Weierstrass points on $X_0^+(p)$ and supersingular $j$-invariants

TL;DR

This paper investigates the arithmetic properties of Weierstrass points on the modular curves $X_0^+(p)$ and establishes a relationship with supersingular elliptic curve $j$-invariants in characteristic $p$, revealing new insights into their interplay.

Contribution

It introduces a novel connection between Weierstrass points on $X_0^+(p)$ and supersingular $j$-invariants, advancing understanding of modular curve arithmetic.

Findings

Weierstrass points are related to supersingular $j$-invariants.

Established a new relationship between geometric and arithmetic properties.

Enhanced understanding of modular curve structures in characteristic $p$.

Abstract

We study the arithmetic properties of Weierstrass points on the modular curves $X_0^+(p)$ for primes $p$. In particular, we obtain a relationship between the Weierstrass points on $X_0^+(p)$ and the $j$-invariants of supersingular elliptic curves in characteristic $p$.

Tables

Table 1. Table 1. Factors arising from elliptic points

Equations

$$0=\text{mbox}{ord}_Q({\omega }_1)<\text{mbox}{ord}_Q({\omega }_2)<\cdots <\text{mbox}{ord}_Q({\omega }_g),$$

$$\mathrm{wt}(Q):=\sum _{j=1}^g(\text{mbox}{ord}_Q({\omega }_j)-j+1).$$

$$\sum _{Q\in M}\mathrm{wt}(Q)=g^3-g.$$

$$\Gamma_{0}(N):=\left\{\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in\Gamma:c\equiv 0\pmod{N}\right\}.$$

$$F_{pM}(x):=\prod _{Q\in Y_0(N)}(x-j(Q){)}^{\mathrm{wt}(Q)},$$

$$S_p(x):={\prod _{E/{\overline{\mathbb{F}}}_p}}_{\mathrm{supersingular}}(x-j(E))\in {\mathbb{F}}_p[x],$$

$$F_p(x)\equiv S_p(x{)}^{g_p(g_p-1)}(\mathrm{mod}p).$$

$${\tilde{S}}_p(x):={\prod _{E/{\overline{\mathbb{F}}}_p\mathrm{supersingular}}}_{j(E)\ne 0,1728}(x-j(E)).$$

$$\displaystyle v(Q):=\begin{cases}2&\mbox{if $w_{p}(Q)=Q$},\\ 1&\mbox{otherwise,}\end{cases}$$

$${\mathcal{F}}_p(x):=\prod _{Q\in Y_0(p)}(x-j(Q){)}^{v(Q)\mathrm{wt}(\overline{Q})},$$

$$S_p^{(l)}(x):={\prod _{E/{\overline{\mathbb{F}}}_p\mathrm{supersingular}}}_{j(E)\in {\mathbb{F}}_p}(x-j(E))\text{mbox}and{S}_p^{(q)}(x):={\prod _{E/{\overline{\mathbb{F}}}_p\mathrm{supersingular}}}_{j(E)\in {\mathbb{F}}_{p^2}\backslash {\mathbb{F}}_p}(x-j(E)),$$

$${\mathcal{F}}_p(x)\equiv S_p^{(q)}(x{)}^{g_p^{+}(g_p^{+}-1)}\cdot H(x{)}^2(\mathrm{mod}p).$$

$$E_k(z):=1-\frac{2k}{B_k}\sum _{n=1}^{\infty }{\sigma }_{k-1}(n)q^n,$$

$$\Delta (z):=\frac{E_4(z{)}^3-E_6(z{)}^2}{1728}=q-24q^2+252q^3-1472q^4+\cdots$$

$$\displaystyle\widetilde{E}_{k}(z):=\begin{cases}1&\mbox{if $k\equiv 0\pmod{12}$},\\ E_{4}(z)^{2}E_{6}(z)&\mbox{if $k\equiv 2\pmod{12}$},\\ E_{4}(z)&\mbox{if $k\equiv 4\pmod{12}$},\\ E_{6}(z)&\mbox{if $k\equiv 6\pmod{12}$},\\ E_{4}(z)^{2}&\mbox{if $k\equiv 8\pmod{12}$},\\ E_{4}(z)E_{6}(z)&\mbox{if $k\equiv 10\pmod{12}$},\end{cases}$$

$$\displaystyle m(k):=\begin{cases}\lfloor k/12\rfloor&\mbox{if $k\not\equiv 2\pmod{12}$},\\ \lfloor k/12\rfloor-1&\mbox{if $k\equiv 2\pmod{12}$}.\end{cases}$$

$$\tilde{F}(f,j(z)):=\frac{f(z)}{\Delta (z{)}^{m(k)}{\tilde{E}}_k(z)}$$

$$\displaystyle\widetilde{F}(f^{2},x)=\begin{cases}\widetilde{F}(f,x)^{2}&\mbox{if $k\equiv 0\pmod{12}$},\\ x(x-1728)\widetilde{F}(f,x)^{2}&\mbox{if $k\equiv 2\pmod{12}$},\\ \widetilde{F}(f,x)^{2}&\mbox{if $k\equiv 4\pmod{12}$},\\ (x-1728)\widetilde{F}(f,x)^{2}&\mbox{if $k\equiv 6\pmod{12}$},\\ x\widetilde{F}(f,x)^{2}&\mbox{if $k\equiv 8\pmod{12}$},\\ (x-1728)\widetilde{F}(f,x)^{2}&\mbox{if $k\equiv 10\pmod{12}$}.\end{cases}$$

