by John F. R. Duncan
1. Monstrous moonshine
In this lecture the speaker presented some new developments in group theory, including the conjectured existence of a new nonabelian finite simple group, (tentatively) dubbed the Monster. Notably, the Monster, were it to exist, would not belong to any of the known infinite and naturally occurring families. That is, it would be a sporadic simple group: Not the group of even permutations of a finite set, and not a finite group of Lie type.
A striking feature of the Monster, \( \mathbb{M} \), is its imposing (conjectural) size, which was given precisely by the speaker as follows. \begin{equation} \tag{1.1} \#\mathbb{M}= 2^{46}\cdot3^{20}\cdot5^9\cdot7^6\cdot11^2\cdot13^3\cdot17\cdot19\cdot23\cdot29\cdot31\cdot41\cdot47\cdot59\cdot71. \end{equation} (This works out to be about \( 8\times 10^{53} \), which is apparently a reasonable approximation1 to the number of atoms in Jupiter.)
It is at this moment that moonshine begins, because Andrew Ogg was in the audience that day, and he had come equipped with an alternative point of view. Indeed, in the course of investigating the geometry of modular curves (see [1], [2]), Ogg had recently arrived at the following result.
- All the supersingular \( j \)-values in characteristic \( p \) lie in \( \mathbb{F}_p \).
- \( X_0(p)+p \) has genus zero.
- \( p\leq 31 \) or \( p\in \{41,47,59,71\} \).
Imagine the surprise you would feel, with Proposition 1.1 fresh in your mind, at seeing the cutting-edge conjectural product (1.1) of finite-group theory steadily inscribed upon the board in front of you: 2, 3 and all the primes up to 31 appear, and 37 does not, and 41 does! And beyond that no others than 47, 59 and 71.
Struck by the coincidence between Equation (1.1) and Proposition 1.1, Ogg offered a bottle of Jack Daniels2 (see [2]) in exchange for an explanation.
One of the main morals of this coincidence is that the finite simple Monster group is most naturally represented on an infinite-dimensional space rather than a finite-dimensional one. More specifically, the “natural” representation \begin{equation}\label{eqn:Vnatural} \tag{1.2} V^{\natural} =\bigoplus V^{\natural}_n \end{equation} of the Monster has the normalized elliptic modular invariant \begin{equation}\label{eqn:J1isj} \tag{1.3} T_1(\tau):=j(\tau)-744=q^{-1}+196884q+\cdots \end{equation} as its graded dimension function, \begin{equation}\label{eqn:J1issum} \tag{1.4} T_1(\tau)=\sum \dim(V^{\natural}_n)q^n, \end{equation} and Proposition 1.1 is telling us the graded traces of certain elements of prime order. (In (1.3)–(1.4) and in what follows, \( q=e^{2\pi i \tau} \).)
To explain this last statement we make some definitions. For \( m \) a positive integer define \( \Gamma_0(m) \) to be the group of transformations of the complex upper half-plane \begin{equation} \tag{1.5} \mathbb{H}:=\{\tau\in\mathbb{C}\mid\Im(\tau) > 0\} \end{equation} that take the form \begin{equation}\label{eqn:Gamma0N_xfms} \tag{1.6} \tau\mapsto \frac{a\tau+b}{cm\tau+d}, \end{equation} where \( a,b,c,d\in \mathbb{Z} \), with \( ad-cdm=1 \), and for \( n \) an exact divisor of \( m \) write \( \Gamma_0(m)+n \) for the group composed of the transformations of \( \Gamma_0(m) \) together with those of the form \begin{equation}\label{eqn:Wn_xfms} \tag{1.7} \tau\mapsto \frac{an\tau+b}{cm\tau+dn}, \end{equation} where \( a,b,c,d\in \mathbb{Z} \), but now \( adn-bc\frac mn=1 \). (A divisor \( n \) of \( m \) is called exact if \( n \) and \( \frac mn \) are coprime.) Next, given a discrete group \( \Gamma \) of transformations of \( \mathbb{H} \) that is commensurable with \( SL_2(\mathbb{Z}) \) define \begin{equation}\label{eqn:XGamma} \tag{1.8} X_\Gamma:=\Gamma\backslash(\mathbb{H}\cup\mathbb{P}_1(\mathbb{Q})) \end{equation} to be the natural compactification of the orbit space \( Y_\Gamma:=\Gamma\backslash\mathbb{H} \) (cf., e.g., [e5]), and to ease notation set \begin{equation}\label{eqn:X0m} \tag{1.9} X_0(m):=X_{\Gamma_0(m)}, \quad Y_0(m):=Y_{\Gamma_0(m)}, \quad X_0(m)+n:=X_{\Gamma_0(m)+n}, \quad \text{\&c.} \end{equation} There is a standard way to regard each of these spaces (1.9) as algebraic curves over \( \mathbb{Q} \). The points of \( Y_0(m)\subset X_0(m) \) parameterize isomorphism classes of isogenies (i.e., morphisms) of elliptic curves with cyclic kernel of order \( m \).
