Open Access

Weierstrass points on and supersingular j-invariants

Research in the Mathematical Sciences20174:25

https://doi.org/10.1186/s40687-017-0115-z

Received: 14 February 2017

Accepted: 11 July 2017

Published: 6 December 2017

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.

Keywords

Weierstrass pointsModular curvesSupersingular elliptic curvesModular forms

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 g-dimensional \(\mathbb {C}\)-vector space of holomorphic differentials on M. If \(\{\omega _1,\omega _2, \dots , \omega _g\}\) forms a basis for \(\mathcal {H}^1(M)\) adapted to \(Q\in M\), so that
then we define the Weierstrass weight of Q to be
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
$$\begin{aligned} \sum _{Q\in M}\mathrm {wt}(Q)=g^3-g. \end{aligned}$$
Hence, each Riemann surface of genus \(g\ge 2\) 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 \(\left( {\begin{matrix}a&{}b\\ c&{}d\end{matrix}}\right) z=\frac{az+b}{cz+d}\). If \(N\ge 1\) is an integer, then we define the congruence subgroup
$$\begin{aligned} \Gamma _0(N):=\left\{ \left( \begin{array}{cc}a&{}b\\ c&{}d\end{array}\right) \in \Gamma :c\equiv 0\pmod {N}\right\} . \end{aligned}$$
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 (EC) 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, [36, 12, 13, 15, 17, 20, 22, 23], and [10]). An interesting open question is to determine those N for which the cusp \(\infty \) is a Weierstrass point. Lehner and Newman [15] 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 [20] showed that for modular curves \(X_0(pM)\) where p is a prime with \(p\not \mid M\) and with the genus of \(X_0(M)\) equal to 0, 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
$$\begin{aligned} F_{pM}(x):={\prod _{Q\in Y_0(pM)}}(x-j(Q))^{\mathrm {wt}(Q)}, \end{aligned}$$
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 \(Q=\Gamma _0(pM)\tau \). This is the divisor polynomial for the Weierstrass points of \(Y_0(pM)\). Next, for a prime p we define
$$\begin{aligned} S_p(x):=\mathop {\prod _{E/\overline{\mathbb {F}}_p}}_{\mathrm {supersingular}} (x-j(E))\in \mathbb {F}_p[x], \end{aligned}$$
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, when \(M=1\).

Theorem 1.1

If p is prime, then \(F_p(x)\) has p-integral rational coefficients and
$$\begin{aligned} F_p(x)\equiv S_p(x)^{g_p(g_p-1)}\pmod {p}. \end{aligned}$$
El-Guindy [8] generalized Theorem 1.1 by considering \(F_{pM}\) 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
$$\begin{aligned} \widetilde{S}_p(x):=\mathop {\prod _{E/\overline{\mathbb {F}}_p\;\mathrm {supersingular}}}_{j(E)\ne 0,1728} (x-j(E)). \end{aligned}$$
(1.1)
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 \(\pi :X_0(p)\rightarrow X_0^+(p)\) 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
$$\begin{aligned} v(Q):={\left\{ \begin{array}{ll}2&{}\text{ if } w_p(Q)=Q,\\ 1&{}\text{ otherwise, }\end{array}\right. } \end{aligned}$$
(1.2)
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 \(Y_0(p)\) to preserve the desired p-integrality of the coefficients. Let
$$\begin{aligned} \mathcal {F}_p(x):=\prod _{Q\in Y_0(p)}(x-j(Q))^{v(Q)\mathrm {wt}(\overline{Q})}, \end{aligned}$$
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 \(p\le 883\) 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 [26]. 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 \(X_0^+(p)\).
We recall that a supersingular elliptic curve \(E/\overline{\mathbb {F}}_p\) must have \(j(E)\in \mathbb {F}_{p^2}\). Since those \(j(E)\in \mathbb {F}_{p^2}\backslash \mathbb {F}_p\) occur in conjugate pairs, we define
$$\begin{aligned} S_p^{(l)}(x):=\mathop {\prod _{E/\overline{\mathbb {F}}_p\;\mathrm {supersingular}}}_{j(E)\in \mathbb {F}_p} (x-j(E))\quad \text{ and } \quad S_p^{(q)}(x):=\mathop {\prod _{E/\overline{\mathbb {F}}_p\;\mathrm {supersingular}}}_{j(E)\in \mathbb {F}_{p^2}\backslash \mathbb {F}_p} (x-j(E)), \end{aligned}$$
so that \(S_p(x)=S_p^{(l)}(x)\cdot S_p^{(q)}(x)\) and both factors lie in \(\mathbb {F}_p[x]\). 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 Sect. 4. Computations suggest that most, if not all, such spaces satisfy 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
$$\begin{aligned} \mathcal {F}_p(x)\equiv S_p^{(q)}(x)^{g_p^+(g_p^+-1)}\cdot H(x)^2\pmod {p}. \end{aligned}$$