$$f(z{)}^2=\Delta (z{)}^{2m(k)}{\tilde{E}}_k(z{)}^2\tilde{F}(f,j(z){)}^2,$$

$$f(z{)}^2=\Delta (z{)}^{m(2k)}{\tilde{E}}_{2k}(z)\tilde{F}(f^2,j(z)).$$

$$\tilde{F}(f^2,j(z))=\Delta (z{)}^{2m(k)-m(2k)}\cdot \frac{{\tilde{E}}_k(z{)}^2}{{\tilde{E}}_{2k}(z)}\cdot \tilde{F}(f,j(z){)}^2.$$

$$\displaystyle\widetilde{F}(f^{2},j(z))=\begin{cases}\widetilde{F}(f,j(z))^{2}&\mbox{if $k\equiv 0\pmod{12}$},\\ \Delta(z)^{-2}E_{4}(z)^{3}E_{6}(z)^{2}\widetilde{F}(f,j(z))^{2}&\mbox{if $k\equiv 2\pmod{12}$},\\ \widetilde{F}(f,j(z))^{2}&\mbox{if $k\equiv 4\pmod{12}$},\\ \Delta(z)^{-1}E_{6}(z)^{2}\widetilde{F}(f,j(z))^{2}&\mbox{if $k\equiv 6\pmod{12}$},\\ \Delta(z)^{-1}E_{4}(z)^{3}\widetilde{F}(f,j(z))^{2}&\mbox{if $k\equiv 8\pmod{12}$},\\ \Delta(z)^{-1}E_{6}(z)^{2}\widetilde{F}(f,j(z))^{2}&\mbox{if $k\equiv 10\pmod{12}$},\end{cases}$$

$$\text{mbox}End(E)\cong \{\begin{array}{ll}{\mathcal{O}}_{4p}& \text{mbox}ifp\equiv 1(\mathrm{mod}4),\\ {\mathcal{O}}_p\text{mbox}or{\mathcal{O}}_{4p}& \text{mbox}ifp\equiv 3(\mathrm{mod}4).\end{array}$$

$$H_p(x):={\prod _{\tau \in {\Gamma }_0(p)\backslash \mathbb{H}}}_{v(Q_{\tau })=2}(x-j(\tau )),$$

$$H_p(x)=\{\begin{array}{ll}{\mathcal{H}}_{4p}(x)& \text{mbox}ifp\equiv 1(\mathrm{mod}4),\\ {\mathcal{H}}_p(x)\cdot {\mathcal{H}}_{4p}(x)& \text{mbox}ifp\equiv 3(\mathrm{mod}4).\end{array}$$

$$H_p(x)\equiv S_p^{(l)}(x{)}^2(\mathrm{mod}p).$$

$$2g_p^{+}=g_p+1-\frac{\sigma }{2},$$

$$2g_p^{+}=g_p+1-\mathrm{deg}(S_p^{(l)}(x)).$$

$$\mathrm{deg}(T(x))=\frac{\mathrm{deg}(H_p(x))}{2}=\mathrm{deg}(S_p^{(l)}(x)).$$

$$f_1(z)$$

Full text

1. Introduction

A Weierstrass point on a compact Riemann surface $M$ of genus $g$ is a point $Q\in M$ at which some holomorphic differential $\omega$ vanishes to order at least $g$. Weierstrass points can be identified by observing their weight. Let $\mathcal{H}^{1}(M)$ be the $\mathbb{C}$-vector space of holomorphic differentials on $M$ of dimension $g$. If $\{\omega_{1},\omega_{2},\dots,\omega_{g}\}$ forms a basis for $\mathcal{H}^{1}(M)$ adapted to $Q\in M$, so that

[TABLE]

then we define the Weierstrass weight of $Q$ to be

[TABLE]

We see that $\mathrm{wt}(Q)>0$ if and only if $Q$ is a Weierstrass point of $M$. The Weierstrass weight is independent of the choice of basis, and it is known that

[TABLE]

Hence each Riemann surface of genus $g\geq 2$ must have Weierstrass points. For these and other facts, see Section III.5 of [9].

We will consider Weierstrass points on modular curves, a class of Riemann surfaces which are of wide interest in number theory. Let $\mathbb{H}$ denote the complex upper half-plane. The modular group $\Gamma:=\mathrm{SL}_{2}(\mathbb{Z})$ acts on $\mathbb{H}$ by linear fractional transformations $\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)z=\frac{az+b}{cz+d}$. If $N\geq 1$ is an integer, then we define the congruence subgroup

[TABLE]

The quotient of the action of $\Gamma_{0}(N)$ on $\mathbb{H}$ is the Riemann surface $Y_{0}(N):=\Gamma_{0}(N)\backslash\mathbb{H}$, and its compactification is $X_{0}(N)$. The modular curve $X_{0}(N)$ can be viewed as the moduli space of elliptic curves equipped with a level $N$ structure. Specifically, the points of $X_{0}(N)$ parameterize isomorphism classes of pairs $(E,C)$ where $E$ is an elliptic curve over $\mathbb{C}$ and $C$ is a cyclic subgroup of $E$ of order $N$.