Note that \( \Gamma_0(1)=\Gamma_0(1)+1 \) is the modular group, and \( X_0(1)=X_0(1)+1 \) has genus zero as a Riemann surface. Note also that the normalized elliptic modular invariant (1.3) defines an isomorphism \[ X_0(1)\xrightarrow{\sim} \mathbb{C}\cup\{\infty\}. \] More generally, if \( X_0(m)+n \) has genus zero then there is a unique \( \Gamma_0(m)+n \)-invariant holomorphic function \begin{equation}\label{eqn:Tm+n} \tag{1.10} T_{m+n}(\tau)=q^{-1}+O(q) \end{equation} on \( \mathbb{H} \) that induces a counterpart isomorphism, \( X_0(m)+n\xrightarrow{\sim}\mathbb{C}\cup\{\infty\} \). Call \( T_{m+n} \) as in (1.10) the normalized principal modulus, or normalized Hauptmodul of \( \Gamma_0(m)+n \).
The illicit aspect of monstrous moonshine is that the notion of normalized principal modulus allows us to directly compute traces of elements of the Monster on \( V^{\natural} \), without any knowledge of how to compute with elements themselves. (How can this be legal?) For an operative example let \( p \) be a prime that divides \( \#\mathbb{M} \). Then \( X_0(p)+p \) has genus zero according to Equation (1.1) and Proposition 1.1, and it develops that there is an order \( p \) element \( g_{p+p}\in\mathbb{M} \) such that the McKay–Thompson series \begin{equation}\label{eqn:Tg} \tag{1.11} T_{g}(\tau):=\sum \operatorname{tr}(g\,|\,V^{\natural}_n)q^n, \end{equation} for \( g=g_{p+p} \), is none other than the normalized principal modulus \( T_{p+p} \) for \( \Gamma_0(p)+p \). That is, \begin{equation} \tag{1.12} T_{g_{p+p}}=T_{p+p}. \end{equation}
The hypothesis that all the McKay–Thompson series \( T_g \) associated to the action of \( \mathbb{M} \) on \( V^{\natural} \) should be normalized principal moduli is presented by Thompson in [e9], in the same issue of the same journal in which Conway–Norton give their precise predictions [e8] for what all these normalized principal moduli should be. This hypothesis is strong, because the Monster is big (see (1.1)) and normalized principal moduli are scarce (see [e30], [e31]). On the other hand, it is hard to imagine how anyone could possibly predict all the trace functions \( T_g \), as Conway–Norton did, without such a strong hypothesis to help.
We have some sense, now, of the second condition of Proposition 1.1. To put the first condition in context we recall that an elliptic curve is called supersingular if its endomorphism ring, like that of the elliptic curve over \( \mathbb{F}_3 \) specified by \begin{equation}\label{eqn:E8|4} \tag{1.13} y^2=x^3-x, \end{equation} has rank 4 as a \( \mathbb{Z} \)-module. (The map \( \iota:(x,y)\mapsto (-x,iy) \) acts with order 4 on the solutions to (1.13) in any characteristic other than 2. In characteristic 3 the map \( (x,y)\mapsto (x+1,y) \) also acts, with order 3, and does not commute with \( \iota \).)
