manuscripta math. 95, 33-45 (1998) manuscripta mathematica ~ Sprmger-Verlag 1998

A N o t e on W e i e r s t r a s s P o i n t s of Bie l l ip t ic Curves

J i h u n P a r k

Department of Mathematics, Seoul National University. $eoul, Korea.

Received October 2 l, 1996; in revised form April 22, 1997

The family of all bielliptic curves of genus 9 -> 6 can be divided into the subfamilies of all bielliptic curves of genus g with s Weierstrass points whose non-gap sequences are

{4 ,6 ,8 , . . . , 2 g - 4 , 2 g - 3 , 2 9 - 2 ,29 ,29+ 1 ,2g+ 2 , . . . },

where s = 0, 1 ,2 , . . . , 2 g - 2 . We will prove that for any nonnegative integers

g _> 6, 0 < s < 2 9 - 2 and s # 2 , 2 g - 3, there exists a bielliptic curve of

genus 9 with s Weierstrass points whose non-gap sequences are as above.

1. I n t r o d u c t i o n

Let C be a compac t R i e m a n n surface of genus 9 and let A/[(C)

be the field of m e r o m o r p h i c funct ions on C. For a point P E C, we

define the Weierstrass non-gap sequence Hp C N by

Hp = {n E 1N : there exists f E M ( C ) with (f)o~ = n P } ,

where N is the addi t ive semigroup of nonnegat ive integers. It is

obvious t ha t Hp is a subsemigroup of N. We call its complement

"This work has arisen from author's master thesis written at Seoul National University in 1996 under the guidence of Prof. C. Keem.

34 Jihun Park

Gp -= 1~ - Hp the Weierstrass gap sequence of P . Each element of Gp (resp. HF) is called a gap (resp. non-gap) of P . Using Riemann- Roch theorem, we can easily check tha t

d i m l n P I = d i m l ( n - 1 )P I

d i m i n P I = d i m i ( n - 1 )P I + 1

and

if n E Gp.

if n • Gp,

cardinal i ty of G p = g.

A point P E C is called a Weierstrass point of C i f G p ~ { 1 , 2, 3..-- , g}. Now, let C be a bielliptic curve of genus g >_ 6 with the bielliptic

involution lr : C * E, where E is an elliptic curve. We will check in Proposi t ion 2.3 tha t a point P of C is a ramification point of n if and only if Hp is ei ther

{4, 6 , 8 , . . . ,2g - 4,2g - 3,2g - 2,2g ,} . . . . . . (h)

o r

{ 4 , 6 , 8 , . . - , 2 g - 4 , 2 g - 2 , 2 g - l , 2 g ,} . . . . . . (1),

where "2g )" denotes all integers larger t han or equal to 2g. Weier- strass points whose non-gap sequences are one of the above sequences play an impor tan t role. In [G], [K], [Ka] , and [go] , we can find the result tha t a compact R iemann surface of genus g :> 8 which has a Weierstrass point whose non-gap sequence is one of (h) and (1) is a bielliptic curve.

We may classify the bielliptic curves of genus g _> 6 according to the number of ramification points whose non-gap sequences are sequence (h). Since there exist 2g - 2 ramificat ion points of ~ by Riemann-Hurwi tz formula, we may write

2g-2

gC(g) = LJ 13C(s, g) for each g _> 6, $~0

where BC(g) denotes the family of all bielliptic curves of genus g and I3C(s, g) denotes the family of all bielliptic curves of genus g which have s points whose non-gap sequences are sequence (h). According to [Ko], there exists a bielliptic curve of genus g which has a Weier- s trass point whose Weierstrass non-gap sequence is one of the above sequences. But it gives no idea whether or not 13C(s, g) is nonempty. In this paper, we will prove the following theorem.

M a i n T h e o r e m . If s <_ 2 g - 2 is a nonnegative integer, and s 2, 2g - 3, then BC(s, g) is nonempty.

Weierstrass Points of Bielliptic Curves 35

2. D e f i n i t i o n s a n d G e n e r a l O b s e r v a t i o n s on B i e l l i p t i c C u r v e s

T h r o u g h o u t this section, C denotes a bielliptic curve of genus g _> 6 wi th the bielliptic involution 7r : C , E, where E is an elliptic curve.

