by Victor Manuel Aricheta
1. Ogg’s observation
In 1975 Andrew Ogg wrote the following remark (here translated from the French) in [1]:
In his inaugural lecture at the Collège de France on 14 January 1975, J. Tits mentioned the Fischer group, the monster, which, if it exists, is a sporadic simple group of order \[ 2^{46} \cdot 3^{20} \cdot 5^9 \cdot 7^6 \cdot 11^2 \cdot 13^3 \cdot 17 \cdot 19 \cdot 23 \cdot 29 \cdot 31 \cdot 41 \cdot 47 \cdot 59 \cdot 71, \] i.e., precisely divisible by the fifteen prime numbers listed in the corollary. A bottle of Jack Daniels is offered to anyone who can explain this coincidence.
The corollary that is being referred in this remark concerns supersingular elliptic curves and genus-zero groups. We will state this result more precisely as Theorem 1, which we will refer to as Ogg’s theorem in this note, after briefly discussing some notions and notations.
For every positive integer \( N \), the congruence subgroup \[ \Gamma_0(N) := \left\{ \begin{pmatrix}a & b \\ cN & d \end{pmatrix} : a,b,c,d \in \mathbb{Z}, ad - bcN = 1 \right\} \] acts on the complex upper half-plane \( \mathbb{H} \) by linear fractional transformations. The modular curve \( X_0(N) \) is obtained by compactifying the orbit space \( \Gamma_0(N) \backslash \mathbb{H} \) in a process of adjoining finitely many cusps. The noncuspidal points of \( X_0(N) \) represent isomorphism classes of complex elliptic curves with a prescribed cyclic subgroup of order \( N \).
The modular curve \( X_0(N) \) is an algebraic curve, which has good reduction modulo any prime \( p \nmid N \). A point of \( X_0(N) \) modulo \( p \) is called a supersingular point if the underlying elliptic curve is supersingular. For a prime \( p \) not dividing \( N \), it is known that there are only finitely many supersingular points of \( X_0(N) \) modulo \( p \) and that these are all defined over \( \mathbb{F}_{p^2} \) [e1]. Nothing precludes the possibility of a supersingular point being defined over the prime field \( \mathbb{F}_p \). In fact, for some exceptional cases it turns out that all the supersingular points of \( X_0(N) \) modulo \( p \) are defined over \( \mathbb{F}_p \). Ogg’s theorem and its generalization give characterizations of the primes \( p \) for which this exceptional behavior occurs. We recall some more definitions and notations before stating Ogg’s theorem.
An exact divisor of a positive integer \( N \) is an integer \( e \) such that \( e|N \) and \( (e,N/e)=1 \). Note that the set of positive exact divisors of \( N \) forms a group under the multiplication given by \( e*f := ef/(e,f)^2 \). Now, to each positive exact divisor \( e \) of \( N \) we can define an element of the normalizer of \( \Gamma_0(N) \) in \( \text{SL}_2(\mathbb{R}) \), called an Atkin–Lehner involution. This element (which is unique modulo \( \Gamma_0(N) \)) is denoted \( w_e \). More explicitly, one may take \( w_e \) to be any matrix of the form \[ \frac{1}{\sqrt{e}}\begin{pmatrix}ae & b \\ cN & de \end{pmatrix}, \] where \( a,b,c,d \) are integers such that \( ade - bcN/e = 1 \). Given two Atkin–Lehner involutions \( w_e \) and \( w_f \), the product \( w_ew_f \) is equal (modulo \( \Gamma_0(N) \)) to \( w_{e*f} \).