We can relate supersingular \( j \) values in characteristic \( p \) to the curve \( X_0(p)+p \) by noting, as Ogg did (see [2]), that the reduction of \( X_0(p) \) modulo \( p \) is a union of two copies of \( X_0(1)\bmod p \), crossing transversally at the supersingular points of \( X_0(1)\bmod p \). Moreover, the action of the non-trivial coset \begin{equation}\label{eqn:Wp} \tag{1.14} W_p:=\Gamma_0(p)+p\setminus \Gamma_0(p) \end{equation} of \( \Gamma_0(p) \) in \( \Gamma_0(p)+p \) (see (1.7)) interchanges these two copies of \( X_0(1)\bmod p \), and fixes a point of intersection precisely when it is represented by a supersingular elliptic curve that is defined over \( \mathbb{F}_p \). The quotient \( X_0(p)+p\bmod p \) (cf. (1.9)) is therefore a single copy of \( X_0(1)\bmod p \), but with ordinary double points indexed by pairs of supersingular \( j \) values that do not belong to \( \mathbb{F}_p \). In particular, the arithmetic genus of \( X_0(p)+p\bmod p \), which is the same as the topological genus of the complex curve \( X_0(p)+p \), is not zero if and only if there are supersingular \( j \) values in characteristic \( p \) that do not belong to \( \mathbb{F}_p \).
After the concrete algebraic construction [e12] of the Monster by Griess, a concrete algebraic conjectural construction of its “natural” module \( V^{\natural} \) was developed by Frenkel–Lepowsky–Meurman [e14], [e15], [e21]; the conjectural part being the expectation that the trace functions (1.11) it defines agree with the predictions of Conway–Norton. Borcherds was awarded the Fields Medal in 1998 in part for the mathematics he developed in order to solve this problem. (There is also his ensuing theory of automorphic products. See [e24], [e25], [e26].) More specifically, Borcherds initiated the theory of vertex algebras [e17] and the theory of generalized Kac–Moody algebras [e20], [e23], and used these theories in [e24] to reduce the verification to a short calculation.
The work of Borcherds earned him at least half a bottle of Jack Daniels, because it explained “one half” of the coincidence between Equation (1.1) and Proposition 1.1. Namely, it explained why it is that if \( p \) divides the order of the Monster then \( X_0(p)+p \) has genus zero. To this day the other half remains unexplained: We don’t yet know how to go from a modular curve \( X_0(p)+p \) that has genus zero, or the supersingular elliptic curves in characteristic \( p \) for \( p \) as in Proposition 1.1, to an order-\( p \) automorphism of the vertex-algebra structure on \( V^{\natural} \).
2. Umbral moonshine
The group-theoretic aspect of umbral moonshine goes back to Émile Mathieu’s investigation of multiply transitive permutation groups in the mid-nineteenth century: In [e1] Mathieu considers the 2-transitive action of a certain group of semiaffine transformations of \( \mathbb{F}_9 \), and gives a recipe for extending this to a 5-transitive action of a group of permutations on 12 points. (By semiaffine we mean a transformation of the form \( x\mapsto ax+b \), for \( a,b\in \mathbb{F}_9 \), or the composition of such a map with the Frobenius automorphism \( x\mapsto x^3 \).) This 5-transitive group is the one we now call \( M_{12} \); with the benefit of hindsight we recognize it as the first known sporadic simple group. For its order we have \begin{equation}\label{eqn:orderm12} \tag{2.1} \# M_{12} = 2^6\cdot 3^3\cdot 5\cdot 11. \end{equation}
Also in [e1] Mathieu announces (but does not detail) a 5-transitive group of permutations of 24 points. In a subsequent work [e2] he recasts his construction of \( M_{12} \) in terms of the projective linear transformations of the projective line over \( \mathbb{F}_{11} \), and then constructs his 5-transitive permutation group of degree 24 explicitly, by using \( \mathbb{F}_{23} \) in an analogous way. (Conway gives a beautiful account of these constructions in [e6], which is reprinted as Chapter 10 of [e27]). The latter group, being the second known sporadic simple group, is the one we now call \( M_{24} \). Its order is given by \begin{equation}\label{eqn:orderm24} \tag{2.2} \# M_{24} = 2^{10}\cdot3^3\cdot5\cdot7\cdot11\cdot23. \end{equation}
For the modular aspect of umbral moonshine we look back a century or so, to the introduction of mock theta functions by Srinivasa Ramanujan. Seventeen examples of these, including \begin{equation}\label{eqn:Ram-mock} \tag{2.3} f(q):=1+\sum_{n > 0} \frac{q^{n^2}}{(1+q)^2(1+q^2)^2\cdots(1+q^{n})^2}, \end{equation} were written down by Ramanujan in his last letter to Hardy in 1920. (See pp. 354–355 of [e28] and pp. 127–131 of [e22].) Several more examples, including \begin{equation}\label{eqn:Ram-mock2} \tag{2.4} \omega(q):=\frac{1}{(1-q)^2}+\sum_{n > 0} \frac{q^{2n(n+1)}}{(1-q)^2(1-q^3)^2\cdots(1-q^{2n+1})^2}, \end{equation} were discovered decades latter (see [e38]), amongst his handwritten notes from the last year or so of his life.
