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270 13. Elliptic Curves over Finite Fields

This induces another commutative diagram in which the vertical morphisms are
∗
cohomolgy boundary morphisms and F is the Frobenius on E inducing F as the
top vertical composite
F ∗
1
1
1
1
H (E, O E ) −−−−→ H (E (p) , O E (p)) −−−−→ H (E, O E )
  
  
∼ ∼ ∼
F 1 ∗ f p−1
2
2
2
H (P, O P (−3)) −−−−→ H (P, O P (−3p)) −−−−→ H (P, O P (−3)).
The image of
∗
F ((wxy) −1 ) = (wxy) −p
1
2
hasasimage f p−1 (wxy) −p ,and H (O P (−3)) hasasbasis (wxy) −1 times the co-
efficient of (wxy) p−1 in f p−1 . This is the formulation of the Hasse invariant used in
(3.1).
§8. Tate’s Description of Homomorphisms
In 12(6.1) and 12(6.3), we saw that for each prime different from the characteristic
of the ground field k, the induced homomorphisms

T :Hom k (E, E ) ⊗ Z → Hom Gal(k s /k) (T (E), T (E )),

V :Hom k (E, E ) ⊗ Q → Hom Gal(k s /k) (V (E), V (E )),
are injective. In the case of a finite field k which we have been considering in this
chapter, the Galois group Gal(k s /k) is topologically generated by Frobenius π = Fr h
h
where #k = p = q, and the symbol Hom Gal(k s /k) can be written Hom (Fr) , namely
the module of homomorphisms commuting with the action of the Frobenius on the
modules T or V .
(8.1) Theorem (Tate). The homomorphisms T and V , defined above, are isomor-
phisms for different from p the ground field characteristic.
This theorem was proved by Tate [1996] for an abelian varies over a finite field.
The assertion that T is an isomorphism is equivalent to the assertion that V is an
isomorphism since the cokernel of T is torsion free by 12(6.3).
Recall that two elliptic curves E and E are isogenous provided Hom(E, E ) is

nonzero. Just from the injectivity of T and V , we have the following elementary
result using linear algebra.

(8.2) Proposition. If two elliptic curves E and E are isogenous, then their charac-
teristic polynomials f E = f E are equal. Moreover,

2 ifEisordinary,

rank Hom (Fr) (V (E), V (E )) =
4 if E is supersingular.
The zeros of f E are distinct in the ordinary case and equal in the supersingular case.