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

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 $p$.


Introduction
A Weierstrass point on a compact Riemann surface M of genus g is a point Q ∈ M at which some holomorphic differential ω vanishes to order at least g. Weierstrass points can be identified by observing their weight. Let H 1 (M ) be the C-vector space of holomorphic differentials on M of dimension g. If {ω 1 , ω 2 , . . . , ω g } forms a basis for H 1 (M ) adapted to Q ∈ M , so that 0 = ord Q (ω 1 ) < ord Q (ω 2 ) < · · · < ord Q (ω g ), then we define the Weierstrass weight of Q to be wt(Q) := g j=1 (ord Q (ω j ) − j + 1).
We see that 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 Q∈M wt(Q) = g 3 − g.
Hence each Riemann surface of genus g ≥ 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 H denote the complex upper half-plane. The modular group Γ := SL 2 (Z) acts on H by linear fractional transformations ( a b c d ) z = az+b cz+d . If N ≥ 1 is an integer, then we define the congruence subgroup The quotient of the action of Γ 0 (N ) on H is the Riemann surface Y 0 (N ) := Γ 0 (N )\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 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 ∞ is a Weierstrass point. Lehner and Newman [14] and Atkin [5] showed that ∞ is a Weierstrass point for most non-squarefree N , while Atkin [6] proved that ∞ 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 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, ∞ 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 where j(z) = q −1 + 744 + 196884q + · · · is the usual elliptic modular function defined on Γ, and j(Q) = j(τ ) for any τ ∈ H with Q = Γ 0 (N )τ . This is the divisor polynomial for the Weierstrass points of Y 0 (N ). Next, for a prime p we define where the product is over all 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 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 S p (x) µ(M )g pM (g pM −1) , where µ(M ) := [Γ : Γ 0 (M )] and g pM is the genus of X 0 (pM ), and where 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 τ → −1/pτ for τ ∈ H. There is a natural projection map π : X 0 (p) → X + 0 (p) which sends a point Q ∈ X 0 (p) to its equivalence class π(Q) = Q in X + 0 (p). This is a 2-to-1 mapping, ramified at those points Q ∈ X 0 (p) that remain fixed by w p . Therefore we set v(Q) := 2 if w p (Q) = Q, 1 otherwise, (1.2) so that v(Q) is equal to the multiplicity of the map π 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 where wt(Q) is the Weierstrass weight of the image 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 ∞ are interchanged by w p , so that X + 0 (p) has a single cusp at ∞, which may or may not be a Weierstrass point. Atkin checked all primes p ≤ 883 and conjectured that ∞ 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 F p (x) is a polynomial of degree 2((g + p ) 3 − g + p − wt(∞)), where g + p is the genus of X + 0 (p). Recalling that the supersingular polynomial S p (x) factors over F p [x] into linear and irreducible quadratic factors, we separate these factors by defining p (x). Our main theorem gives an analogue of Theorem 1.1 for F p (x). We require an assumption that H 1 (X + 0 (p)) has a good basis, a condition about pintegrality which we define later in Section 4. Computations suggest that most, if not all, such spaces satsify this condition. Indeed, each H 1 (X + 0 (p)) with p < 3200 has a good basis. Theorem 1.2. Let p be prime and suppose that H(X + 0 (p)) has a good basis. Then F p (x) has p-integral rational coefficients, and there exists a polynomial H(x) ∈ 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 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 H 1 (X + 0 (p)). Next, in Section 5 we derive a special cusp form on Γ 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).

Divisor polynomials of modular forms
Let M k (resp. M k (p)) denote the space of modular forms of weight k on Γ (resp. Γ 0 (p)), and let S k (resp. S k (p)) be the subspace of cusp forms. For even k ≥ 4, the Eisenstein series E k ∈ M k is defined as where B k is the kth Bernoulli number, and σ k−1 (n) = d|n d k−1 . Then the function 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 ∈ M k vanishes, excluding those zeros that may occur at the elliptic points i and ρ := e 2πi/3 (for details, see [3] or Section 2.6 of [20]). We define Now let f ∈ M k have leading coefficient 1. Then we note that (2.1) and (2.2) are defined such that the quotient is a polynomial in j(z). Therefore, we define F (f, x) to be the unique polynomial in x satisfying (2.3). Furthermore, if f has p-integral rational coefficients, then so does F (f, x).
Finally, we record a result about the divisor polynomial of the square of a modular form. Then Proof. Using (2. 3) for both f and f 2 yields Thus Then by (2.1) and (2.2) we have , the result follows.