Weierstrass points on $X_{0}(N)$ have been studied by a number of authors (see, for example, [14], [5], [6], [19], [16], [21], [22], [12], [13], [3], [4], and [10]). An interesting open question is to determine those $N$ for which the cusp $\infty$ is a Weierstrass point. Lehner and Newman [14] and Atkin [5] showed that $\infty$ is a Weierstrass point for most non-squarefree $N$, while Atkin [6] proved that $\infty$ is not a Weierstrass point when $N$ is prime.

Most central to the present paper is the connection between Weierstrass points and supersingular elliptic curves. Ogg [19] showed that for modular curves $X_{0}(pM)$ where $p$ is a prime with $p\nmid M$ and with the genus of $X_{0}(M)$ equal to [math], the Weierstrass points of $X_{0}(pM)$ occur at points whose underlying elliptic curve is supersingular when reduced modulo $p$. So in particular, $\infty$ is not a Weierstrass point in these cases, extending [6]. This has recently been confirmed by Ahlgren, Masri and Rouse [2] using a non-geometric proof. Ahlgren and Ono [3] showed for the $M=1$ case that in fact all supersingular elliptic curves modulo $p$ correspond to Weierstrass points of $X_{0}(p)$, and they demonstrated a precise correspondence between the two sets. In order to state their result, we make the following definitions.

For $p$ and $M$ as above, let

[TABLE]

where $j(z)=q^{-1}+744+196884q+\cdots$ is the usual elliptic modular function defined on $\Gamma$, and $j(Q)=j(\tau)$ for any $\tau\in\mathbb{H}$ with $Q=\Gamma_{0}(N)\tau$. This is the divisor polynomial for the Weierstrass points of $Y_{0}(N)$. Next, for a prime $p$ we define

[TABLE]

where the product is over all $\overline{\mathbb{F}}_{p}$-isomorphism classes of supersingular elliptic curves. It is well known that $S_{p}(x)$ has degree $g_{p}+1$, where $g_{p}$ is the genus of $X_{0}(p)$. Ahlgren and Ono [3] proved the following.

Theorem 1.1**.**

If $p$ is prime, then $F_{p}(x)$ has $p$-integral rational coefficients and

[TABLE]

El-Guindy [8] generalized Theorem 1.1 to those cases where $M$ is squarefree, showing that $F_{pM}(x)$ has $p$-integral rational coefficients and is divisible by $\widetilde{S}_{p}(x)^{\mu(M)g_{pM}(g_{pM}-1)}$, where $\mu(M):=[\Gamma:\Gamma_{0}(M)]$ and $g_{pM}$ is the genus of $X_{0}(pM)$, and where

[TABLE]

He also gave an explicit factorization of $F_{pM}(x)$ in most cases where $M$ is prime. Generalizing Theorem 1.1 in a different direction, Ahlgren and Papanikolas [4] gave a similar result for higher order Weierstrass points on $X_{0}(p)$, which are defined in relation to higher order differentials.

In this paper we consider the modular curve $X_{0}^{+}(p)$, the quotient space of $X_{0}(p)$ under the action of the Atkin-Lehner involution $w_{p}$, which maps $\tau\mapsto-1/p\tau$ for $\tau\in\mathbb{H}$. There is a natural projection map $\pi:X_{0}(p)\to X_{0}^{+}(p)$ which sends a point $Q\in X_{0}(p)$ to its equivalence class $\pi(Q)=\overline{Q}$ in $X_{0}^{+}(p)$. This is a 2-to-1 mapping, ramified at those points $Q\in X_{0}(p)$ that remain fixed by $w_{p}$. Therefore we set

[TABLE]

so that $v(Q)$ is equal to the multiplicity of the map $\pi$ at $Q$. We now define a divisor polynomial for the Weierstrass points of $X_{0}^{+}(p)$. We will set our product to be over $X_{0}(p)$ rather than $X_{0}^{+}(p)$ to preserve the desired $p$-integrality of the coefficients. Let

[TABLE]

where $\mathrm{wt}(\overline{Q})$ is the Weierstrass weight of the image $\overline{Q}$ of $Q$ in $X_{0}^{+}(p)$. The zeros of this polynomial capture those non-cuspidal points of $X_{0}(p)$ which map to Weierstrass points in $X_{0}^{+}(p)$. The two cusps of $X_{0}(p)$ at 0 and $\infty$ are interchanged by $w_{p}$, so that $X_{0}^{+}(p)$ has a single cusp at $\infty$, which may or may not be a Weierstrass point. Atkin checked all primes $p\leq 883$ and conjectured that $\infty$ is a Weierstrass point for all $p>389$. Stein has confirmed this for all $p<3000$, and his table of results can be found in [25]. Therefore $\mathcal{F}_{p}(x)$ is a polynomial of degree $2((g_{p}^{+})^{3}-g_{p}^{+}-\mathrm{wt}(\infty))$, where $g^{+}_{p}$ is the genus of $X_{0}^{+}(p)$.

Recalling that the supersingular polynomial $S_{p}(x)$ factors over $\mathbb{F}_{p}[x]$ into linear and irreducible quadratic factors, we separate these factors by defining

[TABLE]

so that $S_{p}(x)=S_{p}^{(l)}(x)\cdot S_{p}^{(q)}(x)$. Our main theorem gives an analogue of Theorem 1.1 for $\mathcal{F}_{p}(x)$. We require an assumption that $\mathcal{H}^{1}(X_{0}^{+}(p))$ has a good basis, a condition about $p$-integrality which we define later in Section 4. Computations suggest that most, if not all, such spaces satsify this condition. Indeed, each $\mathcal{H}^{1}(X_{0}^{+}(p))$ with $p<3200$ has a good basis.

