1 Introduction

The list of 15 primes

$$\begin{aligned} {\mathscr {S}}=\{2,3,5,7,11,13,17,19,23,29,31,41,47,59,71\} \end{aligned}$$

known as the supersingular primes (https://oeis.org/A002267) appears in several different contexts. Here are five of them.

  1. 1.

    \(p\in {\mathscr {S}}\) if and only if p divides the order of the Monster sporadic simple group.

  2. 2.

    \(p\in {\mathscr {S}}\) if and only if \(g(X_0(p)^+)=0\), where \(X_0(p)^+={\mathbb {H}}/\varGamma _0(p)^+\) is the modular curve associated with the group \(\varGamma _0(p)^+<\mathop {\mathrm {GL}}\nolimits (2,{\mathbb {Q}})\).

  3. 3.

    \(p\in {\mathscr {S}}\) if and only if the supersingular values of the j-invariant all lie in \({\mathbb {F}}_p\).

  4. 4.

    \(p\in {\mathscr {S}}\) if and only if the space \(J^{\mathop {\mathrm {cusp}}\nolimits }_{2,p}\) of Jacobi cusp forms of weight 2 and index p is of dimension 0.

  5. 5.

    If \(p\not \in {\mathscr {S}}\) then the moduli space \({\mathscr {A}}_p\) of complex abelian surfaces with a polarisation of type (1, p) is of general type.

Of these, (1)–(3) are described in [13], where the equivalence of the conditions in (2) and (3) is proved. In [13] a prize (a bottle of Jack Daniels) is offered for an explanation of why the condition in (1) is equivalent to those in (2) and (3): it is still unclaimed. Monstrous moonshine [2] provides a route from (1) to (2), but not in the other direction: see [3] for further details and [1] for some recent developments.

This note is primarily about (5). The proof that \({\mathscr {A}}_p\) is of general type for \(p\not \in {\mathscr {S}}\) is due to Erdenberger [5], and specialists in moduli of abelian surfaces are occasionally asked to explain the apparent coincidence [11]. In fact the answer consists of a series of well-known facts, but because they are not all well known to the same people, the question still recurs. The purpose of this note is to set the answer out clearly.

2 Moduli of abelian surfaces

An abelian surface equipped with a polarisation of type (1, d) (for \(d\in {\mathbb {N}}\)) may be thought of as a complex torus \({\mathbb {C}}^2/\varLambda \), where \(\varLambda \subset {\mathbb {C}}^2\) is the subgroup (lattice) generated by the columns of \(\varOmega =(I_d,\tau )\) for

$$\begin{aligned} I_d:=\begin{pmatrix} 1 &{}\quad 0 \\ 0 &{}\quad d \end{pmatrix}, \quad \tau = \begin{pmatrix}\tau _1 &{}\quad \tau _2\\ \tau _2&{}\quad \tau _3 \end{pmatrix} \in {\mathbb {H}}_2=\{\tau ={}^t\tau \in M_{2\times 2}({\mathbb {C}})\mid \mathop {\mathrm {Im}}\nolimits \tau >0\}. \end{aligned}$$

The paramodular group

$$\begin{aligned} \varGamma _d=\left\{ \gamma \in \mathop {\mathrm {GL}}\nolimits (4,{\mathbb {Q}})\mid {}^t\gamma \begin{pmatrix}0 &{}\quad I_d\\ -I_d &{}\quad 0 \end{pmatrix}\gamma = \begin{pmatrix}0 &{}\quad I_d\\ -I_d &{}\quad 0 \end{pmatrix}\right\} \end{aligned}$$

acts on the Siegel upper half-plane \({\mathbb {H}}_2\) by fractional linear transformations

$$\begin{aligned} \begin{pmatrix}A&{}\quad B\\ C&{}\quad D \end{pmatrix}:\tau \longrightarrow (A\tau +B)(C\tau +D)^{-1}. \end{aligned}$$

This group action is properly discontinuous and the quotient \({\mathscr {A}}_d:={\mathbb {H}}_2/\varGamma _d\) is a coarse moduli space for (1, d)-polarised abelian varieties. It is a quasi-projective variety, and one may ask for its Kodaira dimension, or more precisely, for the Kodaira dimension \(\kappa (Y_d)\) of a desingularisation \(Y_d\) of a projective compactification \(\bar{\mathscr {A}}_d\). For more on this and related spaces, see [12].

