Page 259 -

236 12. Endomorphisms of Elliptic Curves

Observe that e : L × L × Z defines by reduction modulo m a symplectic pairing
e m : L/mL×L/mL → Z/m, and for m dividing n, we have a commutative diagram
e n
L/nL × L/nL −−−−→ Z/n
 
 

e m
L/mL × L/mL −−−−→ Z/m.
(2.2) Remark. The symplectic pairings e and e m are nondegenerate. This means
for example that for any linear map u : L → Z there exists a unique y in L with
u(x) = e(x, y) for all x in L.

Using the definitions going into the formula for the dual isogeny in (1.2), we
can obtain a useful formula for e(x, y). First, let sgn(x, y) equal +1, 0, −1 when
Im(x/y) is > 0, = 0, and < 0, respectively. Then it is easy to check that
a(Zx + Zy)
e(x, y) = sgn(x, y) · .
a(L)
(2.3) Remark. With this formula for e, we show that for any isogeny λ : E =

C/L → E = C/L with λL ⊂ L and dual isogeny ˆ λ : E → E that the relation

e L (λx, x ) = e L (x, ˆ λx ) holds for x in L and x in L . For we have the following

inclusions between lattices
λL ⊂ L
∪ ∪
Zλx + Znx ⊂ Zλx + Zx .

Now calculate

a(Zx + Z(n/λ)x )

e L (x, ˆ λx ) = sgn(x, ˆ λx ) ·

a(L)

a(Zλx + Znx )

= sgn(λx, x ) ·
a(λL)
= sgn(λx, x )[λL : Zλx + Znx ]

= sgn(λx, x )[L : Zλx + Zx ]

= e L (λx, x ).

This verifies the formula.
The above discussion takes place on latices. Now we reinterpret the pairing on
the division points N E(C) = (I/N)L/L ⊂ E(C) contained in the complex points
of the curve. This will lead later to the algebraic definition of the symplectic pairing
which is due to A. Weil. The results over the complex numbers are summarized in
the following proposition.