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 Sect. 3.

In Sect. 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 Sect. 3 in order to obtain a key result about the \(w_p\)-fixed points of \(X_0(p)\). In Sect. 4 we describe our good basis condition for \(\mathcal {H}^1(X_0^+(p))\). Next, in Sect. 5 we derive a special cusp form on \(\Gamma _0(p)\) which encodes the Weierstrass weights of points on \(X_0^+(p)\). In Sect. 6, we prove Theorem 1.2, and in Sect. 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 \(k\ge 4\), the Eisenstein series \(E_k\in M_k\) is defined as
$$\begin{aligned} E_k(z):=1-\frac{2k}{B_k}\sum _{n=1}^\infty \sigma _{k-1}(n)q^n, \end{aligned}$$
where \(B_k\) is the kth Bernoulli number, and \(\sigma _{k-1}(n)=\sum _{d\mid n}d^{k-1}\). Then the function
$$\begin{aligned} \Delta (z):=\frac{E_4(z)^3-E_6(z)^2}{1728}=q-24q^2+252q^3-1472q^4+\cdots \end{aligned}$$
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 trivial zeros that are forced to occur at the elliptic points i and \(\rho :=e^{2\pi i/3}\) by the valence formula (for details, see [3] or Section 2.6 of [21]). We define
$$\begin{aligned} \widetilde{E}_k(z):={\left\{ \begin{array}{ll} 1&{}\text{ if } k\equiv 0\pmod {12},\\ E_4(z)^2E_6(z)&{}\text{ if } k\equiv 2\pmod {12},\\ E_4(z)&{}\text{ if } k\equiv 4\pmod {12},\\ E_6(z)&{}\text{ if } k\equiv 6\pmod {12},\\ E_4(z)^2&{}\text{ if } k\equiv 8\pmod {12},\\ E_4(z)E_6(z)&{}\text{ if } k\equiv 10\pmod {12}, \end{array}\right. } \end{aligned}$$
(2.1)
and
$$\begin{aligned} m(k):={\left\{ \begin{array}{ll} \lfloor k/12\rfloor &{}\text{ if } k\not \equiv 2\pmod {12},\\ \lfloor k/12\rfloor -1&{}\text{ if } k\equiv 2\pmod {12}. \end{array}\right. } \end{aligned}$$
(2.2)
Now let \(f\in M_k\) have leading coefficient 1. We note that (2.1) and (2.2) are defined such that the quotient
$$\begin{aligned} \widetilde{F}(f,j(z)):=\frac{f(z)}{\Delta (z)^{m(k)}\widetilde{E}_k(z)} \end{aligned}$$
(2.3)
has weight zero. Then the order of f at the elliptic points, together with the non-vanishing of \(\Delta (z)\) on \(\mathbb {H}\), guarantees that \(\widetilde{F}(f,j(z))\) 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
$$\begin{aligned} \widetilde{F}(f^2,x)={\left\{ \begin{array}{ll} \widetilde{F}(f,x)^2&{}{{if }\;\; k\equiv 0\pmod {12}},\\ x(x-1728)\widetilde{F}(f,x)^2&{}{{if }\; \;k\equiv 2\pmod {12}},\\ \widetilde{F}(f,x)^2&{}{{if }\;\; k\equiv 4\pmod {12}},\\ (x-1728)\widetilde{F}(f,x)^2&{}{{if }\;\; k\equiv 6\pmod {12}},\\ x\widetilde{F}(f,x)^2&{}{{if } \;\;k\equiv 8\pmod {12}},\\ (x-1728)\widetilde{F}(f,x)^2&{} {{if } \;\;k\equiv 10\pmod {12}}. \end{array}\right. } \end{aligned}$$

