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§7. Summary of Criteria for a Curve To Be Supersingular 269

Table 2. Elliptic Curves in char p different from 0 (results mostly of Deuring and
Hasse).
E can be defined over a finite field if and only if E has complex multiplication, i.e.,
0
dim End (E)> 1.
Q
Elliptic curves E defined over a finite field divide into two classes:
†
ordinary (Hasse invariant H = 1; Supersingular (Hasse invariant H = 0;
height = 1 height = 2

Characterization in terms of p-division points
1. p-Rank (E) = 1, i.e., the p division 1. p-Rank (E) = 0, i.e., the p-division
points are Z/pZ. points are 0.
2. Height of the formal group at 0 is 1. 2. Height of the formal group at 0 is 2.
3. Frob: E → E p 2 does not factor through 3. Frob: E → E p 2 factors through
p : E → E. p : E → E.
Corollary: E can be defined over F 2.
p
Characterization in terms of endomorphism rings
m
n
m
n
4. Frob = p for all n, m. 4. Frob = p for some n, m.
E E
0 1 0 0 2
5. dim Q End (E) = 2 (= 2).End (E) 5. dim Q End (E) = 2 (= 4).
0
is an imaginary quadratic extension and End (E) = h(p) is a quaternion algebra
End(E) is a maximal order of index with inv = 0 for = p, ∞ and
prime to p. inv = 1/2 otherwise. End(E)isa
maximal order in the algebra.
Characterization in terms of f a cubic equation for E
6. Coefficient of (wxy) p−1 in 6. Coefficient of (wxy) p−1 in
f (w, x, y) p−1 is = 0 f (w, x, y) p−1 is = 0
1
1
7. Frob on H (O E ) is an isomorphism. 7. Frob on H (O E ) is an zero.
Characterization in terms of the differential form ω = dx/y
8. ω = d log ψ and is not exact. 8. ω = dϕ and is exact.
9. ω = 0 is of the first kind with a p−1 = 0. 9. ω = 0 is of the first kind with a p−1 = 0.

i
a
ω = 0≤i i t dt
Characterization in terms of number n p of points for q = p ≤ 3, i.e., only curves over F p
10. N p = 1 + p, 10. N p = 1 + p.
†
Supersingular elliptic curves in characteristic p are defined over either the prime field F p or
its quadratic extension F 2.
p