Weierstrass points on modular curves X₀(N) fixed by the Atkin–Lehner involutions

Abstract

Purpose

The authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions W_Q on X₀(N) for N ≤ 50 are Weierstrass points or not.

Design/methodology/approach

The design is by using Lawittes's and Schoeneberg's theorems.

Findings

Finding all Weierstrass points on X₀(N) fixed by some Atkin–Lehner involutions. Besides, the authors have listed them in a table.

Originality/value

The Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if one knows a Weierstrass nongap sequence of a Weierstrass point, then one is able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X₀(N).

Citation

Bojakli, M. and Sankari, H. (2023), "Weierstrass points on modular curves X₀(N) fixed by the Atkin–Lehner involutions", Arab Journal of Mathematical Sciences, Vol. 29 No. 1, pp. 63-72. https://doi.org/10.1108/AJMS-01-2021-0001

1. Introduction

Let H be the complex upper half plane and Γ be a congruence subgroup of the full modular group SL2(Z). Denote by X(Γ) the modular curve obtained from compactification of the quotient space Γ\H by adding finitely many points called cusps. Then X(Γ) is a compact Riemann surface.

For each positive integer N, we have a subgroup Γ₀(N) of SL2(Z) defined by:

Γ0(N)=abcd∈SL2(Z);c≡0(modN)

and let X₀(N) = X₀(Γ(N)).

A modular curve X₀(N) of genus g ≥ 2 is called hyperelliptic (respectively bielliptic) if it admits a map φ: X → C of degree 2 onto a curve C of genus 0 (respectively 1). A point P of X₀(N) is a Weierstrass point if there exists a non-constant function f on X₀(N) which has a pole of order ≤ g at P and is regular elsewhere.

The Weierstrass points on modular curves have been studied by Lehner and Newman in [1]; they have given conditions when the cusp at infinity is a Weierstrass point on X₀(N) for N = 4n, 9n, and Atkin [2] has given conditions for the case of N = p²n where p is a prime ≥ 5. Besides, Ogg [3], Kohnen [4, 5] and Kilger [6] have given some conditions when the cusp at infinity is not a Weierstrass point on X₀(N) for certain N. Also, Ono [7] and Rohrlich [8] have studied Weierstrass points on X₀(p) for some primes p. And Choi [9] has shown that the cusp 12 is a Weierstrass point of Γ₁(4p) when p is a prime > 7. In addition, Jeon [10, 11] has computed all Weierstrass points on the hyperelliptic curves X₁(N) and X₀(N). Recently Im, Jeon and Kim [12] have generalised the result of Lehner and Newman [1] by giving conditions when the points fixed by the partial Atkin–Lehner involution on X₀(N) are Weierstrass points and have determined whether the points fixed by the full Atkin–Lehner involution on X₀(N) are Weierstrass points or not. In this paper, we have determined which of the points fixed by W_Q on X₀(N) are Weierstrass points and found Weierstrass points on modular curves X₀(N) for N ≤ 50 fixed by the partial and the full Atkin–Lehner involutions. The Weierstrass points have played an important role in algebra. For example, in algebraic number theory, they have been used by Schwartz and Hurwitz to determine the group structure of the automorphism groups of compact Riemann surfaces of genus g ≥ 2. Whereas in algebraic geometric coding theory, if we know a Weierstrass nongap sequence of a Weierstrass point, then we are able to estimate parameters of codes in a concrete way. Finally, the set of Weierstrass points is useful in studying arithmetic and geometric properties of X₀(N).

2. Points fixed by the Atkin–Lehner involutions

For each divisor Q|N with (Q,NQ)=1, consider the matrices of the form QxyNzQw with x,y,z,w∈Z and determinant Q. Then each of these matrices defines a unique involution on X₀(N), which is called the Atkin–Lehner involution and denoted by W_Q. In particular, if Q = N, then W_N is called the full Atkin–Lehner involution (Fricke involution). We also denote by W_Q a matrix of the above form.

Let X0Q(N) be the quotient space of X₀(N) by W_Q. Let g₀(N) and g0Q(N) be the genus of X₀(N) and X0Q(N) respectively. Then g0Q(N) is computed by the Riemann–Hurwitz formula as follows:

g0Q(N)=14(2g0(N)+2−v(Q)),

where v(Q) = v(Q; N) is the number of points on X₀(N) fixed by W_Q. It is given by:

Proposition 2.1.

