactly one twist E where t is any nonsquare
in F q . Moreover
t
#E(F q ) + #E (F q ) = 2q + 2.
Proof. The first assertion follows from 3(8.3) and the fact that for a finite field k
∗ 2
∗
the quotient k /(k ) has two elements. Next we count the points (x, y)onthe two
curves. Firstly, note that g(x) = 0 if and only if g t (x) = 0 giving one point on each
curve for such a value of x. Secondly, note that the element g(x) is a square (resp. a
nonsquare) in k if and only if g t (x) is a nonsquare (resp. square) in k = F q , that is
t
giving two points in either E or E for each such a value of x.Whenweadd thetwo
points at infinity for zero of the curves, we have
t
#E(F q ) + #E (F q ) = 2(#of x ∈ P(F q )) = 2(q + 1).
This proves the proposition.

§2. Generalities on Zeta Functions of Curves over a Finite Field

Let C be an algebraic curve over a field k 1 = F q . We wish to study the number of
points on C over k n = F q for every n. For every point P on C(k 1 ) we have P in
¯
n