Chamiz, Fernando

Fuertes, Yolanda

Osaka City University Repository

Chamizo, F. and Fuertes, Y.
Osaka J. Math.
48 (2011), 743–748
THE NUMBER OF MAPPINGS
BETWEEN COMPACT RIEMANN SURFACES
FERNANDO CHAMIZO and YOLANDA FUERTES†
(Received August 28, 2009, revised March 15, 2010)
Abstract
We give bounds for the number of morphisms f W X ! Y where X and Y are
compact Riemann surfaces. The target surface Y is not necessarily fixed.
1. Introduction
A classical result due to Hurwitz [7] asserts that the number of automorphisms
f W X ! X of a compact Riemann surface is bounded by 84(g   1) where g  2 is
the genus of X . A natural extension is the study of the number of (non-constant holo-
morphic) morphisms onto a compact Riemann surface X 0 of genus g0, 1 < g0 < g. Let
us denote this number by N ,
N D N (X, X 0) D #{ f morphism X ! X 0}.
De Franchis [2] proved in 1913 that N is finite and that, in fact,
I D I(X ) D
X
X 0
N (X, X 0)
is also finite when X 0 runs over all possible target surfaces up to isomorphisms (see
Remark 3.4 in [5] for a short modern proof).
Since then several authors [6], [8], [9], [11], [12] (see generalizations in [1]) have
given effective bounds for N and I in terms of g and g0. In contrast with the linear
behaviour when X D X 0, all of these bounds show an exponential growth in g. The
best known result for N is due to Naranjo and Pirola [10] who proved
(1.1) N  8(g   1)

2g
1

(2)2g 1 C

2g
3

(2)2g 3 C   

where  D (g   1)=(g0   1). The second named author has proved N  2(2g0 C 2)2gC2
when X is hyperelliptic with a different approach based on Weierstrass points [4].
2000 Mathematics Subject Classification. 30F10, 30F30.
Partially supported by the grant MTM2008-03880.
†Partially supported by the grants MTM2006-01859 and MTM2006-14688.
744 F. CHAMIZO AND Y. FUERTES
On the other hand Tanabe got in [12]
(1.2) I  (2g   3)(g   2)(g   1)22g 3(2g   1)g 1(2g)4g .
In this paper we give new bounds for N and I simplifying and sharpening [11]
and [12]. Our results are easier to write in term of the degree of the morphisms (in
fact the same applies for previous results although they are not always written in this
way) therefore we define
Nd D Nd (X, X 0) D #{ f morphism X ! X 0 with deg( f )  d}
and
Id D Id (X ) D
X
X 0
Nd (X, X 0).
Theorem 1.1. We have
Nd  8(g   1)(2d)2g .
Theorem 1.2. We have
Id 

2g   2
d

(2d)2gC1(2g   1)d .
2. Proof of the results
First of all we introduce some notation and recall some basic facts following [3].
For a Riemann surface X of genus g we consider the 2g-dimensional real vec-
tor space H(X ) generated by real harmonic forms on X , and choose a basis of this
space, {!1, !2, : : : , !2g}, being dual to a canonical basis for H1(X, Z). We shall use
ZX to denote the 2g-dimensional lattice generated by this basis, H(X ) D ZX 
 R. In
H(X ) there is a natural inner product given by h!, i D R  ^ !, let k  k denote the
corresponding norm.
Any morphism f W X ! X 0 of degree d induces a linear map, its pullback,
f W H(X 0) !H(X ) that also maps ZX 0 into ZX . In fact by the definition of the degree
k f !k D
p
dk!k for any ! 2 H(X 0)
(note that norms indicated with the same symbol are defined in different spaces). By
duality there is another linear map, the pushforward, f

W H(X ) ! H(X 0), still preserv-
ing the lattices, satisfying ( f

Æ f )(!) D d! for ! 2 H(X 0) and with the same norm
operator, in particular
k f

k 
p
dkk for any  2 H(X ).
MAPPINGS BETWEEN COMPACT RIEMANN SURFACES 745
Proof of Theorem 1.1. Fix a non-zero form e0 2 ZX 0 of minimal norm and con-
sider the equivalence relation in the set of morphisms X ! X 0 of degree  d given by
f  h if f e0 D he0. By Corollary 3.2 of [10] the cardinal of each class is at most
8(g   1). It remains to prove that the number of equivalence classes is  (2d)2g .
Recall that f e0 2 ZX and dimZX D 2g then the coordinates of f e0 form a vector
in Z2g and there are at most (2d)2g possibilities for its reduction (mod 2d). Assume
that there were more than (2d)2g equivalence classes, then pigeonhole principle assures
the existence of two non-equivalent morphisms f and h such that f e0   he0 D 2d
with  2 ZX   {0}. By the positivity of ( f   h)( f    h) its kernel coincides with
that of f    h and we deduce   (Ker f