The \( q \)-series (2.3)–(2.4) are similar to \( q \)-series that define modular forms. For example, we obtain the partition generating function \begin{equation}\label{eqn:ptn} \tag{2.5} \pi(q):=1+\sum_{n > 0} \frac{q^{n^2}}{(1-q)^2(1-q^2)^2\cdots(1-q^{n})^2} \end{equation} by swapping addition for subtraction in the denominators of (2.3), and \( \eta(\tau):=q^{\frac1{24}}\pi(q)^{-1} \) is a (weakly holomorphic) modular form (with multiplier system) of weight \( \frac12 \) for \( \Gamma_0(1) \). (See [e32] for a combinatorial interpretation of the coefficients of (2.3).) But Ramanujan’s mock theta functions are not related to modular forms in such a simple way, and it was not until the doctoral work of Zwegers in 2002 [e29] that a modular form-like theory for them would begin to take hold.
The mock theta functions are now regarded (once suitably modified, cf. (2.12)–(2.16)) as special cases of mock modular forms, which we may define by introducing a certain twisting of the usual modular action that defines modular forms. To motivate this twisting choose a cusp form \( g\in S_2(\Gamma) \) of weight 2, for some group \( \Gamma \) as in (1.8). Then \( g(z)\,{\mathrm d}z \) is \( \Gamma \)-invariant, in the sense that \begin{equation}\label{eqn:gdz_inv} \tag{2.6} g(z)\,{\mathrm d}z= g(\gamma z)\,{\mathrm d}\gamma z \end{equation} for \( \gamma\in \Gamma \), and \( g(z)\,{\mathrm d}z \) descends to a holomorphic 1-form on the complex curve \( X_\Gamma \). We can try now to construct a function on \( X_\Gamma \) by integrating this 1-form. Proceeding with an open mind we define \( F(\tau) \) for \( \tau\in\mathbb{H} \) by setting \begin{equation}\label{eqn:f=int} \tag{2.7} F(\tau):=\int_{\tau}^\infty g(z)\,{\mathrm d}z, \end{equation} where the integration is over any path that starts at \( \tau \) and tends to \( i\infty \). We promptly check if it worked or not: Using (2.6) we have \begin{align} F(\tau)-F(\gamma\tau)&= \int_{\tau}^\infty g(z)\,{\mathrm d}z - \int_{\gamma\tau}^\infty g(z)\,{\mathrm d}z \nonumber\\ &= \int_{\tau}^\infty g(z)\,{\mathrm d}z - \int_{\tau}^{\gamma^{-1}\infty} g(z)\,{\mathrm d}z \nonumber\\ &= \int_{\gamma^{-1}\infty}^\infty g(z)\,{\mathrm d}z \tag{2.8}\label{eqn:f-fgamma} \end{align} for \( \gamma\in \Gamma \), which is generally not zero.
So we failed to construct a function on \( \mathbb{H} \) that is invariant for the usual action of \( \Gamma \), but we succeeded in constructing a function on \( \mathbb{H} \) that is invariant for a “\( g \)-twisted” action \begin{equation}\label{eqn:Fslashggamma} \tag{2.9} (F|_{g}^{\prime}\gamma)(\tau):=F(\gamma\tau) + \int^\infty_{\gamma^{-1}\infty} g(z)\,{\mathrm d}z. \end{equation} In classical language the \( F \) in (2.7)–(2.9) is an abelian integral (cf., e.g., [e3]). In more modern language it is a mock modular form of weight 0.