Theorem 1.2**.**

Let $p$ be prime and suppose that $\mathcal{H}(X_{0}^{+}(p))$ has a good basis. Then $\mathcal{F}_{p}(x)$ has $p$-integral rational coefficients, and there exists a polynomial $H(x)\in\mathbb{F}_{p}[x]$ such that

[TABLE]

Note*.*

From computational evidence, it appears that $H(x)$ is always coprime to $S_{p}(x)$, so that contrary to the situation on $X_{0}(p)$, only those supersingular points with quadratic irrational $j$-invariants correspond to Weierstrass points of $X_{0}^{+}(p)$. We give a heuristic argument for this phenomenon in Section 3.

In Section 2 we start by reviewing some preliminary facts about divisors of polynomials of modular forms. We then consider the reduction of $X_{0}(p)$ modulo $p$ in Section 3 in order to obtain a key result about the $w_{p}$-fixed points of $X_{0}(p)$. In Section 4 we describe our good basis condition for $\mathcal{H}^{1}(X_{0}^{+}(p))$. Next, in Section 5 we derive a special cusp form on $\Gamma_{0}(p)$ which encodes the Weierstrass weights of points on $X_{0}^{+}(p)$. In Section 6, we prove Theorem 1.2, and in Section 7, we demonstrate Theorem 1.2 for the curve $X_{0}^{+}(67)$.

2. Divisor polynomials of modular forms

Let $M_{k}$ (resp. $M_{k}(p)$) denote the space of modular forms of weight $k$ on $\Gamma$ (resp. $\Gamma_{0}(p)$), and let $S_{k}$ (resp. $S_{k}(p)$) be the subspace of cusp forms. For even $k\geq 4$, the Eisenstein series $E_{k}\in M_{k}$ is defined as

[TABLE]

where $B_{k}$ is the $k$th Bernoulli number, and $\sigma_{k-1}(n)=\sum_{d\mid n}d^{k-1}$. Then the function

[TABLE]

is the unique normalized cusp form in $S_{12}$.

We briefly recall how to build a divisor polynomial whose zeros are exactly the $j$-values at which a given modular form $f\in M_{k}$ vanishes, excluding those zeros that may occur at the elliptic points $i$ and $\rho:=e^{2\pi i/3}$ (for details, see [3] or Section 2.6 of [20]). We define

[TABLE]

and

[TABLE]

Now let $f\in M_{k}$ have leading coefficient 1. Then we note that (2.1) and (2.2) are defined such that the quotient

[TABLE]

is a polynomial in $j(z)$. Therefore, we define $\widetilde{F}(f,x)$ to be the unique polynomial in $x$ satisfying (2.3). Furthermore, if $f$ has $p$-integral rational coefficients, then so does $\widetilde{F}(f,x)$.

Finally, we record a result about the divisor polynomial of the square of a modular form.

Lemma 2.1**.**

Let $f\in M_{k}$. Then

[TABLE]

Proof.

Using (2.3) for both $f$ and $f^{2}$ yields

[TABLE]

and

[TABLE]

Thus

[TABLE]

Then by (2.1) and (2.2) we have

[TABLE]

Since $j(z)=\frac{E_{4}(z)^{3}}{\Delta(z)}$ and $j(z)-1728=\frac{E_{6}(z)^{2}}{\Delta(z)}$, the result follows. ∎

3. Modular curves modulo $p$

Here we recall the undesingularized reduction of $X_{0}(p)$ modulo $p$, due to Deligne and Rapoport [7]. The description below closely follows one given by Ogg [18]. The model of $X_{0}(p)$ modulo $p$ consists of two copies of $X_{0}(1)$ which meet transversally in the supersingular points (Figure 1).

The Atkin-Lehner operator $w_{p}$ is compatible with this reduction. It gives an isomorphism between the two copies of $X_{0}(1)$ which preserves the supersingular locus, by fixing the points corresponding to supersingular curves defined over $F_{p}$, and interchanging those defined over $\mathbb{F}_{p^{2}}\backslash\mathbb{F}_{p}$ with their conjugates. Therefore, dividing out by the action of $w_{p}$ glues together the two copies of $X_{0}(1)$. The singularities at the linear supersingular points are thus resolved, while the conjugate pairs of quadratic supersingular points are glued together. This results in a model for the reduction modulo $p$ of $X_{0}^{+}(p)$ consisting of one copy of $X_{0}(1)$ which self-intersects at each point representing a pair of conjugate quadratic supersingular points (Figure 2). This resolution at the linear supersingular points may explain their absence among the Weierstrass points of $X_{0}^{+}(p)$.

To make the correspondence between fixed points and linear supersingular $j$-invariants more precise, let $\mathcal{O}_{D}=\mathbb{Z}[\frac{1}{2}(D+\sqrt{-D})]$ be the order of the imaginary quadratic field $\mathbb{Q}[\sqrt{-D}]$ with discriminant $-D<0$. The Hilbert class polynomial $\mathcal{H}_{D}(x)\in\mathbb{Z}[x]$ is the monic polynomial whose zeros are exactly the $j$-invariants of the distinct isomorphism classes of elliptic curves with complex multiplication by $\mathcal{O}_{D}$, and its degree is $h(-D)$, the class number of $\mathcal{O}_{D}$.