D e f i n i t i o n 2.1. A point on a compact Riemann surface of genus g is said to be of heavy type (respectively, of light type), if its Weier- strass non-gap sequence is {4, 6, 8,. • • , 2g - 4, 29 - 3, 2g - 2, 2g , } (respectively, { 4 , 6 , 8 , . . . ,2g - 4,2g - 2,2g - 1,2g ,})

D e f i n i t i o n 2.2. A b~elliptic curve C o/genus g >_ 6 is said to be of (s, g)-type if it has s points of heavy type.

With simple computa t ion , we can obtain the following proposition.

P r o p o s i t i o n 2.3. A point of C is a ramification point of ~ if and only if it is of either heavy or light type.

Proof . Since an elliptic curve has no Weierstrass point, we can easily check tha t a ramificat ion point of ~r is of either heavy or light type.

Suppose t ha t a point P of C is of heavy or light type. Then there exists meromorphic funct ion f on C such tha t ( f ) ~ = 4P. Since g > 6, there is a meromorphic function h on E such tha t f = ~*h ( [Ko, Corol lary 1.2]). This implies tha t P is a ramification point of T~. []

R e m a r k . We can find a generalization of Proposi t ion 2.3 in [T]. In the case of bielliptic curve, Proposi t ion 2.3 shows tha t Scholium 3.7 in [T] holds for genus >_ 6.

By Riemann-Hurwi tz formula, there exist 2g - 2 ramification points of ~. Thus, Proposi t ion 2.3 implies tha t there exist exactly 29 - 2 points on C which is of heavy or light type. In other words, for a fixed g _> 6, every bielliptic curve of genus g is of (0,g), (1, g ) , . - . ,

or (29 - 2, g) type. Now, let 's recall t ha t there exists only one complete linear series

9-2 on a compact R iemann surface, which is called the canonical linear series. Using this fact, we can obtain the useful proposition.

P r o p o s i t i o n 2.4. Two ramification points P and Q o / ~ are of the same type if and only if dim 1(2g - 2)PI = dim 1(29 - 2)Q I. Further- more, if P and q are of heavy type, then 1(2g - 2)P I = ](2g - 2)Q I.

36 Jihun Park

Proof . If a ramification point P of 7r is of heavy type, then dim ](2g- 2)P] = g - 1 . If it is of light type, dim 1(2g-2)P I = g - 2 . These imply the first assertion. If P and Q are of heavy type, then ](2g- 2)P] and l(2g - 2)Q I are the canonical linear series and hence l(2g - 2)P I =

1(2g - 2)QI . []

3. E x i s t e n c e o f Bie l l ip t ic C u r v e s w i t h C e r t a i n P r o p e r t i e s

Let E be an elliptic curve and let .A4 (E) be the field of meromor- phic functions on E. In this section, we will frequently use the sheaf (DE of holomorphic functions on E. We will write (9 for OE.

First, we will study the topological space associated to the sheaf CO. Let

iol = U P 6 Z

be the disjoint union of all stalks. We define a map p : ]01 , E in the canonical way, that is, assign to each element of OF the point P on E.

Now, we introduce a topology on 1(.91 as follows. For any open subset U of E and element f 6 (D(U), let

[v, f] = { f p e I(Dl: p e u},

where f p is the equivalence class of (Dp represented by f . It is well known that the system of all sets [U, f] forms a basis for a topology on 1(9[. Furthermore, with this topology, the projection p : IOl ) E is a local homeomorphism ( [F] ). From now on, we view ](DI as the topological space with this topology. In general, the topological space associated to a sheaf is not a Hausdorff space. But Identity theorem on Riemann surface guarantees that the topological space IOI is a Hausdorff space.

The following theorem will be a main tool in this paper. Its proof is a slight modification of that of IF, Theorem 8.9].

T h e o r e m 3.1. Let P1, P 2 , ' " , P2g-2 be 2g - 2 distinct given points on an elliptic curve E, where g > 3. Suppose that there exists a meromorphic function f 6 ~,4(E) such that

( f ) =/92 + P3 + ' " + P29-2 - (2g - 3)/°1

o r

(Y) = 3P2 + P3 + . . - + P2g-2 - (2g - 1)P~.

