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242 12. Endomorphisms of Elliptic Curves

(4.4) Theorem (Hasse). For λ in End(E) the characteristic polynomial has values
c λ (r) ≥ 0 for any rational number and

λ + ˆ λ ≤ 2 deg(λ).

Proof. For r = n/m we see that
2
2
2
m c λ (n/m) = n + nmT (λ) + m deg(λ) = deg(n + mλ) ≥ 0.
2
Hence, the discriminant 4 deg(λ) − T (λ) is positive. Now take the square root of
2
2
T (λ) = (λ + ˆ λ) ≤ 4deg(λ) to obtain the result.
(4.5) Definition. An elliptic curve E over k has complex multiplication provided
Z = End(E).
(4.6) Theorem. Assume that End(E) is commutative, then either End(E) = Z or
End(E) ⊗ Z Q is an imaginary quadratic extension of Q. In the second case End(E)
0
is an order in the imaginary quadratic field End (E) = End(E) ⊗ Z Q.
0
Proof. Every element in End (E) satisfies a quadratic equation over the subfield Q
0 0
by (4.3), and this means that either End (E) has degree 1 over Q, that is End (E) =
0
0
Q,orEnd (E) has degree 2 over Q, that is, End (E) = Q(α), where α is a quadratic
irrationality. From the positivity of the quadratic equation for α, see (4.4), the field
Q(α) is an imaginary quadratic number field. By (4.1) and (4.3) the endomorphism
ring embeds as a subring of the ring of integers in End(E). Moreover, it is of finite
0
index in the ring of integers and it contains Z, i.e., End(E) is an order in End (E).
This proves the theorem.
(4.7) Proposition. Let E be an elliptic curve over the complex numbers with com-
plex points C/L τ . Then End(E) is commutative. The curve has complex multiplica-
tion if and only if τ is an imaginary quadratic number. In the case of complex multi-
plication End(C/L τ )⊗Q is Q(τ) the imaginary quadratic number field obtained by
adjoining τ to Q, and as a subring of complex numbers End(E) is contained in L τ .

Proof. Since End(E) is naturally isomorphic to the subring of complex numbers λ
satisfying λL τ ⊂ L τ it is commutative. For such a λ with λL τ ⊂ L τ = Zτ + Z we
see that λ = a + bτ is contained in L τ and the relation c + dτ = λτ = aτ + bτ 2
holds for integers a, b, c,and d. This shows that τ satisfies a quadratic equation over
2
Q. Conversely aτ = bτ +c with a, b,and c integers implies that (aτ)L τ ⊂ L τ ,and
thus C/L τ has complex multiplication. Finally, End(C/L τ )⊗Q = Q(τ) holds from
the quadratic relations for τ over the rational numbers. This proves the proposition.
(4.8) Remark. For an elliptic curve over the complex numbers we can see from the
3
above proposition that Aut(E) is {+1, −1} if and only if j = 0or12 .The case
j = 0is E isomorphic to E(ρ) and Aut(E) ={±1, ±ρ, ±¯ρ}, and the case j = 12 3
is E isomorphic to E(i) and Aut(E) ={±1, ±i}.