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240 12. Endomorphisms of Elliptic Curves

The two possibilities for the group of p-division points will be considered further
in the next chapter. A curve is called supersingular provided its group of p-division
¯
points over k reduces to 0.
The proofs of Theorem (3.5) and (3.6) can be found in the book by Mumford,
Abelian Varieties. They are true for higher-dimensional complete group varieties, but
messy proofs for elliptic curves can be carried out using the Weierstrass equation.
The prevailing wisdom in the mathematics community is that the reader who has
gotten this far in the theory of elliptic curves should start with the theory of abelian
varieties. An other possibility is the see the books of Silverman. In this line our
discussion in this chapter from §3onisonlyasketchofresults designedtogivethe
reader an overview of the results.
The pairing e N : N E × N E → Z/NZ of (2.4) has an algebraic meaning and

an algebraic definition can be given. In fact, for any isogeny λ : E → E there is a
pairing e λ :ker(λ) × ker( ˆ λ) → µ N (k), where N = #ker(λ). This uses divisors as in

(3.4). Recall that associated to λ is an inclusion k(E ) → k(E), and for x ∈ ker( ˆ λ)

we have λ −1 (x ) = 0, or as divisors λ −1 ((x )−(0)) = ( f ) for some f ∈ k(E),and
N

thus ( f ) = ( f λ).Now f N is invariant by translates by x ∈ E, and for a general
point y of E, we can define
f (y + x)

e λ (x, x ) = ∈ µ N (k).
f (y)
(3.7) Remark. In the special case λ = N, there is another formula for the pairing.
We need the notation h( n P P) = h(P) n P for a divisor n P P which
P P P

has no common prime factor with the divisor (h) of the function h.For x, x ∈ N E

choose divisors D and D differing from (x)−(0) and (x )−(0) up to the divisor of a
function and having no prime factors in common. Since ND = ( f ) and ND = ( f ),

it can be shown that e N (x, x ) = f (D)/f (D ).
This definition works because of the:
Reciprocity Law. When ( f ) and (g) have no common prime factors, then
f ((g)) = g(( f )).
1
This reciprocity law holds for functions on P by a direct calculation for f (x) =
(x − a)/(x − b) and g(x) = (x − c)/(x − d) and the multiplicative character of the
formula. Then it holds on any curve by mapping onto the projective line by a finite
ramified covering.
Now we summarize the basic properties of this pairing.

(3.8) Theorem. Let E and E be elliptic curves over a perfect field k and λ : E →

E an isogeny. The pairing e λ :ker(λ) × ker( ˆ λ) → µ N (k) is biadditive and nonde-
generate, and for an automorphism σ of k we have
σ
σ
σ
e λ (x , x ) = e λ (x, x ) .
σ