The general picture (see [e34] or [e37] for reviews, and [e46] for a comprehensive reference) is that the (weakly holomorphic) mock modular forms \( \mathfrak{M}^{\operatorname{wh}}_k(\Gamma) \) of weight \( k\in \frac12\mathbb{Z} \) for \( \Gamma \) fit into a short exact sequence \begin{equation}\label{eqn:ses} \tag{2.10} 0\to M_k^{\operatorname{wh}}(\Gamma)\to\mathfrak{M}_k^{\operatorname{wh}}(\Gamma)\to \overline{M_{2-k}(\Gamma)}\to 0, \end{equation} where \( M_k(\Gamma) \) denotes the space of usual modular forms of weight \( k \) for \( \Gamma \), and \( M_k^{\operatorname{wh}}(\Gamma) \) denotes the space of weakly holomorphic modular forms of weight \( k \) for \( \Gamma \). For general \( k \) and \( g\in M_{2-k}(\Gamma) \) it is convenient to formulate the corresponding twist of the usual weight-\( k \) action of \( \Gamma \) by setting \begin{equation}\label{eqn:fslashkggamma} \tag{2.11} (F|_{k,g}\gamma)(\tau):=(c\tau+d)^{-k}F(\gamma\tau) + C\int^\infty_{-\gamma^{-1}\infty} (\tau+z)^{-k}\overline{g(-\bar{z})}\,{\mathrm d}z \end{equation} for \( \gamma=\left(\begin{smallmatrix}*&*\\c&d\end{smallmatrix}\right)\in\Gamma \), for a suitable constant \( C \). The \( g \) in (2.11) is called the shadow of \( F \) when \( F|_{k,g}\gamma=F \) for \( \gamma\in \Gamma \).
It develops that there are few examples of mock modular forms that remain bounded near the cusps of their invariance groups. For this reason it is common to use “mock modular form” as a shorthand for “weakly holomorphic mock modular form”. We will adopt that convention here, and use the qualifier “holomorphic” to describe mock modular forms which actually do remain bounded near cusps (cf. (3.3)). To formulate the mock modularity of Ramanujan’s mock theta functions \( f \) and \( \omega \) we introduce the vector-valued function \begin{equation}\label{eqn:H6plus2} \tag{2.12} H^{(6+2)}(\tau):= \begin{pmatrix} -2q^{-\frac1{24}}f(q) \\ 4q^{-\frac16}(\omega(q)-\omega(-q)) \\ 0 \\ 4q^{\frac13}(\omega(q)+\omega(-q)) \\ 2q^{-\frac1{24}}f(q) \end{pmatrix}, \end{equation} where \( f \) and \( \omega \) are as in (2.3)–(2.4). Then \( H^{(6+2)} \) is a vector-valued mock modular form of weight \( \frac12 \) for \( \Gamma_0(1) \), with shadow a certain vector-valued unary theta function of weight \( \frac32 \) (see [e53]).
It is time to describe umbral moonshine (“moonshine with shadows”), which first emerged in 2011 [e40]. Armed with subsequent works, especially [e53], [e41], [e48], [e44], [e45], we may formulate its main features as follows.
- To each non-Fricke genus-zero curve of the form \[ X_0(m)+n,n^{\prime},\dots \] is attached a distinguished vector-valued mock modular form \( H^{(m+n,n^{\prime},\dots)} \) of weight \( \frac12 \), all of whose coefficients are integers.
- If the genus-zero curve in
part (1) takes the form
- \( X_0(m) \),
- \( X_0(m)+\frac m2 \),
- \( X_0(12)+4 \) or
- \( X_0(30)+6,10,15 \),
then the associated mock modular form \( H^{(m+n,n^{\prime},\dots)} \) serves as the graded dimension of an infinite-dimensional module \( K^{(m+n,n^{\prime},\dots)} \) for an associated finite group \( G^{(m+n,n^{\prime},\dots)} \).
The elements of (1.7) form the Atkin–Lehner coset \( W_n \) of \( \Gamma_0(m) \) in the full group of isometries of \( \mathbb{H} \) (cf. (1.14)). For the statement of Theorem 2.1 we extend the notation of (1.9) by writing \( X_0(m)+n,n^{\prime},\dots \) for the natural compactification of the quotient of \( \mathbb{H} \) defined by the action of the group \( \Gamma_0(m)+n,n^{\prime},\dots \) obtained by adjoining several Atkin–Lehner cosets \( W_n, W_{n^{\prime}}, \dots \) to \( \Gamma_0(m) \). Also, we say that \[ X_0(m)+{n,n^{\prime},\dots} \] is non-Fricke if \( m \) does not appear amongst the \( n,n^{\prime},\dots \).
