GEOMETRIC ASPECTS OF THE ADDITION ALGORITHM ON THE PICARD GROUP OF A C ab CURVE
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0 −−−→ H −−−→ Pic(C) ∼ = Pic 0 (C ∗ ) −−−→ π∗
g a = dim H 1 (C ∗ , O C∗
n Pi
u i = Y i
In particular, we take polynomials c(x), d(x) ∈ k[X] such that c(X)f 1 (X) + d(X)f 2 (X) = 1. Furthermore, let β ∈ k be an element such that V ((f 1 (X), Y 2
u(X, Y ) = c(X)f 1 (X)u 2 (X) + d(X)f 2 (X)(Y 2
(X − α σi
For each point P ∈ V (X − α) ∩ V ( a ), if P is not a ramification point over Speck[X], there exists a polynomial f P (X) such that I(V (Y − f P (X)) ∩ C alg ; P) = m P ( a ), and in this case we put u P (X, Y ) = Y − f P (X). If P is a ramification point, then we put u P (X, Y ) := (Y − Y (P)) mP
f i (X) := f (X)/(X − α σi
f i (X) e u σ αi
a σ 0i
(X, Y ) := 0 (X)Y a− 1 + 1 (X)Y a− 2 + . . . + a (X) (deg i (X) ≤ deg f (X) − 1) m(X, Y ) := m 0 (X)Y a− 1 + m 1 (X)Y a2
f 1 , (Y 1
5. M 0 u := b(X)f 2 (X)u 1 + a(X)f 1 (X)(Y 1
f 2 , (Y 2
7. M 0 u := b(X)f 2 (X)(Y 2
d(X) := (f 1 , f 2 ) f 12 := (f 1 , d 1
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