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248    12. Endomorphisms of Elliptic Curves

        we can decompose µ = b 1 λ 1 +· · ·+b m λ m .Now    m  a j λ j =      m  b j λ j , which
                                                 j=1           j=1
        implies that the prime   divides all a j which is a contradiction. This proves that T
        is injective.
                     (T   (E), T   (E )) is of finite rank, in fact, of rank at most 4. By the

           Now Hom Z
                                              0

        injectivity of T   on M ⊗ Z   , we see that Hom (E, E ) has dimension at most four
        and finally that Hom(E, E ) is finitely generated. This proves the theorem.

                                        Q   , and then the theorem has the following
           We can define V   (λ) = T   (λ) ⊗ Z
        immediate corollary.
        (6.2) Corollary. For a prime   unequal to the ground field characteristic the natural
        map

                     V   :Hom(E, E ) ⊗ Q   → Hom Q   (V   (E), V   (E ))
        is injective.

           For elliptic curves E and E defined over k the Galois group Gal(k s /k) acts

        on Hom(E, E ) on the right by the formula λ σ  = σ −1 λσ. The fixed subgroup
                   Gal(k s /k)

        Hom(E, E )       = Hom k (E, E ) is the subgroup of homomorphisms defined
        over k.For λ in Hom(E, E ) and x in V   (E) we have the relation

                                       σ    σ σ
                                    (λx) = λ x
                         σ
        for the right action x equal to σ  −1 x from the left action.
        (6.3) Remark. The monomorphism of (6.1) and (6.2) restrict to monomorphisms


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


                    Hom k (E, E ) ⊗ Q   → Hom Gal(k s /k) (V   (E), V   (E )).
        Hence homomorphisms and isogenies over k are distinguished by their action on
        Tate modules viewed as Galois modules. In the next chapter we will see that there
        are cases where these monomorphisms are isomorphisms.

        (6.4) Remark. The cokernel of the monomorphism T   is torsion free. This follows

        since every k-homomorphism f : E → E equal to zero on the  -division points is

        of the form  g for some k-homomorphism g : E → E .

        §7. Expansions Near the Origin and the Formal Group

        The formal group of an elliptic curve is used in the in the next two chapters, and
        appendix III. We give now a brief introduction. In the equation for an elliptic curve E
        in normal form we introduce new variables t =−x/y and s =−1/y. The equation
        in the affine (t, s)-plane becomes

                                                     2
                                                           3
                                              2
                                       2
                           3
                       s = t + a 1 ts + a 2 t s + a 3 s + a 4 ts + a 6 s ,
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