There are 23 genus-zero curves appearing in part (2), and they are in natural correspondence with the unimodular even lattices of rank 24 with roots. See [e41] for the details of this. For a curve \( X_0(m)+n,n^{\prime},\dots \) as in part (2) we have \begin{equation} \tag{2.13} G^{(m+n,n^{\prime},\dots)}=\operatorname{Aut}(L)/\operatorname{Weyl}(L), \end{equation} where \( L=L^{(m+n,n^{\prime},\dots)} \) is the corresponding lattice and \( W(L) \) is the group generated by reflections in the roots of \( L \). Concrete algebraic constructions have been given for several of the \( K^{(m+n,n^{\prime},\dots)} \) (see [e51], [e52], [e54], [e47], [e49]), but many, including \( K^{(2)} \) and \( K^{(3)} \) (see below), are yet to be found. At the time of writing it also remains to be determined whether or not the integer-coefficient mock modular forms associated to the non-Fricke genus-zero curves of part (1) that do not appear in part (2), e.g., \( H^{(6+2)} \), serve as graded-dimension functions of modules for finite groups in a natural way.
The pursuit of umbral moonshine was precipitated by comments of Don Zagier on a 2010 observation of Eguchi–Ooguri–Tachikawa (see [e36]) that related mock modular forms to \( M_{24} \) (cf. (2.2)), and subsequently became known as Mathieu moonshine. More recently, Aricheta observed [e50] that Ogg found clues for this in 1975 as well. For example, Ogg points out in [4] that the primes \( p\neq 2 \) for which every supersingular point of \( X_0(2) \) in characteristic \( p \) is defined over \( \mathbb{F}_p \) are 3, 5, 7, 11 and 23, because \( X_0(2p)+p \) has genus zero exactly for these primes. Comparing with (2.2) we see that these are exactly the odd primes that divide the order of \( M_{24} \). Sure enough, \( X_0(2) \) is non-Fricke and genus zero, and \( G^{(2)}=M_{24} \) is the finite group attached to \( X_0(2) \) by umbral moonshine (and Mathieu moonshine turns out to be the \( X_0(2) \)-case of umbral moonshine).
Ogg also points out in [4] that the primes \( p\neq 3 \) for which every supersingular point of \( X_0(3) \) in characteristic \( p \) is defined over \( \mathbb{F}_p \) are 2, 5 and 11, because \begin{equation}\label{eqn:X03p+p} \tag{2.14} X_0(3p)+p \end{equation} has genus zero exactly for these primes. These are also exactly the primes other than 3 that divide the order of \( M_{12} \) according to (2.1), and the group \( G^{(3)} \) attached to \( X_0(3) \) by umbral moonshine turns out to be the unique perfect 2-fold cover \begin{equation}\label{eqn:ses2M12} \tag{2.15} 0\to\mathbb{Z}/2\mathbb{Z}\to 2.M_{12}\to M_{12}\to 1 \end{equation} of \( M_{12} \). (Note that this 2-fold cover (2.15) of \( M_{12} \) is the Schur cover of \( M_{12} \), and the Schur cover of \( M_{24} \) is none other than \( M_{24} \) itself. See, e.g., [e16].)
Just as in monstrous moonshine, Ogg’s work is telling us the McKay–Thompson series (cf. (1.11)) associated to the actions of elements of prime order in the groups \( G^{(m+n,n^{\prime},\dots)} \) on the modules \( K^{(m+n,n^{\prime},\dots)} \). (In other words, the illicit aspect of moonshine “extends to the shadows”.) For example, the mock modular form \( H^{(6+2)} \) that we get from part (1) of Theorem 2.1 and the fact that \( X_0(3p)+p \) has genus zero for \( p=2 \), reappears, repackaged, as the McKay–Thompson series \begin{equation}\label{eqn:H32B} \tag{2.16} H^{(3)}_{2B}(\tau)= \left( \begin{matrix} -2q^{-\frac{1}{12}}f(q^2) \\ -4q^{\frac23}\omega(-q) \end{matrix} \right) \end{equation} (cf. (2.12)) associated to the action of an (noncentral) element of order \( p=2 \) in \( G^{(3)}=2.M_{12} \) on \( K^{(3)} \).
It is crucial for the above that the mock modular forms of Theorem 2.1 are distinguished from general mock modular forms in a way that is directly analogous to how the principal moduli (1.10) are distinguished from general modular functions. (Otherwise there would be little hope of specifying them explicitly, as was done in [e40], [e41].) We refer to Section 4 of [e43] and Section 1 of [e48] for detailed discussions of this. The connection to genus zero curves \( X_0(m)+n,n^{\prime},\dots \) is explained in [e53].
