Let X be a Riemann surface of genus g ≥ 0 and K(X) be the field of meromorphic functions on X

All the Riemann surfaces in this note are assumed to be connected and com- pact.

Let X be a Riemann surface of genus g ≥ 0 and K(X) be the field of meromorphic functions on X. The free abelian group generated by points of X is denoted by Div(X).

Elements of Div(X) are called divisors on X. They are of the form

(1.1) D = n1P1+ · · · + n_kP_k

for some n₁, · · · , n_k ∈ Z and for some points P1, · · · , P_k ∈ X and for some k ≥ 1. Or equivalently, D can be written as

D = X

P ∈X

n_PP,

where n : X → Z is a function so that nP = 0 except a finite number of P ∈ X.

Given a divisor D of the form (1.1), its degree denoted by deg D is defined to be

(1.2) deg D = X

P ∈X

n_P.

The map deg : Div(X) → Z sending a divisor D to its degree deg D is a group homomor- phism.

Let f be a nonconstant meromorphic function on X and P be a point of X. In a neigh- borhood of P, we write f = g/h with f, g holomorphic. We set

ord_P f = ord_Pg − ord_Ph.

We say that f has a zero of order ord_Pf if ord_Pf > 0 and that f has a pole of order ord_Pf if ordPf < 0. If P is neither a zero nor a pole, we set ordPf = 0. We define a divisor

(f ) = X

P ∈X

(ord_Pf )P.

If f = g/h with g, h holomorphic and relatively prime, the divisors of zero (f )0 is defined to be

(f )₀=X

P

(ord_Pg)P and the divisor of poles is defined to be

(f )∞= X

P ∈X

(ord_P h)P.

Clearly, these divisors are well-defined as long as we require that g, h are relatively prime, and

(f ) = (f )0− (f )_∞.

It is clear that the map (·) : K(X)^∗ → Div(X) sending f to its associated divisor (f ) is a group homomorphism. A divisor D is called principal if there exists f ∈ K(X)^∗ such that D = (f ).

Lemma 1.1. Let f ∈ K(X)^∗. Then deg(f ) = 0.

Proof. If f is a constant function, (f )0 = (f )∞= 0. Hence deg(f ) = 0 is obvious. Assume that f is not a constant function. Since X is compact, there are only finitely many zeros and poles of f on X. Write (f )0= n1P1+ · · · + nkPkand (f )∞= m1Q1+ · · · + mlQl where

1

P1, · · · , P_k, Q1, · · · , Q_l ∈ X and n_i, mj ≥ 1. Since a nonconstant meromorphic function f on X is a holomorphic mapping f : X → P¹, the degree of f is defined to be

deg f = X

P ∈f⁻¹(Q)

e_P.

Here e_P is the ramification index of f at P (or the multiplicity). When we choose Q = 0, we get deg f = n1+ · · · + nk. When we choose Q = ∞, we obtain deg f = m1+ · · · + ml. Then

deg(f ) = (n₁+ · · · + n_k) − (m₁+ · · · + m_l) = deg f − deg f = 0.

 A meromorphic one-form ω on X is also called an abelian differential. Given an abelian differential ω, we can also define its associated divisor

(ω) = X

P ∈X

(ord_Pω)P.

A divisor of this form is called a canonical divisor. In the next lemma, we are going to see that any two canonical divisors are linearly equivalent.

Lemma 1.2. Let ω₁ and ω₂ be two abelian differentials on X with ω₁ 6= 0. Then there is a unique geomorphic function f on X such that ω2 = f ω1. Hence (ω1) is linearly equivalent to (ω₂).

Proof. Let φ : U ⊂ X → V ⊂ C be a local chart on X. Write ωi◦ φ(z) = g_i(z)dz for some meromorphic function g_i on V. Set h = g₂/g₁. Then f = h ◦ φ is a meomorphic function on U. Then f can be glued to a global meromorphic function on X such that ω2 = f ω1. This implies that (ω2) = (f ) + (ω1). In other words, (ω2) − (ω1) = (f ) is a principal divisor.

