1. Abel’s Theorem
In this section, we determine the necessary and sufficient conditions for a divisor of degree zero to be principal.
Let X be a compact Riemann surface1 of genus g. The homology group H1(X, Z) is a free abelian group of rank 2g. A basis {a1, · · · , ag, b1, · · · , bg} is called a canonical basis if
#(ai, aj) = #(bi, bj) = 0, #(ai, bj) = δij, where #(a, b) denote the intersection number of cycle a, b on X.
Given a canonical homology basis {a1, · · · , ag, b1, · · · , bg} for H1(X, Z), we choose a basis {ω1, · · · , ωg} for the space of holomorphic differentials H0(X, KX) such that for all 1 ≤ j, k ≤ g,
Z
ak
ωj = δjk For each 1 ≤ j, k ≤ g, we denote
πjk = Z
bk
ωj. For each 1 ≤ k ≤ g, set
πk=
π1k
π2k ... πgk
∈ Cg.
The period matrix Ω is a g × 2g matrix defined by Ω = [I|π],
where the g × g complex matrix π = (πjk) is symmetric with Im π > 0. Let Λ be the lattice generated by the column vectors of the period matrix Ω. The quotient Cg/Λ is called the Jacobian variety and denoted by J (X). Choose a base point p0 ∈ X, we define a map
A : X → J (X) by
A(p) =
Z p p0
ω1, · · · , Z p
p0
ωg
. This map is called the Abel Jacobi map.
Proposition 1.1. A is a well-defined holomorphic mapping of X into J (X). It has maximal rank.
Proof. Let c1, c2 be two paths from p0 to p. Then γ = c1c−2 is a loop at p. Then γ is homologous to a cyclePg
k=1(mkak+ nkbk). Hence Z
γ
(ω1, · · · , ωg) =
g
X
k=1
mkek+ nkπk
≡ 0 mod Λ.
This shows that A(p) is independent of choice of paths; it is well-defined. Since {ω1, · · · , ωg} are holomorphic differentials, A is a holomorphic mapping. Let (z, U ) be a local chart
1All the Riemann surfaces are assumed to be connected.
1
2
around p. Represents ωiby fi(z)dz for some holomorphic function fi on U. The holomorphic tangent map of A at p denoted by Tp0(A) : Tp0X → TA(p)0 J (X) is represented by
Tp0(A) =
f1(p) f2(p)
... fg(p)
.
Since {ω1, · · · , ωg} is a basis for H0(X, KX), f1(p), · · · , fg(p) are not identically zero. There-
fore Tp0(A) has rank one.
Here the path from p0 to p in the path integrals are chosen to be the same. The Abel Jacobian map can be extended to a group homomorphism
ϕ : Div(X) → J (X) by
D = X
p∈X
npp 7→ X
p∈X
npA(p).
Theorem 1.1. (Abel Theorem) A divisor D ∈ Div(X) of degree 0 is principal if and only if
A(D) ≡ 0 mod Λ
Corollary 1.1. If X is of genus one, then the Abel Jacobi map A : X → J (X)
is an isomorphism.
Proof. Since A is a nonconstant mapping J (X) is connected, A is surjective. Now, let us prove that the map is injective. To do this, assume A(p) = A(q) for some p, q ∈ X.
Assume that p 6= q. The divisor D = p − q has degree zero and A(D) ≡ 0 mod Λ. By Abel’s theorem, D is principal; hence there is a meromorphic function f ∈ K(X) such that D = (f ). We obtain a meromorphic function f on X with a only simple pole at q. By Riemann-Roch theorem, (or the Weierstrass gap theorem), there is no such a meoromorphic function. Hence p = q. We find A is a univalent mapping from X onto J (X). Then A is biholomorphic, i.e. A is an isomorphism.