The points $Q\in Y_{0}(p)$ that are fixed by $w_{p}$ correspond to pairs $(E,C)$ such that $E$ admits complex multiplication by $\sqrt{-p}$, or in other words, $\mathbb{Z}[\sqrt{-p}]$ embeds in $\mbox{End}(E)$, the endomorphism ring of $E$ over the complex numbers (see e.g. [16]). Since $\mbox{End}(E)$ must be an order in an imaginary quadratic field, we have

[TABLE]

Now define

[TABLE]

the monic polynomial whose zeros are precisely the $j$-invariants of the $w_{p}$-fixed points of $Y_{0}(p)$. Then we have

[TABLE]

The following result is due independently to Kaneko and Zagier.

Proposition 3.1**.**

For $p$ prime, there exists a monic polynomial $T(x)\in\mathbb{Z}_{p}[x]$ with distinct roots such that $H_{p}(x)\equiv T(x)^{2}\pmod{p}$.

Proof.

The result follow from Kronecker’s relations on the modular equation $\Phi_{p}(X,Y)$, and may be found in the appendix of [11].∎

We can now prove the following.

Theorem 3.2**.**

Let $p$ be prime. Then we have

[TABLE]

Proof.

By the discussion above, each zero of $H_{p}(x)$ is of the form $j(E)$ where $E$ is a supersingular elliptic curve defined over $\mathbb{F}_{p}$. Then by Proposition 3.1, $T(x)\mid S_{p}^{(l)}(x)$. We will show that $T(x)$ and $S_{p}^{(l)}(x)$ have the same degree, proving that $T(x)=S_{p}^{(l)}(x)$. The result then follows again by Proposition 3.1.

By the Riemann-Hurwitz formula (see, for example, Section I.2 of [9]), we have

[TABLE]

where $\sigma$ is the number of points of $X_{0}(p)$ at which the projection $\pi:X_{0}(p)\to X_{0}^{+}(p)$ is ramified, or in other words, the number of $w_{p}$-fixed points of $X_{0}(p)$. We note that the cusps are not ramified since $w_{p}$ exchanges [math] and $\infty$, so $\sigma=\mathrm{deg}(H_{p}(x))$. On the other hand, Ogg explains in [17] that $g_{p}^{+}$ is equal to the number of conjugate pairs of supersingular $j$-invariants in $\mathbb{F}_{p^{2}}\backslash\mathbb{F}_{p}$. Since there are $g_{p}+1$ total supersingular $j$-invariants, we have

[TABLE]

Then Proposition 3.1, (3.2) and (3.3) imply that

[TABLE]

∎

4. A Good Basis for $\mathcal{H}^{1}(X_{0}^{+}(p))$

For ease of notation we will let $g:=g_{p}^{+}$ for the rest of the paper, and assume that $g\geq 2$. Recall that $g$ is the dimension of $\mathcal{H}^{1}(X_{0}^{+}(p))$, the space of holomorphic 1-forms on $X_{0}^{+}(p)$. Let $\{\omega_{1},\omega_{2},\dots,\omega_{g}\}$ be a basis of $\mathcal{H}^{1}(X_{0}^{+}(p))$, where $\omega_{i}=h_{i}(u)du$ for some local variable $u$. In order to take advantage of the correspondence that exists between holomorphic 1-forms on $X_{0}(p)$ and weight 2 cusp forms of level $p$, we pull back each $\omega_{i}$ to a holomorphic 1-form $\pi^{*}\omega_{i}$ on $X_{0}(p)$ via the projection map $\pi:X_{0}(p)\to X_{0}^{+}(p)$ (see, for example, Chapter 2 of [15]). We can choose a local coordinate $z$ at $Q\in X_{0}(p)$ so that near $Q$, $u=z^{n}$, where $n$ is the multiplicity of $\pi$ at $\mathbb{Q}$, hence $n=v(Q)$ (1.2). Then we have $\pi^{*}\omega_{i}=H_{i}(z)dz$ with $H_{i}(z)=h_{i}(z^{n})nz^{n-1}\in S_{2}(p)$. Since each $H_{i}(z)$ has been pulled back from $X_{0}^{+}(p)$, it must be invariant under $w_{p}$, so it is a member of $S_{2}^{+}(p)$, the subspace of $w_{p}$-invariant cusp forms of weight $2$. In fact, it is straightforward to show that $\{H_{1}(z),H_{2}(z),\dots,H_{g}(z)\}$ forms a basis for $S_{2}^{+}(p)$.

It will be helpful later on to specify a basis for $S_{2}^{+}(p)$ of a particularly nice form. First, we can guarantee a basis with rational Fourier coefficients by the following argument. The space $S_{2}(p)$ has a basis consisting of newforms. Let $f(z)=\sum_{n}a(n)q^{n}$ be a newform for $S_{2}(p)$, and let $\sigma\in\mbox{Gal}(\mathbb{C}/\mathbb{Q})$. Then $f^{\sigma}(z)=\sum_{n}\sigma(a(n))q^{n}$ is also a newform for $S_{2}(p)$, so the action of $\mbox{Gal}(\mathbb{C}/\mathbb{Q})$ partitions the newforms into Galois conjugacy classes. If two newforms are Galois conjugates then they share the same eigenvalue for $w_{p}$. Let $V_{f}$ be the $\mathbb{C}$-vector space spanned by the Galois conjugates of $f$. Standard Galois-theoretic arguments show that $V_{f}$ has a basis consisting of cusp forms with rational coefficients. These are no longer newforms, but as they are linear combinations of the Galois conjugates of $f$, they are still eigenforms for $w_{p}$. Therefore collecting such a basis for each Galois conjugacy class with eigenvalue $1$ for $w_{p}$ yields a basis for $S_{2}^{+}(p)$ with rational Fourier coefficients.