Hence (ω₁) and (ω₂) are linearly equivalent. 

Notice that by deg(f ) = 0, we have

deg(ω2) = deg(f ) + deg(ω1) = deg(ω1).

This formula tells us that every canonical divisor has the same degree. Now, let us compute the degree of a canonical divisor on a compact Riemann surface. In order to do this, we need the following lemma.

Lemma 1.3. Let f : X → Y be holomorphic mapping between Riemann surfaces and ω be a meromorphic one-form on Y. For a point p ∈ X, we have

ord_p(f^∗ω) = (1 + ord_{f (p)}ω)e_p(f ) − 1.

Proof. Let (ψ, U ) and (ϕ, V ) be local charts around p and f (p) with f (U ) ⊂ V, and ψ(p) = 0, and ϕ(f (p)) = 0 such that the local representation F = ϕ ◦ f ◦ ψ⁻¹ of f is given by the local normal form z 7→ zⁿ where m = e_p(f ). Assume that ω is represented by ω_U(w) = (ckw^k+ ck+1w^k+1+ · · · )dw in the local chart (ϕ, V ) with ck6= 0 and k = ord_{f (p)}ω. Hence the local representation of f^∗ω in local chart (ψ, U ) is represented by

F^∗ω_V = F^∗(c_kw^k+ c_k+1w^k+1+ · · · )dF^∗w

= (ckz^nk+ ck+1z^n(k+1)+ · · · )dzⁿ

= (c_kz^nk+ c_k+1z^n(k+1)+ · · · )nzⁿ⁻¹dz

= (nc_kz^nk+n−1+ nc_k+1c^n(k+1)+n−1+ · · · )dz.

This implies that ordp(f^∗ω) = nk + n − 1 = (ord_{f (p)}ω + 1)ep(f ) − 1 which proves our assertion.

 Proposition 1.1. Let X be a compact Riemann surface of genus g and K_X be a canonical divisor. Then deg KX = 2g − 2.

Proof. Let f be a nonconstant meromorphic function on X. Then f : X → P¹ is a noncon- stant holomorphic mapping and thus a ramified covering of P¹. Denote by d the degree of f. The Riemann-Hurwitz formula implies

X

p∈X

b_p(f ) = 2g − 2 + 2d.

Let us define a one-form ω on P¹as follows. Write P¹ = C ∪ {∞}. Let z be the coordinate on C and w = 1/z be a coordinate around ∞. Define ω = dz. Then ω = −dw/w² as a pole of order two around ∞ and no other zeros or poles. Hence (ω) = −2∞. Denote η = f^∗ω.

Then η is a meromorphic one form on X. Now let us compute the degree of (η).

deg(η) = X

p∈X

(ordpη)

= X

p∈X

(ord_pf^∗ω)

= X

p∈X

((1 + ord_{f (p)}ω)e_p(f ) − 1)

= X

p∈X, f (p)6=∞

((1 + ord_{f (p)}ω)ep(f ) − 1) + X

p∈X, f (p)=∞

((1 + ord_{f (p)}ω)ep(f ) − 1) We know ord_qω = 0 if q 6= ∞ and ord_qω = −2 if q = ∞. Hence we have

deg(η) = X

p∈X, f (p)6=∞

(e_p(f ) − 1) + X

p∈X, f (p)=∞

(−e_p(f ) − 1)

= X

p∈X, f (p)6=∞

(e_p(f ) − 1) − X

p∈X, f (p)=∞

(e_p(f ) + 1)

= X

p∈X, f (p)6=∞

(ep(f ) − 1) + X

p∈X,f (p)=∞

(ep(f ) − 1) − X

p∈X,f (p)=∞

(ep(f ) − 1) − X

p∈X, f (p)=∞

(ep(f ) + 1)

=X

p∈X

(ep(f ) − 1) − 2 X

p∈f⁻¹(∞)

ep(f )

=X

p∈X

bp(f ) − 2d

= 2g − 2 + 2d − 2d

= 2g − 2.

