2.6. Elliptic curves over finite fields 39
proposition is the discriminant of the polynomial f(x). The discrim-
inant of E/Q, ΔE as in Definition 2.6.3, is a multiple of D; in fact,
ΔE = 16D. This fact together with Proposition 2.6.8 yield the fol-
lowing corollary:
Corollary 2.6.9. Let E/Q be an elliptic curve with coefficients in
Z. Let p 2 be a prime. If E has bad reduction at p, then p | ΔE.
In fact, if E is given by a minimal model, then p | ΔE if and only if
E has bad reduction at p.
Example 2.6.10. The discriminant of the elliptic curve E1 : y2 =
x3 + 35x + 5 of Example 2.6.7 is ΔE1 = −2754800 = −24 · 52 · 71 · 97
(and, in fact, this is the minimal discriminant of E1). Thus, E1 has
good reduction at 7 but it has bad reduction at 2, 5, 71 and 97. The
reduction at 71 and 97 is multiplicative.
Let E be an elliptic curve defined over a finite field Fq with q
elements, where q = pr and p 2 is prime. Notice that E(Fq)
P2(Fq), and the projective plane over Fq only has a finite number
of points (how many?). Thus, the number Nq := |E(Fq)|, i.e., the
number of points on E over Fq, is finite. The following theorem
provides a bound for Nq. This result was conjectured by Emil Artin
(in his thesis) and was proved by Helmut Hasse in the 1930’s:
Theorem 2.6.11 (Hasse; [Sil86], Ch. V, Theorem 1.1). Let E be
an elliptic curve defined over Fq. Then
q + 1 2

q Nq q + 1 + 2

q,
where Nq = |E(Fq)|.
Remark 2.6.12. Heuristically, we expect that Nq is approximately
q+1, in agreement with Hasse’s bound. Indeed, let E/Q be an elliptic
curve given by y2 = x3 + Ax + B, with A, B Z, and let q = p be a
prime for simplicity. There are p choices of x in Fp. For each value
x0, the polynomial f(x) =
x3
+Ax+B gives a value f(x0) Fp. The
probability that a random element in Fp is a perfect square in Fp is
1/2 (notice, however, that f(x0) is not random; this is just a heuristic
argument). If f(x0) is a square modulo p, i.e., if there is a y0 Fp
such that f(x0) y0
2
mod p, then there are two points (x0, ±y0) in
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