Following the notations of Conway–Norton in [e2], if \( \{e,f,\dots\} \) is a subgroup of the group of positive exact divisors of \( N \) then the symbol \( \Gamma_0(N)+e,f,\dots\, \) denotes the subgroup of \( \text{SL}_2(\mathbb{R}) \) generated by \( \Gamma_0(N) \) and the Atkin–Lehner involutions \( w_e,w_f, \dots\thinspace \), that is, \[ \Gamma_0(N)+e,f,\dots \mathrel{\mathop :}= \langle \Gamma_0(N), w_e, w_f, \dots\thinspace\rangle. \] As a particular example, if \( N = p \) is prime, then \[ \Gamma_0(p)+p := \left\langle \Gamma_0(p), \dfrac{1}{\sqrt{p}}\begin{pmatrix}0 & -1 \\ p & \phantom{-}0 \end{pmatrix} \right \rangle. \]
An Atkin–Lehner involution \( w_e \) induces an involution of \( X_0(N) \) which we also denote by \( w_e \). The genus of \( \Gamma_0(N)+e,f, \dots \) is defined to be the genus of the quotient \( X_0(N)/\langle w_e,w_f,\dots\thinspace \rangle \). We now state Ogg’s theorem.
- The genus of \( \, \Gamma_0(p) + p \) is zero.
- The supersingular points of \( X_0(1) \) modulo \( p \) are all defined over \( \mathbb{F}_p \).
- The prime \( p \leq 31 \) or \( p=41, 47, 59, 71 \).
As noted by Ogg, the primes listed in this result are precisely the primes that divide the order of the monster. In this note we refer to this as Ogg’s observation.
At the time when Ogg made his observation, a connection between genus-zero groups and the monster group had not yet been proposed, let alone established. Ogg’s observation is significant for it provided the first hint that such a link exists. This link, now established and called monstrous moonshine, is the first example of moonshine, a term used to describe unexpected connections between finite groups and modular objects. We will say more about monstrous moonshine in Section 3.
2. A generalization of Ogg’s observation
Ogg’s observation provides an unexpected relationship between genus-zero groups and the monster group. In this section we discuss how similar relationships hold between genus-zero groups and other sporadic groups. To explain how one may obtain such striking connections we first recount Ogg’s proof of Theorem 1.
One starts with the Deligne–Rapoport model for the singular curve \( X_0(p) \) modulo \( p \), which is given by two copies of \( X_0(1) \) modulo p that intersects transversally at the supersingular points of \( X_0(1) \) modulo \( p \). In this model the involution \( w_p \) gives an isomorphism between the two copies of \( X_0(1) \) modulo \( p \) and acts as the Frobenius on the intersection, which consists of the supersingular points of \( X_0(1) \) modulo \( p \). Thus, one obtains a model for \( X_0(p)/\langle w_p \rangle \) modulo \( p \): a single copy of \( X_0(1) \) modulo \( p \) which intersects itself \( Q(p) \) times, where \( 2Q(p) \) is the number of supersingular points of \( X_0(1) \) modulo \( p \) that are not defined over \( \mathbb{F}_p \). Applying the specialization principle yields the following equation: \[ \operatorname{genus}(X_0(p)/\langle w_p \rangle) = \operatorname{genus}(X_0(1)) + Q(p). \] Since the genus of \( X_0(1) \) is zero, the genus of \( X_0(p)/\langle w_p \rangle \) is also zero if and only if \( Q(p)=0 \), that is, when all the supersingular points of \( X_0(1) \) modulo \( p \) are defined over \( \mathbb{F}_p \). This is Ogg’s theorem.
This proof started with the curve \( X_0(p) \) modulo \( p \), but one may as well begin with the singular curve \( X_0(Np) \) modulo \( p \) — as Ogg did in [2] — or with \( X_0(Np)/ \langle w_e,w_f, \dots\thinspace \rangle \) modulo \( p \). The same argument works and one obtains analogous results. For simplicity we only provide two such results below, and only note that more general results exist ([e9], Theorem 1.1).
- The genus of \( \, \Gamma_0(2p) + p \) is zero.
- The supersingular points of \( X_0(2) \) modulo \( p \) are all defined over \( \mathbb{F}_p \).
- The prime \( p \leq 11 \) or \( p = 23 \).
- The genus of \( \, \Gamma_0(2p) + 2, p, 2p \) is zero.
- The supersingular points of \( X_0(2)/\langle w_2 \rangle \) modulo \( p \) are all defined over \( \mathbb{F}_p \).
- The prime \( p \leq 23 \) or \( p = 31, 47 \).