Apart from relationships to representations of groups, the integer-coefficient property in part (1) of Theorem 2.1 is significant in that it stands in contrast to the general expectation (see [e35], [e39], [e53]) that the general mock modular form has transcendental coefficients. This property, which of course holds for all of Ramanujan’s examples, is perhaps part of the reason that it took so long to incorporate them into a convenient theory. From the point of view of umbral moonshine, the reason why the mock theta functions (2.3)–(2.4) exist, and in particular the reason why they have integer coefficients, is that \( X_0(6)+2 \) has genus zero. (One can check that all of the mock theta functions of Ramanujan appear as components of, or linear combinations of components of, the mock modular forms of Theorem 2.1. See [e53].)
3. Arithmetic
Let \( J_0(m) \) denote the Jacobian of \( X_0(m) \). Ogg’s torsion conjecture (see Conjecture 2 of [3]), proven by Mazur (see Theorem 1 of [e7]), is the statement that the torsion subgroup \( J_0(p)(\mathbb{Q})_{\mathrm{tor}} \) of the group of \( \mathbb{Q} \)-rational points on \( J_0(p) \) is — for \( p \) prime — cyclic with order \begin{equation}\label{eqn:J0Ntor} \tag{3.1} \# J_0(p)(\mathbb{Q})_{\mathrm{tor}} = n_p := \operatorname{num}\left(\frac{p-1}{12}\right). \end{equation} (Write \( \operatorname{num}(\alpha) \) for the numerator of a rational number \( \alpha \).) At the level of modular forms, (3.1) implies the existence of an integer-coefficient cusp form \( g\in S_2(\Gamma_0(p)) \) such that \begin{equation}\label{eqn:eiscspcng} \tag{3.2} pE_2(p\tau)-E_2(\tau) \equiv g(\tau) \bmod n_p, \end{equation} where \( E_2(\tau):=1-24\sum_{n > 0}nq^n(1-q^n)^{-1} \) is the quasimodular Eisenstein series.
In [e56] this congruence (3.2) is used to establish an infinite family of relationships between mock modular forms and (abelian) finite simple groups. The presence of the cusp form in (3.2) entails arithmetic consequences.
To motivate the methods of [e56] we focus first on the case that \( p=11 \). This is natural in that \( p=11 \) is the smallest prime for which \( S_2(\Gamma_0(p)) \) is not trivial. It is also of interest on account of a special relationship to the \( X_0(3) \) case of umbral moonshine (cf. Theorem 2.1). Recall from Section 2 that the finite group associated to \( X_0(3) \) is \( G^{(3)}=2.M_{12} \) (cf. (2.15)). In [e55] an operation on mock modular forms is presented that transforms each McKay–Thompson series \( H^{(3)}_g \) for \( g\in 2.M_{12} \) (cf. (2.16)) into an integer-coefficient holomorphic mock modular form \begin{equation}\label{eqn:msH2M12g} \tag{3.3} \mathcal{H}_g^{2.M_{12}}(\tau)=-2+\sum_{D < 0}C^{2.M_{12}}_g(D)q^{|D|} \end{equation} of weight \( \frac32 \) for \( \Gamma_0(4o(g)) \). Moreover, it is shown that this operation lifts to the level of modules. That is, there exists a virtual graded \( 2.M_{12} \)-module \begin{equation}\label{eqn:W2M12} \tag{3.4} W^{2.M_{12}}=\bigoplus_{D < 0} W^{2.M_{12}}_D \end{equation} such that \begin{equation}\label{eqn:C2M12} \tag{3.5} C^{2.M_{12}}_g(D)=\operatorname{tr}(g\,|\,W^{2.M_{12}}_D) \end{equation} (cf. (3.3)) for each \( D < 0 \). (For us a virtual module is an integer combination of irreducible modules, and a virtual graded module is an indexed collection of virtual modules.)
For \( D \) an integer write \( \mathcal{Q}(D) \) for the set of integer-coefficient binary quadratic forms \begin{equation}\label{eqn:Qxy} \tag{3.6} Q(x,y)=Ax^2+Bxy+Cy^2 \end{equation} of discriminant \( D=B^2-4AC \), and for \( Q\in \mathcal{Q}(D) \) let \( \Gamma_0(1)_Q \) denote the stabilizer of \( Q \) in \( \Gamma_0(1) \), where the action is given by \begin{equation}\label{eqn:Qabcd} \tag{3.7} \left(Q\left|\left(\begin{smallmatrix}a&b\\c&d\end{smallmatrix}\right)\right.\right)(x,y):=Q(ax+by,cx+dy). \end{equation} For \( D < 0 \) the Hurwitz class number of \( D \) is \begin{equation}\label{eqn:HD} \tag{3.8} H(D):=\sum_{Q\in \mathcal{Q}(D)/\Gamma_0(1)}\frac1{\#\Gamma_0(1)_Q}, \end{equation} and the Hurwitz class number generating function \begin{equation}\label{eqn:msH} \tag{3.9} \mathcal{H}(\tau):=-\frac1{12}+\sum_{D < 0} H(D)q^{|D|} \end{equation} is the unique (up to scale) holomorphic mock modular form of weight \( \frac32 \) for \( \Gamma_0(4) \).