Proof

Using (2.3) for both f and \(f^2\) yields
$$\begin{aligned} f(z)^2=\Delta (z)^{2m(k)}\widetilde{E}_k(z)^2\widetilde{F}(f,j(z))^2, \end{aligned}$$
and
$$\begin{aligned} f(z)^2=\Delta (z)^{m(2k)}\widetilde{E}_{2k}(z)\widetilde{F}(f^2,j(z)). \end{aligned}$$
Thus
$$\begin{aligned} \widetilde{F}(f^2,j(z))=\Delta (z)^{2m(k)-m(2k)}\cdot \frac{\widetilde{E}_k(z)^2}{\widetilde{E}_{2k}(z)}\cdot \widetilde{F}(f,j(z))^2. \end{aligned}$$
Then by (2.1) and (2.2) we have
$$\begin{aligned} \widetilde{F}(f^2,j(z))={\left\{ \begin{array}{ll} \widetilde{F}(f,j(z))^2&{}\text{ if } k\equiv 0\pmod {12},\\ \Delta (z)^{-2}E_4(z)^3E_6(z)^2\widetilde{F}(f,j(z))^2&{}\text{ if } k\equiv 2\pmod {12},\\ \widetilde{F}(f,j(z))^2&{}\text{ if } k\equiv 4\pmod {12},\\ \Delta (z)^{-1}E_6(z)^2\widetilde{F}(f,j(z))^2&{}\text{ if } k\equiv 6\pmod {12},\\ \Delta (z)^{-1}E_4(z)^3\widetilde{F}(f,j(z))^2&{}\text{ if } k\equiv 8\pmod {12},\\ \Delta (z)^{-1}E_6(z)^2\widetilde{F}(f,j(z))^2&{}\text{ if } k\equiv 10\pmod {12}, \end{array}\right. } \end{aligned}$$
Since \(j(z)=\frac{E_4(z)^3}{\Delta (z)}\) and \(j(z)-1728=\frac{E_6(z)^2}{\Delta (z)}\), the result follows. \(\square \)

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 [19]. The model of \(X_0(p)\) modulo p consists of two copies of \(X_0(1)\) which meet transversally in the supersingular points (Fig. 1). (Here we call a point supersingular if its underlying elliptic curve is supersingular.)
Fig. 1

Reduction of \(X_0(p)\)

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 those points with j-invariant in \(\mathbb {F}_p\), and interchanging the pairs of points whose j-invariants in \(\mathbb {F}_{p^2}\backslash \mathbb {F}_p\) are conjugate. 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 (Fig. 2). This resolution at the linear supersingular points may explain their absence among the Weierstrass points of \(X_0^+(p)\).
Fig. 2

Reduction of \(X_0^+(p)\)

To make the correspondence between fixed points and linear supersingular j-invariants more precise, for \(D\equiv 0,3\pmod {4}\), 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 (EC) such that E admits complex multiplication by \(\sqrt{-p}\), or in other words, \(\mathbb {Z}[\sqrt{-p}]\) embeds in \(\text{ End }(E)\), the endomorphism ring of E over the complex numbers (see, e.g., [17]). Since \(\text{ End }(E)\) must be an order in an imaginary quadratic field, we have
$$\begin{aligned} \text{ End }(E)\cong {\left\{ \begin{array}{ll}\mathcal {O}_{4p}&{}\text{ if } p\equiv 1\pmod {4},\\ \mathcal {O}_{p} \text{ or } \mathcal {O}_{4p}&{}\text{ if } p\equiv 3\pmod {4}.\end{array}\right. } \end{aligned}$$
Now define
$$\begin{aligned} H_p(x):=\mathop {\prod _{\tau \in \Gamma _0(p)\backslash \mathbb {H}}}_{v(Q_\tau )=2}(x-j(\tau )), \end{aligned}$$
(3.1)
the monic polynomial whose zeros are precisely the j-invariants of the \(w_p\)-fixed points of \(Y_0(p)\). Then we have
$$\begin{aligned} {\mathbb {H}_p(x)={\left\{ \begin{array}{ll}\mathcal {H}_{4p}(x)&{}\text{ if } p\equiv 1\pmod {4},\\ \mathcal {H}_p(x)\cdot \mathcal {H}_{4p}(x)&{}\text{ if } p\equiv 3\pmod {4}.\end{array}\right. }} \end{aligned}$$
(3.2)
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 follows from Kronecker’s relations on the modular equation \(\Phi _p(X,Y)\) and may be found in appendix of [11]. \(\square \)

We can now prove the following.