Since any two canonical divisors have the same degree, we obtain that deg K_X = 2g − 2.  Let f : X → Y be a nonconstant holomorphic mapping between Riemann surfaces. Given a point q ∈ Y, we define the pull back divisor f^∗(q) by

f^∗(q) = X

p∈f⁻¹(q)

ep(f )p.

In general, given a divisor D =P

q∈Y nqq, the pull back divisor is defined to be f^∗(D) =X

q∈Y

n_qf^∗(q).

Lemma 1.4. Let f : X → Y be a nonconstant holomorphic map between Riemann surfaces.

(1) f^∗: Div(Y ) → Div(X) is a group homomorphism.

(2) The pull back of principal divisor is principal. In fact, if h is a meromorphic function on Y, then f^∗(h) = (h ◦ f ).

(3) If X and Y are compact, so that divisors have degrees, we have deg f^∗D = deg f · deg D.

Proof. It follows from the definition that f^∗is a group homomorphism. Let h be a meromor- phic function on Y. Then h ◦ f is a meromorphic function on X. Write (h) =P

q∈Y(ordqh)q.

Then

f^∗(h) = X

q∈Y

(ordqh)f^∗q = X

q∈Y

X

p∈f⁻¹(q)

(ordqh)ep(f )p

= X

p∈X

ordp(h ◦ f )p = (h ◦ f ).

Here we use the fact that ep(h ◦ f ) = ep(f ) ord_{f (p)}h. Thus (b) is proved.

Since f^∗ is linear, if D =P

q∈Y nqq, then f^∗D = X

q∈Y

nqf^∗q =X

q∈Y

X

p∈f⁻¹(q)

nqep(f )p.

Hence

deg f^∗D =X

q∈Y

X

p∈f⁻¹(q)

n_qe_p(f ) =X

q∈Y

n_q



 X

p∈f⁻¹(q)

e_p(f )



. Since P

p∈f⁻¹(q)e_p(f ) = deg f, we find

deg f^∗D = deg f ·X

q∈Y

n_q = deg f · deg D.

 If f : X → Y is a nonconstant holomorphic mapping, the ramification divisor of f is the divisor on X defined by

Rf =X

p∈X

bp(f )p.

The branched divisor of f is the divisor on Y defined by Bf =X

q∈Y



 X

p∈f⁻¹(q)

bp(f )



q.

Proposition 1.2. Let f : X → Y be nonconstant holomorphic map between Riemann surfaces. Let ω be a nonzero merormorphic one forms on Y. Then

(f^∗ω) = f^∗(ω) + Rf.

Proof. The proof of this equation is based on the fact that

ord_pf^∗ω = (1 + ord_{f (p)}ω)e_p(f ) − 1 = (ord_{f (p)}ω)e_p(f ) + (e_p(f ) − 1).

We obtain

(f^∗ω) = X

p∈X

(ordpf^∗ω)p

= X

p∈X

(ord_{f (p)}ω)ep(f )p +X

p∈X

(ep(f ) − 1)p

=X

q∈Y

X

p∈f⁻¹(q)

(ord_qω)e_p(f )p + R_f

=X

q∈Y

(ord_qω)f^∗q + R_f

= f^∗(ω) + R_f.

 Corollary 1.1. As above, when X and Y are compact, then

2g(X) − 2 = (deg f )(2g(Y ) − 2) + deg(Rf), where g(X) and g(Y ) are genus of X and Y respectively.

Proof. Computing the degree of (f^∗ω) and using deg(ω) = 2g(Y ) − 2, we find deg(f^∗ω) = deg f^∗(ω) + deg R_f

= deg f · deg ω + deg R_f

= (deg f )(2g(Y ) − 2) + deg Rf.

Since ω is a meromorphic one form on Y, f^∗ω is a meromorphic one form on X. Since the degree of any canonical divisor on X has degree 2g − 2, deg(f^∗ω) = 2g − 2. We prove our

assertion. 

Usually we denote deg R_f by B. Corollary 1.1 gives us the second proof of the Riemann- Hurwitz formula:

g(X) = (deg f )(g(Y ) − 1) + 1 + B 2.