[13] For each Q‖N, v(Q) is given by

v(Q)=∏p|N/Qc1(p)h(−4Q)+∏p|N/Qc2(p)h(−Q),ifQ≥4andQ≡3(mod4),+∏p|N/21+−4p,ifQ=2,+∏p|N/31+−3p,ifQ=3,+∏pk|N/4p⌊k2⌋+p⌊k−12⌋,ifQ=4,

where h(−Q) is the class number of primitive quadratic forms of discriminant −Q,•• is the Kronecker symbol and the functions c_i(p) are defined as follows: for i = 1, 2,

ci(p)=1+−Qp,ifp≠2andQ≡3(mod4),1+−4Qp,ifp≠2andQ≢3(mod4),c1(2)=1,ifQ≡1(mod4)and2‖N,0,ifQ≡1(mod4)and4|N,2,ifQ≡3(mod4)and2‖N,3+−Q2,ifQ≡3(mod4)and4‖N,31+−Q2,ifQ≡3(mod4)and8|N,c2(2)=1+−Q2,ifQ≡3(mod4).

Now, we recall the algorithms for finding Γ₀(N)-inequivalent points fixed by W_Q on X₀(N) [14]. For a negative integer D congruent to 0 or 1 modulo 4, we denote by QD the set of positive definite integral binary quadratic forms:

Q(x,y)=[p,q,r]=px2+qxy+ry2

with discriminant D = q² − 4pr. Then Γ(1) acts on QD by

Q◦γ(x,y)=Q(sx+ty,ux+vy)

where γ=stuv. A primitive positive definite form [p, q, r] is said to be in reduced form if

|q|≤p≤r,andq≥0if either|q|=porp=r.

Let QD°⊂QD be the subset of primitive forms, that is,

QD°≔{[p,q,r]∈QD;gcd(p,q,r)=1}.

Then Γ(1) also acts on QD°. As is well known [15], there is a 1-1 correspondence between the set of classes Γ(1)\QD° and the set of reduced primitive definite forms.

Proposition 2.2.

[14] for each β∈Z/2NZ, we define

Q°D,N,β={[pN,q,r]∈QD;β≡q(mod2N),gcd(p,q,r)=1}.

Then we have the following:

1. Define m=gcd(N,β,β2−D4N) and fix a decomposition m = m₁m₂ with m₁, m₂ > 0 and gcd(m₁, m₂) = 1. Let

Q°D,N,β,m1,m2={[pN,q,r]∈Q°D,N,β;gcd(N,p,q)=m1,gcd(N,q,r)=m2}.

Then Γ₀(N) acts on Q°D,N,β,m1,m2 and there is an 1-1 correspondence between

Q°D,N,β,m1,m2/Γ0(N)→Q°D/Γ(1)[pN,q,r]→[pN1,q,rN2]

where N₁N₂ is any decomposition of N into coprime factors such that gcd(m₁, N₂) = gcd(m₂, N₁) = 1. Moreover we have a Γ₀(N)-invariant decomposition as follows:

(1) Q°D,N,β=⋃m=m1m2m1m2>0gcd(m1,m2)=1Q°D,N,β,m1,m2.

1. The inverse image [pN₂, q, r/N₂] of any primitive form [p¯,q¯,r¯] of discriminant D under the 1-1 correspondence in (1) is obtained by solving the following equations:

p=p¯s2+q¯su+r¯u2

q=2p¯st+q¯(sv+tu)+2r¯uv

r=p¯t2+q¯tv+r¯v2.

satisfying p ≡ 0(mod N₁), q ≡ β(mod 2N), r ≡ 0(mod N₂) and stuv∈Γ(1).

1. we have the following Γ₀(N)-invariant decomposition:

(2) QD,N,β=⋃l>0l2|D⋃λ(2N)lλ≡β(2N)λ2≡D/l2(4N)lQ°D/l2,N,λ.

Suppose Q ≥ 5. Since W_Q has a non-cuspidal fixed point on X₀(N), then W_Q is given by an elliptic element, that is,

WQ=QxyNz−Qw.