Weierstrass Points of Bielliptic Curves 37

Then, there exist a bielliptic curve C of genus g and a two-sheeted

branched holomorphic covering map 7r : C , E such that P1, P2." " • ,

P2~-'2 are branch points of 7r and T 2 - 7c*f E M ( C ) [ T ] has zeros in

M ( C ) , where . M ( C ) denotes the field of meromorphic func t ions on

C.

P r o o f . Let A be the support of the divisor of f , tha t is, {P1, P2," • • , P29-2} and let E ' = E \ A . Then the polynomial T 2 - f ( P ) E C[T] has two dist inct zeros for each P E Eq We define a topological subspace

C' of IOi as follows

C t = { ~ E IOl : ~ E Op for some P E E ' a n d ~2 _ fp ___ 0 in Op}.

We let 7r ~ = p I c': C~ , E' . Since IO] is a Hausdorff space, so is C ~.

C l a i m 1. The map ~' : C ~

sheeted covering map.

E t is a proper unbranched two-

By Hensel 's lemma, for every point P E E ~ there exist a connected open neighborhood U of P in E ~ and a holomorphic function h E

O ( U ) such tha t

T 2 - f = (T - h ) ( T + h) on U.

This implies tha t

I_ 1 (u) = [u, h] u [U,-h].

Since h and - h are distinct holomorphic functions on U, [U, h] and [U , -h ] are disjoint. It is obvious tha t bo th ~r' ][U,h]: [U,h] , U

and lr' [[U,-h]: [U, -h] , U are homeomorphisms. Thus, the map ld : C ~ ~ E ~ is an unbranched two-sheeted covering map. We can

easily check tha t 7r ~ is a proper map.

Since the map n ~ is an unbranched covering map, there exists a complex s t ructure on C t wi th which 7r ~ is holomorphic ([F, The- orem 4.6]). Now, we give this complex s t ructure to our Hausdorff

topological space C .

Now, suppose tha t the elliptic curve E is defined by a lattice A = Z'~I + Z~2, where "yl. 72 E C are linearly independent over N.

38 Jihun Park

Let p : C , C / A ( = E) denote the quot ient m a p and Pi E C a represen ta t ive of Pi for each i = 1, 2,. • • , 29 - 2. For each pi, we ma y

take an ri > 0 such tha t the image of the disc Ui = {z E C :[ z - p i [< ri} unde r p contains no other points of A and p [ c , : Ui , p(Ui) is

a b iho lomorph ic map.

C l a i m 2. The preirnage of p(Ui)\{Pi} under 7c' is connected for each i = 2, 3 , . . . . 2 9 - 2.

We will prove the claim in the case of i = 2. Th e o the r cases

can be proved in the same way. Suppose tha t f has a zero of order 2 n + l at P2, where n = 0 or 1. We consider the ho lomorphic funct ion

p* f on the open disc U := U2. Since p*f is a ho lomorphic funct ion

which has a zero of order 2n + 1 at P2, there exists a nonvanishing ho lomorph ic funct ion kl on U such t ha t p ' f (z) = (z - p 2 ) 2 n + l k l ( Z )

on U. Since kl does not vanish on U and U is s imply connec ted ,

the re exists a holomorphic function k : U , C such t h a t k 2 = kl.

T h u s we obta in

p*f = (z - p2){(z - p2)nk} 2.

Suppose tha t 0 < A < r2, 0 E R, and let ( = P2 + Ae°i . By Hensel 's

l emma, there exists a holomorphic germ ~< E Ou,< such t h a t ~ =

p*f and ~<(( ) = vr-Ae~,U~en°ik((). It is obvious tha t - ~ is the

ana ly t ic cont inua t ion of 7)4 along the closed pa th c = P2 + Ae(t+°)i, 0 < t < 27r. Since p [u: U , p(U) is a b iholomorphic map, we m a y

conc lude tha t there exists a holomorphic funct ion germ ~p(<) E Op(~) such t ha t -2 ~p(<) = f and such tha t -@(<) is the analyt ic con t inua t ion

of q3p(<) along the closed pa th poe. This means t h a t 7r'- l(p(U)\{P2}) is connec ted .

We claim tha t C I is connected and hence a R i e ma n n surface.

