Divisors - Elliptic Curves: Number Theory and Cryptography - page 350

5. e n ( σS, σT )= σ ( e n ( S, T )) for allautom orphism s σ of K su ch that σ is
the identity m ap on the coe cientsof E (if E isinWe erstra ss form ,
thismeansthat σ ( A )= A and σ ( B )= B ).

6. e n ( α ( S ) ,α ( T )) = e n ( S, T ) deg( α ) for all separable endom orphism s α of
E .If he coe cientsof E liein a finitefie d F q ,then the statem ent
also holds w hen α isthe Frobenius endom orphism φ q .(Ac ually, the
statem ent holds for allendom orphism s α , separableornot.See[38].)

PROOF (1) Since e n is independent of the choice of P , we use (11.6) with
P and with P + S 1 to obtain

e n ( S 1 ,T ) e n ( S 2 ,T )= g ( P + S 1 )

g ( P + S 1 + S 2 )
g ( P + S 1 )
= g ( P + S 1 + S 2 )
g ( P )
= e n ( S 1 + S 2 ,T ) .

g ( P )

This proves linearity in the first variable.
Suppose T 1 ,T 2 ,T 3 ∈ 3, let f i ,g i be

the functions used above to define e n ( S, T i ). By Theorem 11.2, there exists a
function h such that

E [ n ]with T 1 + T 2 = T 3 .For1 ≤ i ≤

div( h )=[ T 3 ] − [ T 1 ] − [ T 2 ]+[ ∞ ] .

Equation (11.5) yields
div f 3
f 1 f 2

= n div( h )=div( h n ) .

K × such that Therefore, there exists a constant c ∈

f 3 = cf 1 f 2 h n .

This implies that

g 3 = c 1 /n ( g 1 )( g 2 )( h ◦ n ) .

The definition of e n yields

e n ( S, T 1 + T 2 )= g 3 ( P + S ) = g 1 ( P + S )

g 1 ( P )

g 2 ( P + S )
g 2 ( P )

h ( n ( P + S ))
h ( nP )

g 3 ( P )

= e n ( S, T 1 ) e n ( S, T 2 ) ,

since nS = ∞ ,so h ( n ( P + S )) = h ( nP ). This proves linearity in the second
variable.