Theorem 3.2

Let p be prime. Then we have
$$\begin{aligned} H_p(x)\equiv S_p^{(l)}(x)^2\pmod {p}. \end{aligned}$$

Proof

The prime p is ramified in both \(\mathbb {Q}(\sqrt{-p})\) and \(\mathbb {Q}(\sqrt{-4p})\), so a result of Deuring (see, e.g., Theorem 12 in §13.4 of [14]) together with (3.2) implies that the reduction modulo p of each root of \(H_p(x)\) must be a supersingular j-invariant. Since the roots of \(H_p(x)\) also correspond to fixed points of \(w_p\), these supersingular j-invariants must lie in \(\mathbb {F}_p\), so by Proposition 3.1, we have \(T(x)\mid S_p^{(l)}(x)\). 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
$$\begin{aligned} 2g_p^+=g_p+1-\frac{\sigma }{2}, \end{aligned}$$
(3.3)
where \(\sigma \) is the number of points of \(X_0(p)\) at which the projection \(\pi :X_0(p)\rightarrow X_0^+(p)\) 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 0 and \(\infty \), so \(\sigma =\mathrm {deg}(H_p(x))\). On the other hand, Ogg explains in [18] 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
$$\begin{aligned} 2g_p^+=g_p+1-\mathrm {deg}(S_p^{(l)}(x)). \end{aligned}$$
(3.4)
Then Proposition 3.1, (3.3), and (3.4) imply that
$$\begin{aligned} \mathrm {deg}(T(x))=\frac{\mathrm {deg}(H_p(x))}{2}=\mathrm {deg}(S_p^{(l)}(x)). \end{aligned}$$
\(\square \)

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 \(g\ge 2\). 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 \(\pi :X_0(p)\rightarrow X_0^+(p)\) (see, for example, Chapter 2 of [16]). 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 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 \(\sigma \in \text{ Gal }(\mathbb {C}/\mathbb {Q})\). Then \(f^\sigma (z)=\sum _n \sigma (a(n))q^n\) is also a newform for \(S_2(p)\), so the action of \(\text{ Gal }(\mathbb {C}/\mathbb {Q})\) 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
$$\begin{aligned} f_1(z)&=q^{c_1}+O(q^{c_g+1})\\ f_2(z)&=q^{c_2}+O(q^{c_g+1})\nonumber \\&\vdots \nonumber \\ f_g(z)&=q^{c_g}+O(q^{c_g+1})\nonumber \end{aligned}$$
(4.1)
where
$$\begin{aligned} c_1<c_2<\dots <c_g. \end{aligned}$$
(4.2)

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
$$\begin{aligned} W(h_1,h_2,\dots ,h_g)(u):= \left| \begin{array}{cccc}h_1&{}h_2&{}\cdots &{}h_g\\ h_1'&{}h_2'&{}\cdots &{}h_g'\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ h_1^{(g-1)}&{}h_2^{(g-1)}&{}\cdots &{}h_g^{(g-1)} \end{array}\right| . \end{aligned}$$
(5.1)
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,
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
(5.2)
Alternatively, we could pull back each \(\omega _i\) individually to \(\pi ^*\omega _i=H_i(z)dz\) as in Sect. 4. Then we can form the Wronskian \(W(H_1,H_2,\dots ,H_g)(z)\) (defined analogously to (5.1)). 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
$$\begin{aligned} W(H_1,H_2,\dots ,H_g)(z)(dz)^{g(g+1)/2} =W(h_1,h_2,\dots ,h_g)(z^n)(nz^{n-1}dz)^{g(g+1)/2}. \end{aligned}$$
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 (5.2),
(5.3)
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
$$\begin{aligned} W(f_1,f_2,\dots ,f_g)=(2\pi i)^{g(g-1)/2}\left| \begin{array}{cccc}f_1&{}f_2&{}\cdots &{}f_g\\ \theta f_1&{}\theta f_2&{}\cdots &{}\theta f_g\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ \theta f_1^{(g-1)}&{}\theta f_2^{(g-1)}&{}\cdots &{}\theta f_g^{(g-1)} \end{array}\right| . \end{aligned}$$
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
$$\begin{aligned} V:=\left| \begin{array}{cccc}1&{}1&{}\cdots &{}1\\ c_1&{}c_2&{}\cdots &{}c_g\\ \vdots &{}\vdots &{}\vdots &{}\vdots \\ c_1^{(g-1)}&{}c_2^{(g-1)}&{}\cdots &{}c_g^{(g-1)} \end{array}\right| =\prod _{1\le j<k\le g} (c_k-c_j). \end{aligned}$$
(5.4)
It now suffices to show that p does not divide the leading coefficient. By Sturm’s bound [27] for the order of vanishing modulo p for modular forms of weight 2 on \(\Gamma _0(p)\), we have \(1\le c_i\le \frac{p+1}{6}<p\) for each \(1\le i\le g\), so \(1\le c_k-c_j \le p-1\) for all \(j<k\). Therefore, the lemma is proved. \(\square \)