Suppose tha t C ~ is disconnected. Since rd : C I , E ~ is an un-

b ranched two-sheeted covering map, C ~ has two connec ted compo-

nents C{ and C~. Fur thermore , the rest r ic t ions re' [c[: C[ , E ' and

7r' Ic;: C[ , E ' are biholomorphic. T h e n 7r ' - l (p(g) \{P2}) has to be d isconnected . This contradicts Cla im 2. Thus C ~ is a connec ted

complex manifold of complex dimension 1, t ha t is, a R i e m a n n surface.

T h e covering 7r' : C ' , E ~ can be cont inued to a p rope r holo-

Weierstrass Points of Bielliptic Curves 39

morphic map ~r of a R iemann surface C onto E ([F, Theorem 8.4]). Since 7r is a proper map and E is compact , C is a compact R iemann surface.

C l a i m 3. The points P1,"" , Pvg-2 are the branch points o] 7r.

If some point Pi (i = 2 , 3 , . . . ,2g - 2) is not a branch point, then there exists an open disc U centered at Pi E C such tha t 7r- l (p(g) \{Pi}) has two components . But 7r-l(p(U)\{Pi}) is con- nected for sufficiently small open disc U centered at Pi by Claim 2. Thus, for each i = 2, 3 , - . - , 2g - 2, the point Pi on E is a branch point of ~r. Since 7r is two-sheeted, the number of branch points of 7r is even by Riemann-Hurwi tz formula. Since P1 is the only remaining candidate for branch point of 7r, the point Pt is a branch point of ~.

Using Riemann-Hurwi tz formula and Claim 3, we can conclude tha t the genus of C is ½(2+the number of branch points of 7r) = g. Thus, C is a bielliptic curve of genus g and r : C ~ E is a two-sheeted branched covering map such tha t P1, P2,"" , P2g-2 are branch points of 7r.

Let h : C' ~ C be a function defined by h(~) = ~(Tr'(~)). Clearly, h is a holomorphic funct ion on C ~. From the definition of h

we obta in h(~) 2 - (Tr'*f)(~) = 0 for all ~ E C'.

The holomorphic function h on C ~ can be extended to a meromorphic function H on C such tha t H 2 - 7r*f = 0 on C ([F, Theorem 8.2]). []

4. M e r o m o r p h i c F u n c t i o n s o n a n E l l i p t i c C u r v e

In this section, we will find the points and meromorphic functions on an elliptic curve E which satisfy certain properties. Let E be the elliptic curve defined by a lattice A -- "rlZ + 72Z, tha t is, E = C/A, where ~1 and ~2 are complex numbers linearly independent over R.

We will need the following l emma which is an application of Abel 's

theorem.

L e m m a 4.1. ( [ H ] , [F] ) A divisor D = E~=lniPi e Div(E) is principal if and only if deg D = 0 and ~,i~=l n{Pi is zero in the group C/A

4o Jihun Park

T h e o r e m 4.2. There exist 2g - 2 dist inct points Po, P I , " " .Ps-1, Q 1 , " " , Qt on E and meromorphie funct ions f , f~, h 5 E A4 (E) such

that

( f ) = P1 + P2 + " " + Ps-1 + Q1 + " " + Qt - (2g- 3)P0, (f i ) = (g - 1)(P0 - Pi) for each i = 1 , 2 , . . . , s - 1,

and (h)) = (g - 2)(P0 - Qj) for each j = 1 , 2 , . . . , t ,

where s + t = 2g - 2, s >_ 1, s ~ 2, t >_ O, t ~ l , and g >_6.

P r o o f . Let

ml m2 /~1 ---~ { g--'-~9,1 " } - _ _ g - 1 9 , 2 : m l , m 2 = O, 1, 2 , - . . ,g - 2}\{0},

n l n2 and A2 = {~--Z-~_ 29,, + q _---~9,2 : n l , n 2 = 0, 1 , 2 , . . . ,g - 3}\{0}.