Theorem 2 has already been proven by Ogg; it is obtained by starting with \( X_0(2p) \) modulo \( p \). Meanwhile, Theorem 3 is obtained by beginning with \( X_0(2p)/\langle w_2 \rangle \) modulo \( p \). In this note we will call these Ogg’s theorem for \( \Gamma_0(2) \) and Ogg’s theorem for \( \Gamma_0(2)+2 \), respectively, for reasons that we hope are clear.
The primes listed in Theorem 2 and Theorem 3 are familiar to finite group theorists. The largest Mathieu group \( M_{24} \), a sporadic group that occurs as the automorphism group of the Golay code, has order \[ |M_{24}| = 2^{10} \cdot 3^3 \cdot 5 \cdot 7 \cdot 11 \cdot 23. \] Meanwhile, the baby monster group \( \mathbb{B} \), the second largest sporadic group in terms of size, has order \[ |\mathbb{B}| = 2^{41} \cdot 3^{13} \cdot 5^6 \cdot 7^2 \cdot 11 \cdot 13 \cdot 17 \cdot 19 \cdot 23 \cdot 31 \cdot 47. \] Paralleling Ogg’s observation we note that the primes listed in Theorem 2 and Theorem 3 are identical to the odd primes that divide the order of the largest Mathieu group and the baby monster group, respectively. In other words, Ogg’s observation, which relates supersingular elliptic curves to the monster group, extends to other sporadic groups.
As mentioned earlier, analogues of Ogg’s theorem for other genus-zero groups exist. In particular, remarkable connections to finite groups also exist in general [e9]. In the following table we present a few more coincidences evocative of Ogg’s observation. An arithmetic proof of some of these coincidences, using class numbers of imaginary quadratic fields, is given in [e10].
\[ \begin{array}{lcl} \hline & \text{Primes } p \nmid \ell & \text{Sporadic group with}\\ \Gamma = \Gamma_0(\ell) + e & \text{that satisfy the equivalent} & \text{order divisible precisely}\\ \text{where } e = 1 \text{ or }\ell & \text{statements in} & \text{by } \ell \text{ and the primes}\\ & \text{Ogg’s theorem for }\Gamma & \text{in the second column}\\ \hline \hline \Gamma_0(3) & 2, 5, 11& \text{Mathieu group } M_{12}\\ \Gamma_0(3)+3 & 2, 5, 7, 11, 13, 17, 23, 29 & \text{largest Fischer group}\\ \Gamma_0(5)+5 & 2, 3, 7, 11, 19 & \text{Harada--Norton group}\\ \Gamma_0(7)+7 & 2, 3, 5, 17 & \text{Held group}\\ \hline \end{array} \]
3. Ogg’s observation and moonshine
In the previous section we observed several relationships between genus zero groups and sporadic groups. We elaborate in this section how monstrous moonshine provides a partial explanation for Ogg’s original observation. Additionally, we discuss how the new observations from the previous section offer fresh insights into both monstrous and umbral moonshine. For more details on moonshine, we refer the reader to the survey articles [e8], [e11].
Monstrous moonshine — postulated by John Conway and Simon Norton [e2], following numerical calculations by John McKay and John Thompson, and proven by Richard Borcherds [e3] — asserts the existence of a \( \mathbb{Z} \)-graded infinite-dimensional complex representation \( V^{\natural} \) of the monster with exceptional properties. If \[ V^{\natural} = V_0^{\natural} \oplus V_2^{\natural} \oplus V_3^{\natural} \oplus V_4^{\natural} \oplus \cdots, \] where each \( V_i^{\natural} \) is a finite-dimensional complex representation of the monster, one of these exceptional properties of \( V^{\natural} \) is that for each element \( g \) of the monster, the corresponding function \[ T_g(\tau) = \chi_{V_0^{\natural}}(g)q^{-1} + \chi_{V_2^{\natural}}(g)q + \chi_{V_3^{\natural}}(g)q^{2} + \chi_{V_4^{\natural}}(g)q^{3} + \cdots \hspace{15pt} (q = e^{2\pi i \tau}), \] called a McKay–Thompson series, generates the function field of some genus zero modular curve. Here, \( \tau \) is an element of the complex upper half-plane \( \mathbb{H} \) and \( \chi_V \) is the character of the representation \( V \).