Then

(3)τ=2Qx+−4Q2Nz

is a point fixed by W_Q. Conversely, every point fixed by W_Q has the form (3).

We note that each fixed point in (3) can be considered as the Hegneer point of a quadratic form [Nz, −2Qx, −y]. So, if we can find Γ₀(N)-inequivalent quadratic forms [Nz, −2Qx, −y] (by using Proposition 2.2), then we can produce Γ₀(N)-inequivalent points which are fixed points as in (3).

Regarding the computation of points of X₀(N) fixed by W_Q, we can follow the next algorithms:

Algorithm 2.3.

[14] The following steps implement as algorithm to find Γ₀(N)-inequivalent points fixed by W_Q where Q ≠ N:

• Step I We search β(mod 2N) such that β² ≡−4Q(mod 4N) with β ≡−2Qx(mod 2N) where x∈Z.

• Step II We set the decomposition as in (1) and (2) with D = −4Q.

• Step III For each factor in the decomposition in Step II, we find the quadratic form representations and taking the inverse of reduced form under the map which is described in Proposition 2.2(2).

• Step IV We form the elliptic elements corresponding to quadratic form representations obtained in Step III and find their Heegner points.

Algorithm 2.4.

[12] When Q = N, the four steps above come as the following:

• Step I Set (Q, β) = (4N, 0) or (Q, β) = (N, N) when (N ≡ 3(mod 4)).

• Step II Starting from a reduced form Qred, we first find a quadratic form [a, b, c] which in SL2(Z)-inquivalent with Qred and gcd(a, N) = 1.

• Step III Set [A,B,C]=[a,b,c]◦K−110 where K is a solution to the linear congruence equation 2aX + b ≡−β(mod 2N). Then [A, B, C] belongs to QQ,N,β.

• Step IV Let τ=−B+−Q2A. Then Γ₀(N)τ gives a point fixed by W_N.

3. Weierstrass points

In this section, we have computed Weierstrass points on X₀(N) for N ≤ 50 fixed by all the partial and the full Atkin–Lehner involutions in three cases:

1. Modular curves of genus g₀(N) ≤ 1.

2. Hyperelliptic modular curves.

3. Modular curves for N = 34, 38, 42, 43, 44, 45.

The number n of Weierstrass points is finite and satisfies

2g+2≤n≤g3−g,

with n = 2g + 2 if and only if X is hyperelliptic.

Next theorems help us to find Weierstrass points on modular curves X₀(N).

Theorem 3.1.

(Schoeneberg). [16] Let X be a Riemann surface of genus g ≥ 2. Let P be a point fixed by an automorphism T of X, of order p > 1, let g^T be the genus of X^T = X/(T) . If gT≠⌊gp⌋, the greatest integer of gp, then P is a Weierstrass point of X.

Theorem 3.2.

[17] Let X be a Riemann surface of genus g ≥ 2. Let T be an automorphism with 5 or more fixed points. Then, each fixed point is a Weierstrass point.

Theorem 3.3.

[17] If P is not a Weierstrass point and T(P) = P, then there are at least two and at most four points fixed by T and the genus g^T of X^T = X/(T) is given by gT=⌊gp⌋, the greatest integer of gp. Writing g = g^Tp + r there are only three possible cases:

1. r = 0, g = g^Tp, v(T) = 2.

2. r=p−12,g=(gT+12)p−12,v(T)=3.

3. r = p − 1, g = (g^T + 1)p − 1, v(T) = 4.

where v(T) is the number of points fixed by T.

Theorem 3.4.

[12] The points fixed by W_N for N ≤ 50 are Weierstrass points on X₀(N) with g₀(N) > 1 except possibly for the following values

N=22,28,30,33,34,37,40,42,43,45,46,48.

First, only a finite number of Weierstrass points can exist on X₀(N), and if g₀(N) ≤ 1, then are no such points at all. So we have the following theorem:

Theorem 3.5.

The modular curves X₀(N) for N = 1 − 21, 24, 25, 27, 32, 36, 49 have no Weierstrass points.

Second, Let g₀(N) ≥ 2 and X₀(N) be hyperelliptic modular curves. Then there are 19 values of N, which belong to the set

{22,23,26,28,29,30,31,33,35,37,39,40,41,46,47,48,50,59,71}.