Since g - 1 and g - 2 are relatively prime, A1 and A2 are disjoint modulo A. We define a subset /~1 of A1 x A1 and a subset A2 of A 2 x A 2 as follows;

ml

U { (gm--~31 9,1 + 19,2, g - g - 1

2 g - 2 U { ( ~ 9,1+ .q- 19,2' g----l- 1 9,1+

g - 1 9,1 : r o t = 1 , 2 , . . . ,

:m2 = 1 ,2 , . - . g - 1

m3 ( g - 1 ) - m 3 ( g - 1 ) - m 3 "~ 9,1 + 22) :

g - - 1

g

2/-., ) ~2 ---- "71, g -- 2 9,1 : nl = 1, 2 , ' ' ' ,

U g_--~9,2, g - 2 9,2 : n 2 = 1 , 2 , " '" ,

9'1 + 2 9,2, 9,1 + "t2 : g - g - 2 g - 2

{ \ g - z g - - g - 2

1 g - 4 g - 329,2 ) U { (2-'-~9,1 "f- g---~ "}'2 '~ g---~ 9,1 -~- }" \ g - z - g

Weierstrass Points of Bielliptic Curves 41

For every dist inct pair (x, y), (x', y') E /~1, tile four entries are pair- wise dis t inct modulo A. Of course, the same assertion holds for ~2. Note tha t

the cardinal i ty of ~1 = 3[~-~] + 1 >__ g, and the cardinali ty of As = 3 [ @ ] + 2 _> g - 1.

C a s e 1) s is even. Since s is even and s ¢ 2, we have 4 _< s <__ 2 g - 2 . Prom the

equa t ion s + t = 2g - 2, it follows that t is even and 0 < t < 2g - 6. t dist inct elements Since the cardinal i ty of/~2 _> g - 1, we can choose

(Yl, g2), (Y3,Y4),"", ( Y t - l , Y t ) i n /~2. There exist 3 [ @ ] + 1 elements

in ~1 . Thus , the cardinal i ty of zX~ >_ g - 3, where

1 9 - 2 1 1 g - 2 _ , I 'Yl + 13'2 , g---~'Yl

+ g - 21,~2), 1 g - 2 )} ( g - - ~ ? 2 , g---~3'2 • g _ _ - - - -

Since - ~ < g - 3, we can choose ~A elements (x , ,x2) , ( x 3 , x 4 ) . ' " ,

(xs-5 , xs -4 ) in A t. Set

1 x 0 : O, X s - 3 = ~ _ 1 ~ 1 ,

1 ~-~ + ~ and xs-1 g_-=-r3'2. X s _ 2 : g _ l " f l g_l"Y2,

Then, it follows that

s - 1 t ~i=1 xi + ~ j = l YJ - (2g - 3)xo -- 0 (mod A), (g - 1)(xo - Xi) ~ 0 (mod A) for i = 1, 2 , . . . , s - 1.

and (g - 2)(xo - yj) - 0 (mod A) for j = 1 , 2 , . . . , t .

Let Pi and Qj be the points on E corresponding to xi and yj respec- tively. Then , by L e m m a 4.1 there exist meromorphic functions f , fi, and hj o n E such that

( f ) = P1 + P2 + " " + Ps-1 + Q1 + " " + Qt - (2g - 3)Po, ( f i ) = (g - 1)(P0 - Pi) for each i = 1, 2 , . . . , s - 1,

and (h i ) = (g - 2)(Po - Qj) for each j = 1 ,2 , - . . , t .

Clearly, P o , " , Vs-1, Q 1 , " ' , Qt are dist inct points on E.

C a s e 2) s is odd.

42 Jihun Park

We know 1 < s <_ 2 g - 5 since t ¢ 1. It is obvious that t is odd and tha t 3 < t __% 2 g - 3 . Since the cardinality of/k1 _> g and s-1 < g - 3 , we can choose ~-~ distinct elements (xl, x2), (x3, x4),. • •

( x s -2 , x s -1 ) in '-~1. Because the cardinality of A2 is 3[~-~] + 2 > g - 1 and t2---~3 < g - 3 , we can take t2----~3 distinct elements (gl, y~), (Y3, Y4)," " ",

2 1 g_~ + 9 - 3 (yt-4,yt-3) in /k2 different from (9_-~71 + g_-~32, g_271 g_272) and 2 g-S g-4 (g_--~"fl 4- g_-~3'2, g_231 4- g_232)'

Set _ 2

= 4- o---~'~2, xo O, Yt-2 -- ~-L-~2/1 1

1 g gg_~ ~1 4. gg4_2~ 72. g-s +9_-~72, a n d y t = _ Y t - 1 ---- g_2~l

Then, we obtain

s--1 t ~i=1 x~ + ~ j = l YJ - (2g - 3)x0 = 0 (rood A), ( g - 1 ) ( x o - x i ) = _ O (rood A) for i = 1 ,2 , . . . , s - l ,

and (g - 2)(x0 - yj) = 0 (mod A) ibr j = 1 ,2 , - . . ,t.

