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

© The Author(s) 2017

**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 points
- Modular curves
- Supersingular elliptic curves
- Modular forms

## 1 Introduction

*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 thatthen we define the

*Weierstrass weight*of

*Q*to beWe 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

*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, [3–6, 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.

*p*and

*M*as above, let

*p*we define

### Theorem 1.1

*p*is prime, then \(F_p(x)\) has

*p*-integral rational coefficients and

*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

*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.

*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

*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 \(X_0^+(p)\).

*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

*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

### 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

*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

*k*th Bernoulli number, and \(\sigma _{k-1}(n)=\sum _{d\mid n}d^{k-1}\). Then the function

*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

*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

### Proof

*f*and \(f^2\) yields

## 3 Modular curves modulo *p*

*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.)

*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 \(X_0^+(p)\).

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 \(\mathcal {O}_D\).

*E*,

*C*) 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

*j*-invariants of the \(w_p\)-fixed points of \(Y_0(p)\). Then we have

### 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

*p*be prime. Then we have

### 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.

*j*-invariants in \(\mathbb {F}_{p^2}\backslash \mathbb {F}_p\). Since there are \(g_p+1\) total supersingular

*j*-invariants, we have

## 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.

## 5 Wronskians and *p*-integrality

*z*is near

*Q*, we haveAlternatively, 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

### 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

*p*-integral coefficients, with leading coefficient given by the Vandermonde determinant

*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 \(p\ge 67\).

*f*(

*z*) defined on \(\mathbb {H}\) and any integer

*k*, we define the

*slash*operator \(\mid _k\) by

*f*is a modular form of weight

*k*.

### Lemma 6.1

### 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

*p*is a prime such that \(X_0^+(p)\) has genus at least 2, define

*p*-integral rational coefficients, and

### Proof

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

*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

*V*is coprime to

*p*, then by Lemma 6.2 we have

*p*-integral rational coefficients such that

### Lemma 6.3

*p*-integral coefficients, we have

### Theorem 6.4

*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

### Proof

*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 haveWhen \(\tau _0\mathop {\sim }\limits ^{\Gamma }i\), 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 haveFinally, 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 toTherefore, by (3.1), (6.5), (6.6), (6.7), and (6.8), we have

*p*modulo 12.

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 | \(x^{\lceil \frac{g^2-g}{3}\rceil }(x-1728)^{(g^2-g)/2}\) | \(x(x-1728)\cdot \widetilde{S}_p^{(l)}(x)\) |

*x*and \((x-1728)\) are coprime to \(\widetilde{S}_p(x)\), we have

*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

*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 \(X_0^+(67)\)

### 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

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