In practice one expects that \({\mathscr {A}}_d\) is of general type, i.e. \(\kappa (Y_d)=3\), except for some small values of d. Very loosely, this is because k-fold differential forms on \(Y_d\) correspond to suitable modular forms of weight 3k for \(\varGamma _d\), and these become abundant as d grows at least for k sufficiently divisible. However, not every modular form of weight 3k will do: obstructions come from the boundary \(\bar{\mathscr {A}}_d\setminus {\mathscr {A}}_d\) and from the branching of \({\mathbb {H}}_2\rightarrow {\mathscr {A}}_d\).

The obstructions at the boundary may be overcome by using the low-weight cusp form trick [7]: if we can find a cusp form \(f_2\) of weight 2 for \(\varGamma _d\) then we may consider modular forms f of weight 3k of the form \(f=f_2^kf_k\), where \(f_k\) is a modular form of weight k. These are also abundant, if d and k are large enough, and because they vanish to high order at the boundary, the associated differential forms extend.

The branching behaviour has to be analysed separately, and it depends on the factorisation of d. For that reason much work in this direction has concentrated, for simplicity, on the case \(d=p\) prime. The case \(d=p^2\) has some simplifying features and was treated in [8, 14].

By this method, it was shown in [15] that \({\mathscr {A}}_p\) is of general type for \(p>173\). Because of the inefficient compactification used there, the effective constraint on p came from the branching, so all that was necessary was to verify that a weight 2 cusp form exists for all \(p>173\). Such a form may be obtained by lifting a Jacobi cusp form of weight 2 and index p according to Gritsenko ([7, Theorem 3], [6]). The dimension of the space of Jacobi cusp forms is computed in [4, 17] and in this case it takes the form [7, 15]

$$\begin{aligned} \dim J^{\mathop {\mathrm {cusp}}\nolimits }_{2,p}=\sum _{j=1}^p\left\lfloor \frac{1+j}{6}\right\rfloor -\delta _6(j)-\left\lfloor \frac{j^2}{4p}\right\rfloor , \end{aligned}$$

where \(\delta _6(j)=1\) if 6|j and 0 otherwise. This is positive for all \(p>173\).

Erdenberger [5] found a better compactification and was able to reduce the condition imposed by the branching from \(p>173\) to \(p\ge 37\), so that the existence of the Jacobi form becomes the effective constraint. Then, it is easy to compute from the formula above that \(p\in {\mathscr {S}}\) exactly when no Jacobi cusp form of weight 2 and index p exists, i.e. when the condition in (4) holds.

It is not necessarily to be expected that \({\mathscr {A}}_p\) is of general type exactly when \(p\not \in {\mathscr {S}}\). The method of proof of [5] fails for \(p\in {\mathscr {S}}\), as we shall see, and it is known that \({\mathscr {A}}_d\) is unirational (so in particular not of general type) for some small values of d, including all primes \(p\le 11\): see [9]. However, if \(p\ge 13\), nothing currently excludes the possibility that \({\mathscr {A}}_p\) is of general type.

On the other hand, the possibility that \({\mathscr {A}}_p\) is unirational, other than in the known cases \(p\le 11\), is excluded. Gritsenko [6] showed that \({\mathscr {A}}_d\) has non-negative Kodaira dimension, so is not uniruled, for all \(d\ge 13\), prime or not, except possibly for \(d=14\), 15, 16, 18, 20, 24, 30, 36. Of these, the cases \(d=14\), 16, 18, 20 have since been settled in [10] (such \({\mathscr {A}}_d\) are in fact unirational) and only for \(d=15\), 24, 30, 36 is nothing known about the Kodaira dimension of \({\mathscr {A}}_d\).

