Cryptography Reference
In-Depth Information
There is a natural map φ on divisors that is called the pushforward (it is called the
divisor-norm map in Section VII.7 of Lorenzini [ 355 ]).
Definition 8.3.4 Let φ : C 1
C 2 be a non-constant morphism of curves. Define the
pushforward
φ :Div
k
( C 1 )
Div
( C 2 )
k
by φ ( P )
=
φ ( P ) and extend to the whole of Div
( C 1 ) by linearity.
k
( C 2 ) that corresponds (in the sense of property
4 of Theorem 8.3.8 ) to the pushforward. This is achieved using the norm map with respect
to the extension
It remains to find a map from
k
( C 1 )to
k
( C 1 ) (
k
k
( C 2 )). As we will show in Lemma 8.3.13 , this norm satisfies,
= φ ( P ) = Q f ( P ) e φ ( P ) .
for f
∈ k
( C 1 ) and Q
C 2 (
k
), N k ( C 1 ) ( k ( C 2 )) ( f )( Q )
Definition 8.3.5 Let C 1 ,C 2 be curves over
C 2 be a non-constant rational
map. Let N k ( C 1 ) k ( C 2 ) be the usual norm map in field theory (see Section A.6 ). Define
k
and let φ : C 1
k
→ k
φ :
( C 1 )
( C 2 )
( φ ) 1 (N k ( C 1 ) ( k ( C 2 )) ( f )).
by φ ( f )
=
φ (
Note that the definition of φ ( f ) makes sense since N k ( C 1 ) k ( C 2 ) ( f )
k
( C 2 )) and
so is of the form h
φ for some h
∈ k
( C 2 ). So φ ( f )
=
h .
1
x 2 . Then
Example 8.3.6
Let C 1 =
C 2 = A
and φ : C 1
C 2 be given by φ ( x )
=
φ (
( x 2 ) and
φ (
x 2 / ( x
k
( C 2 ))
= k
k
( C 1 )
=
k
( C 2 ))( x ). Let f ( x )
=
1). Then
x 2
x ) 2
x 4
(
N k ( C 1 ) k ( C 2 ) ( f )
=
f ( x ) f (
x )
=
1) =
1 ,
( x
1)
(
x
x 2
+
X 2 / (1
which is h
φ for h ( X )
=
X ). Hence, φ ( f ( x ))
=−
f ( x ).
V ( y 2
x 2
2 , C 2 = A
1
Exercise 8.3.7 Let C 1 =
=
+
1)
⊆ A
and let φ : C 1
C 2 be given
by φ ( x,y )
=
x .Let f ( x,y )
=
x/y . Show that
x 2
N k ( C 1 ) k ( C 2 ) ( f )
=
x 2
+
1
X 2 / ( X 2
and so φ ( f )
=
h ( X ) where h ( X )
=−
+
1).
We now state the main properties of the pullback and pushforward.
Theorem 8.3.8 Let φ : C 1
C 2 be a non-constant morphism of curves over
k
. Then:
1. deg( φ ( D ))
=
deg( φ )deg( D ) for all D
Div( C 2 ) .
2. φ (div( f ))
div( φ f ) for all f
( C 2 ) .
=
∈ k
3. deg( φ ( D ))
=
deg( D ) for all D
Div( C 1 ) .
( C 1 ) .
4. φ (div( f ))
=
div( φ ( f )) for all f
∈ k
5. φ ( φ ( D ))
=
deg( φ ) D for D
Div( C 2 ) .
 
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