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 \(g\ge 2\), in which case we have \(p\ge 67\).

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
$$\begin{aligned} f(z)|_k\gamma :=(\det \gamma )^{k/2}(cz+d)^{-k}f(\gamma z), \end{aligned}$$
where \(\gamma :=\left( {\begin{matrix}a&{}b\\ c&{}d\end{matrix}}\right) \) 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 \(f\mapsto f|_k \left( {\begin{matrix}0&{}-1\\ p&{}0\end{matrix}}\right) \) when f is a modular form of weight k.

Lemma 6.1

We have
$$\begin{aligned} \mathcal {W}_p(z)|_{g(g+1)}\left( {\begin{matrix}0&{}-1\\ p&{}0\end{matrix}}\right) =\mathcal {W}_p(z). \end{aligned}$$

Proof

The proof is identical to Lemma 3.2 of [3] except that \(f|_2\left( {\begin{matrix}0&{}-1\\ p&{}0\end{matrix}}\right) =f\) for every newform f in \(S_2^+(p)\). \(\square \)

Lemma 6.2

If p is a prime such that \(X_0^+(p)\) has genus at least 2, define
$$\begin{aligned} \widetilde{\mathcal {W}}_p(z):=\prod _{A\in \Gamma _0(p)\backslash \Gamma }\mathcal {W}_p(z)|_{g(g+1)}A, \end{aligned}$$
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
$$\begin{aligned} \widetilde{\mathcal {W}}_p(z)\equiv \mathcal {W}_p(z)^2\pmod {p}. \end{aligned}$$

Proof

This follows from our Lemma 6.1 exactly as Lemma 3.3 follows from Lemma 3.2 in [3]. \(\square \)