) \ (Ker h

), in particular
ke0k  max(k f

k, kh

k) 
p
dkk.
Connecting this with the triangle inequality
2
p
dke0k  k f e0   he0k  k f e0k C khe0k  2
p
dke0k.
Then inequalities become equalities. The central one implies f e0 D he0,  2 R,
and the last one k f e0k D khe0k (because k f e0k, khe0k  pdke0k). Consequently
 D 1 and f and h are in the same class against our assumption.
Proof of Theorem 1.2. Let Nd be the number of equivalence classes of morphisms
of degree exactly d modulo isomorphisms counted by Id . We are going to prove
(2.1) Nd  2

2g   2
d

(2d)2g(2g   1)d ,
that gives
(2.2) Id  2
d
X
jD1

2g   2
j

(2 j)2g(2g   1) j .
The bound in the statement follows using the monotonicity of binomial coefficients in
the range j  g   1 (which is assured by the Riemann–Hurwitz formula).
To prove (2.1), given f W X ! X 0i and h W X ! X 0j in the set of morphisms of
degree d, we define f  h if f e0i D he0j where e0i 2 ZX 0i and e0j 2 ZX 0j are non-
zero fixed elements of minimal norm. By Lemma 3 of [12] we have
Nd  2

2g   2
d

(2g   1)d E
where E is the number of equivalence classes of , and (2.1) follows if E  (2d)2g .
746 F. CHAMIZO AND Y. FUERTES
Suppose on the contrary that E > (2d)2g , then by pigeonhole principle, there exist
f and h with f e0i ¤ he0j , such that
(2.3) f e0i   he0j D 2d for  2 ZX   {0}.
If  2 Ker f

then (2.3) implies de0i   fhe0j D 0 hence ke0ik  ke0jk. The same
applies for  2 Ker h

and we have
max(ke0ik, ke0jk)  max(k fk, khk) 
p
dkk
whenever   (Ker f

)\ (Ker h

), that we can assume because  2 (Ker f

)\ (Ker h

) D
(Im f )?\(Imh)? D (Im f CImh)? is incompatible with (2.3). Using this inequality
and (2.3) we obtain
2
p
d max(ke0ik, ke0jk)  k f e0i   he0jk  k f e0ik C khe0jk D
p
d(ke0ik C ke0jk)
and this leads to a contradiction for f e0i ¤ he0j .
3. Comparison with previous results
The Riemann–Hurwitz formula implies d   with  D (g 1)=(g0 1) as in (1.1)
and one recovers bounds for N and I just substituting d by  (or better by b) in
our results.
Note that the bound (1.1) can be written as
4(g   1)[(2 C 1)2g   (2   1)2g].
In [10] it is claimed that the leading term of the function between brackets is 4g(2)2g 1
but this is a little misleading because the exponential behaviour in g avoids any no-
ticeable cancellation in the subtraction. The ratio with respect to the bound in The-
orem 1.1 is

2

1C
1
2
2g
 

1  
1
2
2g
  sinh(g0   1)
where  indicates the same asymptotics for large values of . Note that then (1.1)
shows an exponential growth in g0 in comparison with Theorem 1.1.
On the other hand (1.2) divided by our bound in Theorem 1.2 in the worst case
scenario d D g   1 gives
(2g   3)(g   2)(g   1)22g 3(2g)4g
 2g 2
g 1
(2g   2)2gC1 
e2
p
g
2
g2(2g)2g
where we have employed Stirling asymptotic formula N ! N N e N
p
2N giving
 2N
N


22N=
p
N and (g=(g   1))2g ! e2 as g !1.
MAPPINGS BETWEEN COMPACT RIEMANN SURFACES 747
In fact in the proof of (1.2) Tanabe gets a bound depending on  (see p. 3063 in
[12]) to be compared with (2.1). In this case the quotient is
(2g   2g0 C 1)(4d2 C 1)2g
2(2d)2g 
1
2
(2d)2g
that still grows exponentially.
References
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to varieties of general type, Ann. Inst. Fourier (Grenoble) 47 (1997), 801–824.
[2] M. De Franchis: Un teorema sulle involuzioni irrazionali, Rend. Circ. Mat. Palermo 36 (1913),
368.
[3] H.M. Farkas and I. Kra: Riemann Surfaces, second edition, Graduate Texts in Mathematics 71,
Springer, New York, 1992.
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surfaces, Israel J. Math. 109 (1999), 1–12.
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Riemann surfaces, Publ. Mat. 37 (1993), 339–353.
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(1986), 185–198.
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eral type, Amer. J. Math. 129 (2007), 1689–1709.
[11] M. Tanabe: A bound for the theorem of de Franchis, Proc. Amer. Math. Soc. 127 (1999),
2289–2295.
[12] M. Tanabe: Bounds on the number of holomorphic maps of compact Riemann surfaces, Proc.
Amer. Math. Soc. 133 (2005), 3057–3064.
748 F. CHAMIZO AND Y. FUERTES
Fernando Chamizo
Departamento de Matemáticas
Facultad de Ciencias
Universidad Autónoma de Madrid
28049 Madrid
Spain
e-mail: fernando.chamizo@uam.es
Yolanda Fuertes
Departamento de Matemáticas
Facultad de Ciencias
Universidad Autónoma de Madrid
28049 Madrid
Spain
e-mail: yolanda.fuertes@uam.es


The number of mappings between compact Riemann surfaces

Chamizo, Fernando

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The number of mappings between compact Riemann surfaces

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