We can determine such a basis $\{f_{1},f_{2},\dots,f_{g}\}$ uniquely by requiring that

[TABLE]

where

[TABLE]

Definition*.*

We say that $\mathcal{H}^{1}(X_{0}^{+}(p))$ has a good basis if the cusp forms $f_{1},f_{2},\dots,f_{g}$ satisfying (4.1) and (4.2) have $p$-integral Fourier coefficients.

5. Wronskians and $p$-integrality

Given any basis $\{\omega_{1},\omega_{2},\dots,\omega_{g}\}$ for $\mathcal{H}^{1}(X_{0}^{+}(p))$ with $\omega_{i}=h_{i}(u)du$, we define the Wronskian

[TABLE]

Let $\mathcal{W}^{+}(u)$ be the scalar multiple of $W(h_{1},h_{2},\dots,h_{g})(u)$ with leading coefficient 1, so that $\mathcal{W}^{+}(u)$ is independent of the choice of basis. It is well-known that the Wronskian encodes the Weierstrass weights of points in $X_{0}^{+}(p)$ (see [9], page 82). Specifically,

[TABLE]

Since it is advantageous to work on $X_{0}(p)$ instead of $X_{0}^{+}(p)$, we consider the pullback of $W^{+}:=\mathcal{W}^{+}(u)(du)^{g(g+1)/2}$ to $X_{0}(p)$ via $\pi$, which is $\pi^{*}W^{+}=\mathcal{W}^{+}(z^{n})(nz^{n-1}dz)^{g(g+1)/2}$. Recalling that $n=v(Q)$ when $z$ is near $Q$, we have

[TABLE]

Alternatively, we could pull back each $\omega_{i}$ individually to $\pi^{*}\omega_{i}=H_{i}(z)dz$ as in Section 4. Then we can form the Wronskian $W(H_{1},H_{2},\dots,H_{g})(z)$ (defined analogously to (5.5)). Since the $H_{i}$ are cusp forms of weight $2$ for $\Gamma_{0}(p)$ then $W(H_{1},H_{2},\dots,H_{g})(z)$ is a cusp form of weight $g(g+1)$ for $\Gamma_{0}(p)$. It can be shown using basic facts about determinants that

[TABLE]

Now let $\mathcal{W}_{p}(z)$ be the multiple of $W(H_{1},H_{2},\dots,H_{g})(z)$ with leading coefficient 1. Then $\mathcal{W}_{p}(z)$ is independent of the choice of basis for $S_{2}^{+}(p)$, and we have $\mathcal{W}_{p}(z)(dz)^{g(g+1)/2}=\pi^{*}W^{+}$, hence by (5.6),

[TABLE]

We next see the advantage of having a good basis for $\mathcal{H}^{1}(X_{0}^{+}(p))$.

Theorem 5.1**.**

Let $p$ be a prime such that $\mathcal{H}^{1}(X_{0}^{+}(p))$ has a good basis. Then $\mathcal{W}_{p}(z)\in S_{g(g+1)}(p)$ has $p$-integral rational coefficients.

Proof.

Here we closely follow the proof of Lemma 3.1 in [4]. Let $\{f_{1},f_{2},\dots,f_{g}\}$ be a basis for $S_{2}^{+}(p)$ satisfying (4.1) and (4.2). Let $\theta:=q\frac{d}{dq}$ be the usual differential operator for modular forms, so that $\frac{d}{dz}=2\pi i\theta$. Then by properties of determinants, we have

[TABLE]

We see that the Fourier expansion of $\left(\frac{1}{2\pi i}\right)^{g(g-1)/2}W(f_{1},f_{2},\dots,f_{g})$ has rational $p$-integral coefficients, with leading coefficient given by the Vandermonde determinant

[TABLE]

It now suffices to show that $p$ does not divide the leading coefficient. By Sturm’s bound [26] for the order of vanishing modulo $p$ for modular forms on $\Gamma_{0}(p)$, we have $1\leq c_{i}\leq\frac{p+1}{6}<p$ for each $1\leq i\leq g$, so $1\leq c_{k}-c_{j}\leq p-1$ for all $j<k$. Therefore the lemma is proved. ∎

6. Proof of the Main Theorem

Let $p$ be a prime for which $\mathcal{H}^{1}(X_{0}^{+}(p))$ has a good basis. We note that when $g<2$, there are no Weierstrass points on $X_{0}^{+}(p)$. Then $\mathcal{F}_{p}(x)=1$ and $g^{2}-g=0$, so the theorem holds trivially by taking $H(x)=1$. Thus from here on we will assume that $g\geq 2$, in which case we have $p\geq 67$.

We first adapt two lemmas from [3]. For any meromorphic function $f(z)$ defined on $\mathbb{H}$ and any integer $k$, we define the slash operator $\mid_{k}$ by

[TABLE]

where $\gamma:=\left(\begin{smallmatrix}a&b\\ c&d\end{smallmatrix}\right)$ is a real matrix with positive determinant, and $\gamma z:=\frac{az+b}{cz+d}$. In particular, the Atkin-Lehner involution $w_{p}$ is given by $f\mapsto f|_{k}\left(\begin{smallmatrix}0&-1\\ p&0\end{smallmatrix}\right)$ when $f$ is a modular form of weight $k$.

