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

Purpose –The authors have determined whether the points fixed by all the full and the partial Atkin–Lehner involutions WQ 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 andHurwitz 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 concreteway. Finally, the set ofWeierstrass points is useful in studying arithmetic and geometric properties of X0(N).


Introduction
Let H be the complex upper half plane and Γ be a congruence subgroup of the full modular group SL 2 ðZÞ. Denote by X(Γ) the modular curve obtained from compactification of the quotient space ΓnH by adding finitely many points called cusps. Then X(Γ) is a compact Riemann surface.
For each positive integer N, we have a subgroup Γ 0 (N) of SL 2 ðZÞ defined by: and let X 0 (N) 5 X 0 (Γ(N)). A modular curve X 0 (N) of genus g ≥ 2 is called hyperelliptic (respectively bielliptic) if it admits a map f: X → C of degree 2 onto a curve C of genus 0 (respectively 1). A point P of X 0 (N) is a Weierstrass point if there exists a non-constant function f on X 0 (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 X 0 (N) for N 5 4n, 9n, and Atkin [2] has given conditions for the case of N 5 p 2 n 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 X 0 (N) for certain N. Also, Ono [7] and Rohrlich [8] have studied Weierstrass points on X 0 (p) for some primes p. And Choi [9] has shown that the cusp 1 2 is a Weierstrass point of Γ 1 (4p) when p is a prime > 7. In addition, Jeon [10,11] has computed all Weierstrass points on the hyperelliptic curves X 1 (N) and X 0 (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 X 0 (N) are Weierstrass points and have determined whether the points fixed by the full Atkin-Lehner involution on X 0 (N) are Weierstrass points or not. In this paper, we have determined which of the points fixed by W Q on X 0 (N) are Weierstrass points and found Weierstrass points on modular curves X 0 (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 X 0 (N).

Points fixed by the Atkin-Lehner involutions
For each divisor QjN with ðQ; N Q Þ ¼ 1, consider the matrices of the form

Qx y Nz Qw with
x; y; z; w ∈ Z and determinant Q. Then each of these matrices defines a unique involution on X 0 (N), which is called the Atkin-Lehner involution and denoted by W Q . In particular, if Q 5 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 X Q 0 ðNÞ be the quotient space of X 0 (N) by W Q . Let g 0 (N) and g Q 0 ðNÞ be the genus of X 0 (N) and X Q 0 ðN Þ respectively. Then g Q 0 ðN Þ is computed by the Riemann-Hurwitz formula as follows: where v(Q) 5 v(Q; N) is the number of points on X 0 (N) fixed by W Q . It is given by: is the Kronecker symbol and the functions c i (p) are defined as follows: for i 5 1, 2, if p ≠ 2 and Qu3ðmod 4Þ; Now, we recall the algorithms for finding Γ 0 (N)-inequivalent points fixed by W Q on X 0 (N) [14]. For a negative integer D congruent to 0 or 1 modulo 4, we denote by Q D the set of positive definite integral binary quadratic forms: Let Q D 8 ⊂ Q D be the subset of primitive forms, that is, Q D 8df½p; q; r ∈ Q D ; gcdðp; q; rÞ ¼ 1g: Then Γ(1) also acts on Q D 8. As is well known [15], there is a 1-1 correspondence between the set of classes Γð1ÞnQ D 8 and the set of reduced primitive definite forms.
(3) we have the following Γ 0 (N)-invariant decomposition: Suppose Q ≥ 5. Since W Q has a non-cuspidal fixed point on X 0 (N), then W Q is given by an elliptic element, that is, is a point fixed by W Q . Conversely, every point fixed by W Q has the form (3).

AJMS
We note that each fixed point in (3) can be considered as the Hegneer point of a quadratic form [Nz, À2Qx, Ày]. So, if we can find Γ 0 (N)-inequivalent quadratic forms [Nz, À2Qx, Ày] (by using Proposition 2.2), then we can produce Γ 0 (N)-inequivalent points which are fixed points as in (3).
Regarding the computation of points of X 0 (N) fixed by W Q , we can follow the next algorithms: The following steps implement as algorithm to find Γ 0 (N)-inequivalent points fixed by W Q where Q ≠ N: Step I We search β(mod 2N) such that β 2 ≡À4Q(mod 4N) with β ≡À2Qx(mod 2N) where x ∈ Z.
Step II We set the decomposition as in (1) and (2) with D 5 À4Q.
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.
Step II Starting from a reduced form Q red , we first find a quadratic form [a, b, c] which in SL 2 ðZÞ-inquivalent with Q red and gcd(a, N) 5 1.

Weierstrass points
In this section, we have computed Weierstrass points on X 0 (N) for N ≤ 50 fixed by all the partial and the full Atkin-Lehner involutions in three cases: (1) Modular curves of genus g 0 (N) ≤ 1.
(2) Hyperelliptic modular curves. The number n of Weierstrass points is finite and satisfies 2g þ 2 ≤ n ≤ g 3 À g; with n 5 2g þ 2 if and only if X is hyperelliptic. Next theorems help us to find Weierstrass points on modular curves X 0 (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 5 X/(T) .
If g T ≠ " g p # , the greatest integer of g p , then P is a Weierstrass point of X.

Weierstrass points on modular curves
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) 5 P, then there are at least two and at most four points fixed by T and the genus g T of X T 5 X/(T) is given by g T ¼ " g p # , the greatest integer of g p . Writing g 5 g T p þ r there are only three possible cases: (1) r 5 0, g 5 g T p, v(T) 5 2.
where v(T) is the number of points fixed by T. Lewittes [17] proved that if X 0 (N) is a hyperelliptic modular curve, then any involution on X 0 (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 X 0 (N) fixed by the hyperelliptic involution. So we have the following theorem: Theorem 3.6. If X 0 (N) is a hyperelliptic modular curve of genus g 0 (N) ≥ 2, then only 2g 0 (N) þ 2 points fixed by the hyperelliptic involution are Weierstrass points on X 0 (N).
Theorem 3.7. Let X 0 (N) be bielliptic modular curves for N 5 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 X 0 (N) of genus 3, it has 4 5 2g 0 (N) À 2 points fixed by W Q on X 0 (N). And g Q 0 ðN Þ ¼ ). Finally, the modular curve X 0 (42) has no points fixed by W 2 and W 7 . Therefore, X 0 (42) has no Weierstrass points fixed by W 2 and W 7 . Now we will give an example by using Proposition 2.2 and Algorithm 2.3 to find Weierstrass points on X 0 (44) fixed by W 11 . Example 3.9. Consider X 0 (44) which is of genus 4. Since W 11 is a bielliptic involution on X 0 (44) [18], it has six fixed points on X 0 (44