Lewittes [17] proved that if X₀(N) is a hyperelliptic modular curve, then any involution on X₀(N) either has no fixed points or has only non Weierstrass fixed points or is the hyperelliptic involution. Jeon [11] found all Weierstrass points on the hyperelliptic modular curves X₀(N) fixed by the hyperelliptic involution. So we have the following theorem:

Theorem 3.6.

If X₀(N) is a hyperelliptic modular curve of genus g₀(N) ≥ 2, then only 2g₀(N) + 2 points fixed by the hyperelliptic involution are Weierstrass points on X₀(N).

Third, in this case, we will study the modular curves X₀(N) for N = 34, 38, 42, 43, 44, 45.

Theorem 3.7.

Let X₀(N) be bielliptic modular curves for N = 34, 43, 45. Then, all points fixed by any bielliptic involution W_Q are not Weierstrass points.

Proof: Since W_Q is a bielliptic involution of X₀(N) of genus 3, it has 4 = 2g₀(N) − 2 points fixed by W_Q on X₀(N). And g0Q(N)=⌊g0(N)2⌋=1, thus by theorems 3.3 and 3.4, each of these points is not a Weierstrass point.

Theorem 3.8.

The modular curves X₀(N) for N = 38, 42, 44 have Weierstrass points fixed by some W_Q.

Proof: Since W₁₉ is a bielliptic involution of X₀(38) of genus 4, it has six points fixed by W₁₉ on X₀(38). So, by theorem 3.2, all these points are Weierstrass points (similarly X038(38),X014(42),X011(44),X044(44)). While g02(38)=⌊g0(38)2⌋=2. So, by theorem 3.3, the modular curve X₀(38) has non Weierstrass points fixed by W₂ (similarly X03(42),X06(42),X021(42),X042(42),X04(44)). Finally, the modular curve X₀(42) has no points fixed by W₂ and W₇. Therefore, X₀(42) has no Weierstrass points fixed by W₂ and W₇.

Now we will give an example by using Proposition 2.2 and Algorithm 2.3 to find Weierstrass points on X₀(44) fixed by W₁₁.

Example 3.9.

Consider X₀(44) which is of genus 4. Since W₁₁ is a bielliptic involution on X₀(44) [18], it has six fixed points on X₀(44). Applying Step I and Step II we have D = −44 and β ≡ ±22, 66(mod 176). First consider the case of β = 20, then we have decomposition as follows:

Q−44,44,22=Q°−44,44,22=Q°−44,44,22,1,1.

We know that Q°−44/Γ(1)={[1,0,11],[3,2,4],[3,−2,4]}. Applying Step III we obtain by taking the inverse image of reduced forms under the map which is described in Proposition 2.2(2) the following forms:

Q°−44,44,22,1,1/Γ(1)={[132,22,1],[44,22,3]}.

Next, consider the case of β = −22, by the same way, we obtain the following forms:

Q°−44,44,−22,1,1/Γ(1)={[132,−22,1],[44,−22,3]}.

When β = 66, we have the following

Q°−44,44,66,1,1/Γ(1)={[1100,66,1],[44,66,25]}.

Moreover in Step IV, the corresponding elliptic elements are given as follows:

W1=11−1132−11,W2=11−344−11,W3=−11−113211,W4=−11−34411,W5=33−11100−33,W6=33−2544−33.

Then Weierstrass points (fixed points) are:

τ1=−112+−11132,τ2=−14+−1144,τ3=112+−11132,τ4=14+−1144,τ5=−3100+−111100,τ6=−34+−1144.

In next example, we will use Algorithm 2.4 to find Weierstrass points on X₀(38) fixed by W₃₈.

Example 3.10.

Consider X₀(38) which is of genus 4. Since W₃₈ is a bielliptic involution on X₀(38) [18], it has six fixed points. Applying Step I and Step II, we have (Q, β) = (152, 0) and

Q−152red={[1,0,38],[2,0,19],[3,2,13],[3,−2,13],[6,4,7],[6,−4,7]}.

we find quadratic forms [a, b, c] which is SL2(Z)-equivalent with Q−152red and gcd(a, 38) = 1 as follows (respectively):

[a,b,c]={[1,0,38],[21,−38,19],[3,2,13],[3,−2,13],[7,10,9],[7,18,17]}.