In particular the assignment of McKay–Thompson series to elements of the monster establishes a correspondence between conjugacy classes of the monster and genus zero groups. This correspondence is explicitly described in the seminal paper [e2]. For example, the conjugacy class containing the identity, labeled 1A in the ATLAS of Finite Groups, corresponds to the genus zero group \( \Gamma_0(1) \). There are two conjugacy classes of involutions in the monster group, labeled 2A and 2B in the ATLAS. The conjugacy class 2A, the smaller of the two conjugacy classes, corresponds to the genus zero group \( \Gamma_0(2)+2 \), while the conjugacy class 2B corresponds to the genus zero group \( \Gamma_0(2) \). The crucial point for our discussion is that all the groups \( \Gamma_0(p)+p \) for primes \( p \) dividing the order of the monster appear in the monstrous moonshine correspondences.
Thus, monstrous moonshine provides an explanation for the appearance of the primes listed in Theorem 1 in the following way: For a prime divisor \( p \) of the order of the monster, there exists a conjugacy class of the monster whose corresponding genus zero group via monstrous moonshine is \( \Gamma_0(p)+p \). The genus of \( \Gamma_0(p)+p \) is therefore zero; otherwise, the function field of the corresponding modular curve cannot be generated by a single element. While this accounts for the presence of the primes dividing the order of the monster in Ogg’s theorem, this does not explain the absence of the rest of the primes, and so, a complete explanation of Ogg’s observation remains to be found.
The results presented in the previous section may offer a hint toward such an explanation. This is explained in more detail in ([e9], Theorem 1.3), but the insight gained from all of this is the following: obtaining a thorough understanding of Ogg’s observation requires a closer examination of the so-called Fricke elements of prime order, which are the elements of the conjugacy classes of the monster labeled \( pA \) in the ATLAS, for primes \( p \). Theorem 4, particularly the fourth of the equivalent statements, provides an example of this connection between supersingular elliptic curves and monstrous moonshine.
In a somewhat orthogonal direction, the generalizations of Ogg’s theorem also suggest a connection between supersingular elliptic curves and umbral moonshine [e4], [e5], [e6], a more contemporary example of moonshine.
There are 23 instances of moonshine which are collectively known as umbral moonshine, connecting finite groups arising from lattices to vector-valued mock modular forms, with the most prominent case being Mathieu moonshine. Mathieu moonshine assigns to each conjugacy class of the largest Mathieu group \( M_{24} \) a corresponding Mathieu McKay–Thompson series, distinguished in the sense that it is a mock modular form of optimal growth. As the name suggests, mock modular forms are generally not elliptic modular forms; mock modular forms are holomorphic functions that are “almost modular”. We refer the reader to [e7] for the theory of mock modular forms and their applications.
Each of the 23 examples of moonshine in umbral moonshine is identified by some genus zero subgroup of \( \text{SL}_2(\mathbb{R}) \), referred to as its lambency. Mathieu moonshine is the case of umbral moonshine with lambency \( \Gamma_0(2) \). It turns out that for each lambency \( \Gamma \) appearing in umbral moonshine, the corresponding generalization of Ogg’s theorem for \( \Gamma \) provides a way of detecting umbral McKay–Thompson series which are genuinely modular and not only mock modular. The details are in ([e9], Theorem 1.5). Theorem 4, particularly the last of the equivalent statements, gives an example of this phenomenon.
We conclude this note by presenting the following theorem which provides a glimpse of connections — yet to be understood — between supersingular elliptic curves and moonshine, connections that were made manifest through an exploration of Ogg’s observation.
- The genus of \( \, \Gamma_0(2p) + p \) is zero.
- The supersingular points of \( X_0(2)/\langle w_2 \rangle \) modulo \( p \) are all defined over \( \mathbb{F}_p \).
- The prime \( p \leq 11 \) or \( p = 23 \).
- A Fricke element of order \( p \) lies in the centralizer of a 2B-element of the monster.
- An elliptic modular form of level \( p \) occurs as a Mathieu McKay–Thompson series.
Victor Manuel Aricheta is an Assistant Professor at the Institute of Mathematics, University of the Philippines Diliman. His research interests include modular forms, elliptic curves and vertex operator algebras.