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 (5.4). Since the differential operator \(\theta \) preserves congruences, we have
$$\begin{aligned} \left( \frac{1}{2\pi i}\right) ^{g(g-1)/2}W(f_1,f_2,\dots ,f_g)\equiv \left( \frac{1}{2\pi i}\right) ^{g(g-1)/2}W(b_1,b_2,\dots ,b_g)\pmod {p},\end{aligned}$$
and hence
$$\begin{aligned} V\cdot \mathcal {W}_p(z)\equiv V\cdot W(z)\pmod {p}. \end{aligned}$$
Since V is coprime to p, then by Lemma 6.2 we have
$$\begin{aligned} \widetilde{\mathcal {W}}_p(z)\equiv \mathcal {W}_p(z)^2\equiv W(z)^2\pmod {p}. \end{aligned}$$
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
$$\begin{aligned} \widetilde{\mathcal {W}}_p(z)\equiv W(z)^2\cdot E_{p-1}(z)^{g^2-g}\pmod {p}, \end{aligned}$$
(6.1)
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 \(\widetilde{F}(W^2E_{p-1}^{g^2-g},x)\) with p-integral rational coefficients such that
$$\begin{aligned} \widetilde{\mathcal {W}}_p(z)=\Delta (z)^{m(\tilde{k}(p))}\widetilde{E}_{\tilde{k}(p)}(z)\widetilde{F}(\widetilde{\mathcal {W}}_p,j(z)){,} \end{aligned}$$
and
$$\begin{aligned} W(z)^2E_{p-1}(z)^{g^2-g}=\Delta (z)^{m(\tilde{k}(p))}\widetilde{E}_{\tilde{k}(p)}(z)\widetilde{F}(W^2E_{p-1}^{g^2-g},j(z)). \end{aligned}$$
Then by (6.1), we conclude that
$$\begin{aligned} \widetilde{F}(\widetilde{\mathcal {W}}_p,x)\equiv \widetilde{F}(W^2 E_{p-1}^{g^2-g},x)\pmod {p}. \end{aligned}$$
(6.2)
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 \(p\ge 5\) and \(f\in M_k\) with p-integral coefficients, we have
$$\begin{aligned} \widetilde{F}(fE_{p-1},x)\equiv \widetilde{F}(E_{p-1},x)\cdot \widetilde{F}(f,x)\cdot C_p(k;x)\pmod {p} \end{aligned}$$
where
$$\begin{aligned} C_p(k;x):={\left\{ \begin{array}{ll}x&{}{{\text{ if } } (k,p)\equiv (2,5), (8,5), (8,11)\pmod {12}},\\ x-1728&{}{\text{ if } } (k,p)\equiv (2,7), (6,7), (10,7), (6,11), (10,11) \pmod {12},\\ x(x-1728)&{}\text{ if } (k,p)\equiv (2,11)\pmod {12},\\ 1&{}{{\text{ otherwise } }}.\end{array}\right. } \end{aligned}$$
Then using Lemma 6.3 inductively, we have
$$\begin{aligned} \widetilde{F}(W^2\cdot E_{p-1}^{g^2-g},x)\equiv \widetilde{F}(E_{p-1},x)^{g^2-g}\cdot \widetilde{F}(W^2,x)\cdot \mathcal {G}_p(x)\pmod {p}, \end{aligned}$$
where
$$\begin{aligned} \mathcal {G}_p(x):=\prod _{s=1}^{g^2-g}C_p(2g(g+p)+(g^2-g-s)(p-1);x). \end{aligned}$$
A case-by-case computation reveals that
$$\begin{aligned} \mathcal {G}_p(x)={\left\{ \begin{array}{ll}1&{}\text{ if } \,p\equiv 1\pmod {12},\\ x^{\lceil \frac{g^2-g}{3}\rceil }&{}\text{ if } \,p\equiv 5\pmod {12},\\ (x-1728)^{(g^2-g)/2}&{}\text{ if } \,p\equiv 7\pmod {12},\\ x^{\lceil \frac{g^2-g}{3}\rceil }(x-1728)^{(g^2-g)/2}&{}\text{ if } \, p\equiv 11\pmod {12}.\end{array}\right. } \end{aligned}$$
By a result of Deligne (see [24]), and recalling (1.1), we have
$$\begin{aligned} \widetilde{F}(E_{p-1},x)\equiv \widetilde{S}_p(x)\pmod {p}, \end{aligned}$$
and therefore
$$\begin{aligned} \widetilde{F}(W^2 E_{p-1}^{g^2-g},x)\equiv \widetilde{S}_p(x)^{g^2-g}\cdot \widetilde{F}(W^2,x)\cdot \mathcal {G}_p(x)\pmod {p}. \end{aligned}$$
(6.3)
Next, in the following theorem, we evaluate the left-hand side of (6.2). We recall here the definitions
$$\begin{aligned} \mathcal {F}_p(x):=\prod _{Q\in Y_0(p)}(x-j(Q))^{v(Q)\mathrm {wt}(\overline{Q})}, \end{aligned}$$
and
$$\begin{aligned} H_p(x):=\mathop {\prod _{\tau \in \Gamma _0(p)\backslash \mathbb {H}}}_{v(Q_\tau )=2}(x-j(\tau )). \end{aligned}$$

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
$$\begin{aligned} \epsilon _p(i)=\frac{(g^2+g)\left( 1+\left( \frac{-1}{p}\right) \right) }{4}, \end{aligned}$$
and
$$\begin{aligned} \epsilon _p(\rho )=\frac{(g^2+g)\left( 1+\left( \frac{-3}{p}\right) \right) -k^*}{3}, \end{aligned}$$
where \(k^*\in \{0,1,2\}\) with \(k^*\equiv \tilde{k}(p)\pmod {3}\). Then we have
$$\begin{aligned} \widetilde{F}(\widetilde{\mathcal {W}}_p,x)=x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}\mathcal {F}_p(x)H_p(x)^{g(g+1)/2}. \end{aligned}$$