3 Modular forms

Since we have now established a connection between (4) and (5), to achieve a moderately satisfactory explanation of the apparently coincidental appearance of \({\mathscr {S}}\) in (5) we should show, without direct computation, that the conditions in (2) and (4) are equivalent. (A fully satisfactory explanation would also involve (1): this we are not able to give.) This is well known among specialists in Jacobi forms, and follows easily from a small part of [17].

It is shown in [17] that the space \(J_{k,d}\) of Jacobi forms of weight k and index d is isomorphic (even as a Hecke module) to a certain subspace \({\mathfrak {M}}^-_{2k-2}(d)\) of the space \(M_{2k-2}(d)\) of modular forms of weight \(2k-2\) for \(\varGamma _0(d)\). This subspace is defined by \({\mathfrak {M}}^-_{2k-2}(d)=M^-_{2k-2}(d)\cap {\mathfrak {M}}_{2k-2}(d)\), where \(M^-_{2k-2}(d)\) is the space of weight \(2k-2\) modular forms for \(\varGamma _0(d)\) that satisfy an extra condition on the behaviour under the Fricke involution \(w:\tau \mapsto \frac{-1}{{{d}}\tau }\), namely

$$\begin{aligned} f\left( \frac{-1}{{{d}}\tau }\right) =(-1)^k d^{k-1}\tau ^{2k-2}f(\tau ). \end{aligned}$$

In our case (\(k=2\) and \(d=p\)) this is equivalent to saying that f is a modular form of weight 2 for the group \(\varGamma _0(p)^+<\mathop {\mathrm {GL}}\nolimits (2,{\mathbb {Q}})\) generated by \(\varGamma _0(p)\) and \(w=\begin{pmatrix}0&{}\quad 1\\ -d&{}\quad 0 \end{pmatrix}\). See, for example, the definition of automorphic form in [16, Chapter 2]. So if there is a weight 2, index p Jacobi form, then \(\varGamma _0(p)^+\) has a weight 2 modular form.

Conversely, inspecting the definition of \({\mathfrak {M}}\) in [17] we find that there are no other conditions for p prime: simply \({\mathfrak {M}}_{2k-2}(p)=M_{2k-2}(p)\). This can be seen at once, for instance, from [17, Eq. (4), p. 116], since \({\mathfrak {M}}_2(1)\subset M_2(1)=0\). In other words, the space of Jacobi forms in this case is isomorphic exactly to the space of weight 2 modular forms for \(\varGamma _0(p)^+\). Moreover, the isomorphism respects cusp forms: see [17, Theorem 5].

We remark that for \(k=2\) and p square-free (in particular for p prime) there are no Eisenstein series, so the condition (3) is equivalent to the same statement but with \(J^{\mathop {\mathrm {cusp}}\nolimits }_{2,p}\) replaced by \(J_{2,p}\).

However, Ogg [13] shows that the modular curve \(X_0(p)^+\) corresponding to \(\varGamma _0(p)^+\) is of genus 0 precisely for \(p\in {\mathscr {S}}\), i.e. he shows (2). One can compute the dimension of the space of weight 2 forms from the formulae given in [16, Theorem 2.23]: it is \(g+m-1\), where g is the genus of \(X_0(p)^+\) and m is the number of cusps. Because p is prime, the curve \(X_0(p)\) has two cusps, which are interchanged by the Fricke involution; so \(m=1\), and so the space of modular forms for \(\varGamma _0(p)^+\) has dimension g (i.e. they are all cusp forms, as one should also expect from the remark above). So for \(p\in {\mathscr {S}}\), there can be no weight 2, index p Jacobi forms; so we definitely cannot prove that \({\mathscr {A}}_p\) is of general type result by the methods of [5, 15] for any \(p\in {\mathscr {S}}\).