Lemma 6.1**.**

We have

[TABLE]

Proof.

The proof is identical to Lemma 3.2 of [3] except that $f|_{2}\left(\begin{smallmatrix}0&-1\\ p&0\end{smallmatrix}\right)=f$ for every newform $f$ in $S_{2}^{+}(p)$.∎

Lemma 6.2**.**

If $p$ is a prime such that $X_{0}^{+}(p)$ has genus at least 2, define

[TABLE]

normalized to have leading coefficient 1. Then $\widetilde{\mathcal{W}}_{p}(z)$ is a cusp form of weight $g(g+1)(p+1)$ on $\Gamma$ with $p$-integral rational coefficients, and

[TABLE]

Proof.

This follows from our Lemma 6.1 exactly as Lemma 3.3 follows from Lemma 3.2 in [3].∎

We again consider a basis $\{f_{1},f_{2},\dots,f_{g}\}$ for $S_{2}^{+}(p)$ satisfying (4.1) and (4.2). For each $f_{i}$, there is a cusp form $b_{i}\in S_{p+1}$ with $p$-integral rational coefficients for which $f_{i}\equiv b_{i}\pmod{p}$ ([4], Theorem 4.1(c)). Define $W(z)$ to be the multiple of $W(b_{1},b_{2},\dots,b_{g})$ with leading coefficient 1. By the same reasoning as in Theorem 5.1, $\left(\frac{1}{2\pi i}\right)^{g(g-1)/2}W(b_{1},b_{2},\dots,b_{g})$ has $p$-integral rational coefficients and leading coefficient $V$ (5.8). Since the differential operator $\theta$ preserves congruences, we have

[TABLE]

and hence

[TABLE]

Since $V$ is coprime to $p$, then by Lemma 6.2 we have

[TABLE]

We now have two cusp forms $\widetilde{\mathcal{W}}_{p}(z)$ and $W(z)^{2}$ on the full modular group, but $\widetilde{\mathcal{W}}_{p}(z)$ has weight $\tilde{k}(p):=g(g+1)(p+1)$ while $W(z)^{2}$ has weight $2g(g+p)$. Using the fact that the Eisenstein series $E_{p-1}(z)\equiv 1\pmod{p}$, we have

[TABLE]

where the cusp forms on each side of the congruence in (6.1) have the same weight $\tilde{k}(p)$. By (2.3) there exist polynomials $\widetilde{F}(\widetilde{\mathcal{W}}_{p}(x),x)$ and $\widetilde{F}(W^{2}E_{p-1}^{g^{2}-g},x)$ with $p$-integral rational coefficients such that

[TABLE]

and

[TABLE]

Then by (6.1), we conclude that

[TABLE]

We next compute each side of (6.2). To compute the right hand side, we begin with the following.

Lemma 6.3**.**

(Theorem 2.3 in [3])* For a prime $p\geq 5$ and $f\in M_{k}$ with $p$-integral coefficients, we have*

[TABLE]

where

[TABLE]

Then using Lemma 6.3 inductively, we have

[TABLE]

where

[TABLE]

A case-by-case computation reveals that

[TABLE]

By a result of Deligne (see [23]), and recalling (1.1), we have

[TABLE]

and therefore

[TABLE]

Next we evaluate the left-hand side of (6.2). We recall here the definitions

[TABLE]

and

[TABLE]

Theorem 6.4**.**

Let $p$ be a prime such that the genus of $X_{0}^{+}(p)$ is at least 2. Define $\epsilon_{p}(i)$ and $\epsilon_{p}(\rho)$ by

[TABLE]

and

[TABLE]

where $k^{*}\in\{0,1,2\}$ with $k^{*}\equiv\tilde{k}(p)\pmod{3}$. Then we have

[TABLE]

Proof.

If $\tau_{0}\in\mathbb{H}$ and $A\in\Gamma$, then

[TABLE]

so that

[TABLE]

Now recall by (5.7) that for $Q\in Y_{0}(p)$, we have

[TABLE]

Let $\ell_{\tau}\in\{1,2,3\}$ be the order of the isotropy subgroup of $\tau$ in $\Gamma_{0}(p)/\{\pm I\}$, where $\tau$ is an elliptic fixed point if and only if $\ell(\tau)\neq 1$. If $Q_{\tau}\in Y_{0}(p)$ is associated to $\tau\in\mathbb{H}$ in the usual way then we have

[TABLE]

If $\tau_{0}$ is not equivalent to $i$ or $\rho$ under $\Gamma$, then $\{A(\tau_{0})\}_{A\in\Gamma_{0}(p)\backslash\Gamma}$ consists of $p+1$ points which are $\Gamma_{0}(p)$-inequivalent, so by (6.4) and (6.5),

[TABLE]