Proof

If \(\tau _0\in \mathbb {H}\) and \(A\in \Gamma \), then
so that
(6.4)
Now recall by (5.3) that for \(Q\in Y_0(p)\), we have
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 \(\ell (\tau )\ne 1\). If \(Q_\tau \in Y_0(p)\) is associated with \(\tau \in \mathbb {H}\) in the usual way, then we have
(6.5)
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),
When \(\tau _0\mathop {\sim }\limits ^{\Gamma }\rho \), then , and \(\{A(\rho )\}_{A\in \Gamma _0(p)\backslash \Gamma }\) contains \(1+(\frac{-3}{p})\) elliptic fixed points of order 3 which are \(\Gamma _0(p)\)-inequivalent, and \(p-(\frac{-3}{p})\) additional points which are partitioned into \(\Gamma _0(p)\)-orbits of size 3. Then by (6.5) we have
(6.6)
When \(\tau _0\mathop {\sim }\limits ^{\Gamma }i\), then , and \(\{A(i)\}_{A\in \Gamma _0(p)\backslash \Gamma }\) contains \(1+(\frac{-1}{p})\) elliptic fixed points of order 2 which are \(\Gamma _0(p)\)-inequivalent, and \(p-(\frac{-1}{p})\) additional points which are partitioned into \(\Gamma _0(p)\)-orbits of size 2. We then have
(6.7)
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
(6.8)
Therefore, by (3.1), (6.5), (6.6), (6.7), and (6.8), we have
$$\begin{aligned} \widetilde{F}(\widetilde{\mathcal {W}}_p,x)&=x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}\mathcal {F}_p(x)\mathop {\prod _{\tau \in \Gamma _0(p)\backslash \mathbb {H}}}_{v(Q_\tau )=2}(x-j(\tau ))^{g(g+1)/2}\\&=x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}\mathcal {F}_p(x)H_p(x)^{g(g+1)/2}.\end{aligned}$$
\(\square \)
Combining (6.2), (6.3), Theorem 3.2 and Theorem 6.4 now yields
$$\begin{aligned} x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}\mathcal {F}_p(x)S_p^{(l)}(x)^{g^2+g}\equiv \widetilde{S}_p(x)^{g^2-g}\cdot \widetilde{F}(W^2,x)\cdot \mathcal {G}_p(x)\pmod {p}. \end{aligned}$$
(6.9)
We next define
$$\begin{aligned} \widetilde{S}_p^{(l)}(x):=\mathop {\prod _{E/\overline{\mathbb {F}}_p\;\mathrm {supersingular}}}_{j(E)\in \mathbb {F}_p\backslash \{0,1728\}} (x-j(E)). \end{aligned}$$
In Table 1 below, we compare certain factors appearing in (6.9) for each choice of p modulo 12.
Table 1

Factors arising from elliptic points

\(\varvec{p\,(\mathrm {mod}\,{12})}\)

\(\varvec{x^{\epsilon _p(\rho )}}\)

\(\varvec{(x-1728)^{\epsilon _p(i)}}\)

\(\varvec{\mathcal {G}_p(x)}\)

\(\varvec{S_p^{(l)}(x)}\)

1

\(x^{\lfloor \frac{2(g^2+g)}{3}\rfloor }\)

\((x-1728)^{(g^2+g)/2}\)

1

\(\widetilde{S}_p^{(l)}(x)\)

5

1

\((x-1728)^{(g^2+g)/2}\)

\(x^{\lceil \frac{g^2-g}{3}\rceil }\)

\(x\cdot \widetilde{S}_p^{(l)}(x)\)

7

\(x^{\lfloor \frac{2(g^2+g)}{3}\rfloor }\)

1

\((x-1728)^{(g^2-g)/2}\)

\((x-1728)\cdot \widetilde{S}_p^{(l)}(x)\)

11

1

1

\(x^{\lceil \frac{g^2-g}{3}\rceil }(x-1728)^{(g^2-g)/2}\)