Let Pi and Qj be the points on E corresponding to xi and yj respec- tively. Then, they are distinct points and there exist meromorphic functions which satisfy our assertion. []

T h e o r e m 4.3. There exist 2g - 2 distinct points Qo, Q 1 , ' " , Q2g-3 on E and meromorphic functions f , fi E j£4(E) such that

( f ) = Q1 + Q2 + " " + 3Q2g-3 - (2g - 1)Qo, and ( f i ) = (g - 1)(Qo - Qi) for each i = 1 ,2 , . . . ,2g - 3,

where g > 6.

P r o o f . We have at least g elements in A1 (defined in the proof of Theorem 4.2). Thus we may choose g - 4 distinct elements (Yt, Y2),

(Y3, Y4) , ' " , (Y2g-9, Y2g-8) in

-,, / Z _ ~ g - 2 1 2 g - 2 A 1 :---- /7~1\ ~, Y - J ' ( 31, ~ - ~ _ I"Yl), (g _ ~ " Y 1 4. g---~'~32' ~--~__ 1 " f l__

+ g - 31 2), (x, y) }, g

where (x,y) 3 g-4 > 8; otherwise g-4 3 (g_171 ?-z-T- 171 )- = (g_--~31, g _ 1 3 t ) i f g _

Set 1 2

Yo = O, Y2g-7 : g_--L']'71 4- g_--L-~l')'2, 1 g-2 Y2g-5 ~--- 4- 9-132, Y2g-6 = g_----'l.-.31 4- g-132, g-1 31

k :d and y2g-3 = 1 Y2g-4 = g_131, g_-'-i-~31.

Weierstrass Points of Bielliptic Curves 43

Then we obta in

E~g7 4 ~]z + 392g-3 -- (2g -- 1)yo = 0 (,nod A), a n d ( g - 1 ) ( y o - y , ) = O (rood A) for each j = 1 ,2 , . . . ,29 - 3.

Let Q, be the points on E corresponding to Yi. Clearly, Q~ are distinct points which satisfl" our condition. []

5. M a i n R e s u l t

We have reached the state in which we can prove the main result.

Note t h a t 9 is an integer _> 6.

P r o o f o f M a i n T h e o r e m . Let E be an elliptic curve and let M ( E ) be the field of meromorphic functions on E.

C a s e 1) s = 0 By Theorem 4.3. we may choose 2 g - 2 dist inct points Qo, "'" ,Q2g-3

on E for which there exist meromorphic functions f and f~ E .M(E)

such tha t

( f ) = Q1 + Q2 + " " + 3 Q 2 g - 3 - ( 2 g - 1)Qo, and ( f i ) = (g - 1)(Qo - Qi) for each i = 1 , 2 , . . . ,2g - 3.

By T he o r e m 3.1, there exists a bielliptic curve C of genus g and a two-sheeted branched holomorphic covering map ~- : C , E such tha t Qo,"" ,Q2g-3 a r e branch points of u and the polynomial T 2 - 7r*f e M ( C ) [ T ] has a zero in .M(C), where ,~¢I(C) denotes the field of meromorphic functions on C. Let z be a zero of the polynomial T 2 -

7r*f and Qi the preimage of Qi under 7r for each i = 0, 1 , . . . , 2g - 3. From the equation z2 = 7r 'f , we obtain

2(z) = (Tr*f)

= 201 + ' " + 2029-4 + 602g-3 - 2 ( 2 g - 1)0o,

and hence

(z) = 0 1 + ' " + 0 2 g - 4 + 3 Q 2 g - 3 - ( 2 g - 1 ) 0 0 '

This means tha t the point Q0 on C is of light type since (2g - 1) is a non-gap of 00. We have the meromorphic functions fi E j~t(E)

whose divisors are

(f /) = (g - 1)(Qo - Qi) for each i = 1 , . . . ,2g - 3.

44 Jihun Park

Since the points Qi a r e branch points of 7r, we obta in

(rr*fi) = (2g - 2)(@0 - @,) for each i = 1 , . . - ,29 - 3.