Applying Step III we have:

[1,0,38]=[1,0,38]◦0−111,[741,−76,2]=[21,−38,19]◦19−111,[1938,−152,3]=[3,2,13]◦25−111,[546,−80,3]=[3,−2,13]◦13−111,[2737,−256,6]=[7,10,9]◦21−111,[1297,−176,6]=[7,18,17]◦15−111.

From Step IV, the Weierstrass points are (respectively):

τ1=−3838,τ2=239+−38741,τ3=251+−381938,τ4=20273+−38546,τ5=1282737+−382737,τ6=881297+−381297.

We list in Table 1 Weierstrass points on X₀(N) for N ≤ 50 fixed by the Atkin–Lehner involutions. We have used Maple and Wolfram Mathematica for the numerical computations:

Table 1

Weierstrass points on X₀(N) for N ≤ 50 by W_Q

N	g0(N)	WQ	g0Q(N)	v(Q)	Weierstrass points
1 − 21	≤1	None
22	2	W2, W22	1	2	None
		W11	0	6	−12+−1122,±16+−1166,±14+−1144,−112+−11132
23	2	W23	0	6	−2323,12+−2346,±13+−2369,±14+−2392
24 − 25	≤1	None
26	2	W2, W13	1	2	None
		W26	0	6	−2626,±13+−2678,±25+−26130,−531+−26806
27	1	none
28	2	W4, W28	1	2	None
		W7	0	6	±14+−728,±18+−756,±38+−756
29	2	W29	0	6	−2929,12+−2958,±13+−2987,±15+−29145
30	3	W5, W6, W30	1	4	None
		W2, W3, W10	2	0	None
		W15	0	8	−12+−1530,±14+−1560,±16+−15120,−38+−15120,−116+−15240,−334+−15510
31	2	W31	0	6	−3131,12+−3162,±14+−31124,±25+−31155
32	1	None
33	3	W3	2	0	None
		W11	0	8	±13+−1133,±16+−1166,±29+−1199,±112+−11132
		W33	1	4	None
34	3	W2, W17	1	4	None
		W34
35	3	W5	1	4	None
		W7	2	0	None
		W35	0	8	−3535,±14+−35140,−16+−35210,±112+−35420,−512+−35420,−118+−35630
36	1	none
37	2	W37	1	2	none
38	4	W2	2	2	none
		W19	1	6	±12+−7638,±14+−76152,±110+−76380
		W38	1	6	−3838,239+−38741,251+−381938,20273+−38546,1282737+−382737,881297+−381297
39	3	W3	1	4	none
		W13	2	0	none
		W39	0	8	−3939,±14+−39156,±18+−39312,−38+−39312,−516+−39624,−120+−39780
40	3	W5, W8	2	0	none
		W40	1	4	none
		−101−12010	0	8	±14+−520,±112+−560,±512+−560,±536+−5180
41	3	W41	0	8	−4141,12+−4182,±13+−41123,±25+−41205,±16+−41246
42	5	W2, W7	3	0	none
		W3, W6	2	4	none
		W21, W42
		W14	1	8	±13+−1442,±115+−14210,257+−14798,−475+−141050,−4225+−143150,−23+−1442
43	3	W43	1	4	none
44	4	W4	2	2	none
		W11	1	6	±112+−11132,±14+−1144,−34+−1144,−3100+−111100
		W44	1	6	−4444,52687+−44687,259+−442596,115+−44660,4141+−446204,19+−44396
45	3	W5, W9, W45	1	4	none
46	5	W2	3	0	none
		W23	0	12	−12+−2346,±14+−2392,±16+−23138,±18+−23184,±38+−23184,±112+−23276,−124+−23552
		W46	2	4	none
47	4	W47	0	10	−4747,12+−4794,±13+−47141,±14+−47188,±16+−47282,±27+−47329
48	3	W3, W16	2	0	none
		W48	1	4	none
		−6−1−486	0	8	±18+−324,±38+−324,±356+−3168,±1756+−3168
49	1	none
50	2	W2, W25	1	2	none
		W50	0	6	−210,±311+−2110,±117+−2170,−743+−2430