\(x(x-1728)\cdot \widetilde{S}_p^{(l)}(x)\)

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
$$\begin{aligned} \mathcal {F}_p(x)\frac{S_p^{(l)}(x)^{g^2+g}}{\mathcal {G}_p(x)}\equiv \widetilde{S}_p(x)^{g^2-g}\frac{\widetilde{F}(W^2,x)}{x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}}\pmod {p}, \end{aligned}$$
(6.10)
where the two quotients reduce to polynomials.
Now on the left of (6.10), 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
$$\begin{aligned}&\mathcal {F}_p(x)\widetilde{S}_p^{(l)}(x)^{g^2+g}\frac{(x^{\alpha _p(\rho )}(x-1728)^{\alpha _p(i)})^{g^2+g}}{\mathcal {G}_p(x)}\nonumber \\&\quad \equiv \widetilde{S}^{(l)}_p(x)^{g^2-g}S_p^{(q)}(x)^{g^2-g}\frac{\widetilde{F}(W^2,x)}{x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}}\pmod {p}. \end{aligned}$$
(6.11)
Now the quotient on the left of (6.11) must divide \(\widetilde{F}(W^2,x)\). Then canceling \(\widetilde{S}^{(l)}_p(x)^{g^2-g}\) 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
$$\begin{aligned} \mathcal {F}_p(x)\equiv S_p^{(q)}(x)^{g^2-g}H_1(x)\pmod {p}, \end{aligned}$$
where \(H_1(x)\) is the polynomial given in non-reduced form by the quotient
$$\begin{aligned} H_1(x):=\frac{\mathcal {G}_p(x)\widetilde{F}(W^2,x)}{x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}(x^{\alpha _p(\rho )}(x-1728)^{\alpha _p(i)})^{g^2+g}\widetilde{S}_p^{(l)}(x)^{2g}}. \end{aligned}$$
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,
$$\begin{aligned} H_1(x)=\frac{\mathcal {G}_p(x)x^{\delta _p(\rho )}(x-1728)^{\delta _p(i)}}{x^{\epsilon _p(\rho )}(x-1728)^{\epsilon _p(i)}}\cdot \frac{\widetilde{F}(W,x)^2}{(x^{\alpha _p(\rho )}(x-1728)^{\alpha _p(i)})^{g^2+g}\widetilde{S}_p^{(l)}(x)^{2g}}. \end{aligned}$$
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. \(\square \)

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
$$\begin{aligned} f_1=q - 3q^3 - 3q^4 - 3q^5 + q^6 + 4q^7 + 3q^8 + \cdots , \end{aligned}$$
and
$$\begin{aligned} f_2=q^2 - q^3 - 3q^4 + 3q^7 + 4q^8 + \cdots . \end{aligned}$$
The associated Wronskian is
$$\begin{aligned} \mathcal {W}_{67}(z)=q^3 - 2q^4 - 6q^5 + 6q^6 + 15q^7 + 8q^8 +\cdots \in S_6(67). \end{aligned}$$
Then by Lemma 6.2 and (2.3), we have
$$\begin{aligned} \widetilde{F}(\widetilde{\mathcal {W}}_{67},x)\equiv & {} x^4(x+1)^6(x+14)^6(x^2+8x+45)^2(x^2+44x+24)^2\\&\times \,(x^2 +\,10x+62)^2\pmod {67}. \end{aligned}$$
But \(\epsilon _{67}(i)=0\), \(\epsilon _{67}(\rho )=4\), and
$$\begin{aligned} S_{67}(x)=(x+1)(x+14)(x^2+8x+45)(x^2+44x+24). \end{aligned}$$
Therefore, by Theorem 6.4, we have
$$\begin{aligned} \mathcal {F}_{67}(x)&\equiv (x^2+8x+45)^2(x^2+44x+24)^2(x^2+10x+62)^2\pmod {67}\\&\equiv S_{67}^{(q)}(x)^2(x^2+10x+62)^2\pmod {67}. \end{aligned}$$

Note

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

Declarations

Acknowledgements

The author thanks the reviewers for their helpful comments and gratefully acknowledges Scott Ahlgren for his invaluable mentoring and for suggesting this problem in the first place.

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Authors’ Affiliations

(1)
Department of Mathematics, Western Washington University

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