Then the complete linear series [(2g - 2)~)0[ and l(2g - 2)@/] have the same dimensions for each i = 1,- . . , 2 9 - 3. It immedia te ly follows from Proposi t ion 2.4 tha t every point Qi is of light type. Thus, the bielliptic curve C is contained in 13C(O, g).

C a s e 2) s ¢- 0 Let t = ( 2 g - 2 ) - s . Since s ¢ 2, t ¢ 1, it follows from Theo-

rem 4.2 t ha t there exist points Po, P 1 , " , P s - I , Q 1 , ' " , Q t on E and meromorphic functions f , fi, hj E 3,4(E) such tha t

( f ) = P1 + . - - + P , - 1 + Q~ + - - - + Qt - (2g - 3)Po, ( f~) = (g - 1 ) ( P o - P / ) ,

a n d ( h i ) = (g - 2 ) ( P o - Q j ) .

From Theorem 3.1, we obtain a bietliptic curve C of genus g and a two-sheeted branched holomorphic covering map 7r : C , E such tha t the points P0, P 1 , ' " , Ps-1. Q 1 , ' " , Qt are branch points of 7r and the polynomial T 2 - 7r*f E M(C) [T] has a zero in j ~ ( C ) where .M(C) denotes the field of meromorphic functions on C. Let z be a zero of the polynomial T 2 - 7r*f and /5/ and @j the preimages of P/ and Qj under 7r respectively. Then, we obtain

(z) = Pl + . . - + / 5 s - 1 + @1 + . . . + @~ - (2g - 3)Po, (1)

(~*I,) = (2g - 2)(/5o - /5~) for each i = 1 , . . . , s - 1, (2)

and (Tr*hj) = (2g - 4)(/50 - @j) for each j = 1 , - . . , t. (3)

From Equa t ion (1), we know tha t the point /50 on C is of heavy type. Equa t ion (2) implies tha t every p o i n t / 5 / i s of heavy type since the dimensions of [(2g - 2)/5o[ and ](2g - 2)/5/[ are equal for each

i = 1 , - . . , s - 1 . Suppose t ha t the point @j is of heavy type. Then , [(2g - 2)@j[ =

1(2g - 2)/5o[ by Proposi t ion 2.4. Thus there exists a meromorphic

Weiers trass P o i n t s o f Bie l l ip t ic Curves 45

function h E A4(C) such that (h ) = ( 2 g - 2 ) ( ~ ) j - /5o). Then, we obtain

(hT~*h~) = (2g - 2)(03 - /5o ) + (2g - 4)(/5o - (~j)

= 2Q - 2/50.

Since a curve of genus g _> 4 cannot be both hyperelliptic and biel- liptic, this is a contradiction. Therefore, each ~)j is of light type and hence the bielliptic curve C is contained in 13C(s, g). [] R e m a r k . We do not know whether the families BC(2, g) and BC(2g- 3, g) are empty or not. Our method will not show tha t the families are nonempty since there exists no meromorphic function on E sat isfying such condit ions as in Theorem 4.2.

R e f e r e n c e s

[r]

[¢1

[H] [K]

[Kat

lifo]

{T]

Otto Forster, Lectures on Riemann surfaces, Springer-Verlag, 1981.

ArnMdo Garcia, Weights of Weierstrass points zn double coverings of curves of genus one or two, Manuscripta Math., 55 (1986), 419-432.

Robin Hartshorne, Algebrmc geometry, Springer-Verlag, 1977.

Seon Jeong Kim, Weierstrass points on tmgonal curves and ~-gonal curves, PhD thesis, Seoul National University, 1989.

Takao Kato, Non-hyperellipt~c Weierstrass points of maximal weight, Math. Ann., 239 (1979), 141-147.

Jiryo Komeda, On Weierstrass points whose first non-gaps are four, J. reine angew. Math., 341 (1983), 68-86.

Fernando Torres, We~erstrass points and double covemng of curves, Manuscripta Math., 83 (1994), 39-58.

Current address : Johns Hopkins University Department of Mathematics Krieger Hall 3400 N. Charles Street Baltimore, MD. 21218, U.S.A.

E-mail address : jhpark@chow.mat.jhu.edu