When $\tau_{0}\stackrel{{\scriptstyle\Gamma}}{{\sim}}\rho$, then $\mathrm{ord}_{\tau_{0}}(\widetilde{\mathcal{W}}_{p}(z))=\mathrm{ord}_{\rho}(\widetilde{\mathcal{W}}_{p}(z))$, and $\{A(\rho)\}_{A\in\Gamma_{0}(p)\backslash\Gamma}$ contains $1+\left(\frac{-3}{p}\right)$ elliptic fixed points of order 3 which are $\Gamma_{0}(p)$-inequivalent, and $p-\left(\frac{-3}{p}\right)$ additional points which are partitioned into $\Gamma_{0}(p)$-orbits of size 3. Then by (6.5) we have

[TABLE]

When $\tau_{0}\stackrel{{\scriptstyle\Gamma}}{{\sim}}i$, then $\mathrm{ord}_{\tau_{0}}(\widetilde{\mathcal{W}}_{p}(z))=\mathrm{ord}_{i}(\widetilde{\mathcal{W}}_{p}(z))$, and $\{A(i)\}_{A\in\Gamma_{0}(p)\backslash\Gamma}$ contains $1+\left(\frac{-1}{p}\right)$ elliptic fixed points of order 2 which are $\Gamma_{0}(p)$-inequivalent, and $p-\left(\frac{-1}{p}\right)$ additional points which are partitioned into $\Gamma_{0}(p)$-orbits of size 2. We then have

[TABLE]

Finally, we recall that $j(z)$ vanishes to order 3 at $z=\rho$, that $j(z)-1728$ vanishes to order 2 at $z=i$, and that $j(z)-j(\tau_{0})$ vanishes to order 1 at all other points $\tau_{0}\in\Gamma\backslash\mathbb{H}$. Therefore the exponent of $x-j(\tau_{0})$ in $\widetilde{F}(\widetilde{\mathcal{W}}_{p},x)$ is equal to

[TABLE]

Therefore by (3.1), (6.6), (6.7) and (6.8) we have

[TABLE]

∎

Combining (6.2), (6.3), Theorem 3.2 and Theorem 6.4 now yields

[TABLE]

We next define

[TABLE]

In the chart below, we compare certain factors appearing in (6.9) for each choice of $p$ modulo $12$.

Since both $\lceil\frac{g^{2}-g}{3}\rceil$ and $\frac{g^{2}-g}{2}$ are less than $g^{2}+g$, we see from Table 1 that $\mathcal{G}_{p}(x)$ always divides $S_{p}^{(l)}(x)^{g^{2}+g}$. Then since $x$ and $(x-1728)$ are coprime to $\widetilde{S}_{p}(x)$ we have

[TABLE]

where the two quotients reduce to polynomials.

Now on the left, we write $S_{p}^{(l)}(x)=x^{\alpha_{p}(\rho)}(x-1728)^{\alpha_{p}(i)}\widetilde{S}_{p}^{(l)}(x)$ with $\alpha_{p}(\rho),\alpha_{p}(i)\in\{0,1\}$ according to $p$ modulo 12, as in Table 1. On the right, we write $\widetilde{S}_{p}(x)=\widetilde{S}_{p}^{(l)}(x)S^{(q)}(x)$. Then (6.10) becomes

[TABLE]

Now the quotient on the left of (6.11) must divide $\widetilde{F}(W^{2},x)$. Then cancelling $\widetilde{S}^{(l)}_{p}(x)^{g^{2}-g}$ on each side leaves $\widetilde{S}_{p}^{(l)}(x)^{2g}$ on the left, which must then divide $\widetilde{F}(W^{2},x)$ as well. So (6.11) becomes

[TABLE]

where $H_{1}(x)$ is the polynomial given in non-reduced form by the quotient

[TABLE]

It remains to show that $H_{1}(x)$ is a perfect square. By Lemma 2.1, we write $\widetilde{F}(W^{2},x)=x^{\delta_{p}(\rho)}(x-1728)^{\delta_{p}(i)}\widetilde{F}(W,x)^{2}$, where $\delta_{p}(\rho),\delta_{p}(i)\in\{0,1\}$ according to $g(g+p)$ modulo 12. We then decompose $H_{1}(x)$ into a product of two quotients,

[TABLE]

Note that the exponents in the right-hand quotient are all even. The quotient on the left is of the form $x^{a}(x-1728)^{b}$, where $a$ and $b$ are integers, possibly negative. It is sufficient to show that $a$ and $b$ are both even. An examination of the exponents reveals that the parity of $a$ and $b$ depend only on $p$ and $g$ modulo 12. A check of all possible combinations of these values using Table 1 and Lemma 2.1 confirms that $a$ and $b$ are indeed even in all cases, and therefore we can write $H_{1}(x)=H(x)^{2}$ for some polynomial $H(x)\in\mathbb{F}_{p}$. This concludes the proof of Theorem 1.2. ∎

7. The example for $X_{0}^{+}(67)$

Here we compute $\mathcal{F}_{67}(x)$, the divisor polynomial corresponding to the modular curve $X_{0}^{+}(67)$, which has genus 2. A basis for $S_{2}^{+}(67)$ is given by $\{f_{1},f_{2}\}$, with

[TABLE]

and

[TABLE]

The associated Wronskian is

[TABLE]

Then by Lemma 6.2 and (2.3), we have

[TABLE]

But $\epsilon_{67}(i)=0$, $\epsilon_{67}(\rho)=4$, and

[TABLE]

Therefore by Theorem 6.4 we have

[TABLE]

Note*.*

In general, $H(x)$ may not be irreducible.

8. Acknowledgements

The author gratefully acknowledges Scott Ahlgren for his invaluable mentoring and for suggesting this problem in the first place.