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 F p 2 \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 ] be the order of the imaginary quadratic field is the monic polynomial whose zeros are exactly the j-invariants of the distinct isomorphism classes of elliptic curves with complex multiplication by O D , and its degree is h(−D), the class number of O D .
The points Q ∈ Y 0 (p) that are fixed by w p correspond to pairs (E, C) such that E admits complex multiplication by √ −p, or in other words, Z[ √ −p] embeds in End(E), the endomorphism ring of E over the complex numbers (see e.g. [16]). Since End(E) must be an order in an imaginary quadratic field, we have the monic polynomial whose zeros are precisely the j-invariants of the w p -fixed points of Y 0 (p). Then we have The following result is due independently to Kaneko and Zagier.
Proof. The result follow from Kronecker's relations on the modular equation Φ p (X, Y ), and may be found in the appendix of [11].
We can now prove the following.
p (x) 2 (mod p). 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 F p . Then by Proposition 3.1, T (x) | S (l) p (x). We will show that T (x) and S By the Riemann-Hurwitz formula (see, for example, Section I.2 of [9]), we have where σ is the number of points of X 0 (p) at which the projection π : X 0 (p) → 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 ∞, so σ = 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 F p 2 \F p . Since there are g p + 1 total supersingular j-invariants, we have

A Good
Basis for 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 ≥ 2. Recall that g is the dimension of H 1 (X + 0 (p)), the space of holomorphic 1-forms on X + 0 (p). Let {ω 1 , ω 2 , . . . , ω g } be a basis of H 1 (X + 0 (p)), where ω 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 ω i to a holomorphic 1-form π * ω i on X 0 (p) via the projection map π : X 0 (p) → X + 0 (p) (see, for example, Chapter 2 of [15]). We can choose a local coordinate z at Q ∈ X 0 (p) so that near Q, u = z n , where n is the multiplicity of π at Q, hence n = v(Q) (1.2). Then we have π * ω i = H i (z)dz with H i (z) = h i (z n )nz n−1 ∈ 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), . . . , 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) = n a(n)q n be a newform for S 2 (p), and let σ ∈ Gal(C/Q). Then f σ (z) = n σ(a(n))q n is also a newform for S 2 (p), so the action of Gal(C/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 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 , . . . , f g } uniquely by requiring that . . .
Proof. Here we closely follow the proof of Lemma 3.1 in [4]. Let {f 1 , f 2 , . . . , f g } be a basis for S + 2 (p) satisfying (4.1) and (4.2). Let θ := q d dq be the usual differential operator for modular forms, so that d dz = 2πiθ. Then by properties of determinants, we have We see that the Fourier expansion of 1 2πi g(g−1)/2 W (f 1 , f 2 , . . . , f g ) has rational p-integral coefficients, with leading coefficient given by the Vandermonde determinant 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 Γ 0 (p), we have 1 ≤ c i ≤ p+1 6 < p for each 1 ≤ i ≤ g, so 1 ≤ c k − c j ≤ p − 1 for all j < k. Therefore the lemma is proved.

Proof of the Main Theorem
Let p be a prime for which 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 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 ≥ 2, in which case we have p ≥ 67.
We first adapt two lemmas from [3]. For any meromorphic function f (z) defined on H and any integer k, we define the slash operator | k by c d ) is a real matrix with positive determinant, and γz := az+b cz+d . In particular, the Atkin-Lehner involution w p is given by f → f | k 0 −1 p 0 when f is a modular form of weight k.
Proof. The proof is identical to Lemma 3.2 of [3] except that f | 2 0 −1 p 0 = 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 W p (z) := normalized to have leading coefficient 1. Then W p (z) is a cusp form of weight g(g + 1)(p + 1) on Γ with 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].

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