Comparing (3.3) and (3.9) we conclude that \begin{equation}\label{eqn:msH2M121A} \tag{3.10} \mathcal{H}_{1A}^{2.M_{12}}=24\mathcal{H}. \end{equation} Thus the \( X_0(3) \) case of umbral moonshine gives us an interpretation of \( 24H(D) \) as the graded dimension of a virtual module \( W^{2.M_{12}}_D \) for \( 2.M_{12} \), for each Hurwitz class number \( H(D) \).
To get a sense for how the McKay–Thompson series associated to this module structure (3.3)–(3.5) look, we generalize the construction (3.8)–(3.9) as follows. For \( N \) a positive integer let \( \mathcal{Q}_N(D) \) denote the set of discriminant-\( D \) integer-coefficient binary quadratic forms \( Q \) as in (3.6) but with \( A\equiv 0\bmod N \), and observe that the rule (3.7) defines an action of \( \Gamma_0(N) \) on \( \mathcal{Q}_N(D) \). Next define the generalized Hurwitz class number \begin{equation}\label{eqn:HND} \tag{3.11} {H}_N(D):=\sum_{Q\in\mathcal{Q}_N(D)/\Gamma_0(N)}\frac1{\#\Gamma_0(N)_Q}, \end{equation} and the generalized Hurwitz class number generating function \begin{equation}\label{eqn:msHN} \tag{3.12} \mathcal{H}_N(\tau):=-\frac1{12}[\Gamma_0(1):\Gamma_0(N)]+\sum_{D < 0}{H}_N(D)q^{|D|}. \end{equation} Then \( \mathcal{H}_N \) is a holomorphic mock modular form of weight \( \frac32 \) for \( \Gamma_0(4N) \).
Now recall that \( 2.M_{12} \) has an element of order 11, because \( X_0(33)+11 \) has genus zero (cf. (2.14)). By direct computation we find that the corresponding McKay–Thompson series on \( W^{2.M_{12}} \), which we denote \( \mathcal{H}^{2.M_{12}}_{11A} \) (cf. (3.3)), takes the form \begin{equation}\label{eqn:msH2M1211A} \tag{3.13} \mathcal{H}^{2.M_{12}}_{11A}= \frac15(11\mathcal{H}_{11}-12\mathcal{H}) -\frac{11}{5}\varphi_{11} \end{equation} (cf. (3.10)), where \( \mathcal{H}_{11} \) is defined by (3.12) and \( \varphi_{11} \) is the unique integer-coefficient plus-space cusp form of weight \( \frac32 \) for \( \Gamma_0(44) \) with \( \varphi_{11}(\tau)=q^3+O(q^4) \). (A mock modular form of half-integer weight \( k+\frac12 \) is in the Kohnen plus-space if its coefficients are supported on exponents \( n \) such that \( (-1)^kn \) is a square modulo 4.) The series \( \mathcal{H}^{2.M_{12}}_{11A} \) and \( \varphi_{11} \) have integer coefficients, so \( 11\mathcal{H}_{11}-12\mathcal{H} \) has integer coefficients too. We conclude from (3.13) that \begin{equation}\label{eqn:11msH1112msHequivvarphi11} \tag{3.14} 11\mathcal{H}_{11}-12\mathcal{H}\equiv\varphi_{11}\bmod 5, \end{equation} and if not for this congruence (3.14) then \( W^{2.M_{12}} \) would not exist, and neither would umbral moonshine for \( X_0(3) \). Of course \begin{equation} \tag{3.15} \#J_0(11)(\mathbb{Q})_\mathrm{ tor} = n_{11} = \operatorname{num}\left(\frac{11-1}{12}\right) = 5 \end{equation} (cf. (3.1)) according to Mazur’s proof of Ogg’s torsion conjecture.
Work of Waldspurger (see [e10], [e11]) implies that the coefficients of \( \varphi_{11} \) are proportional to special values of \( L \)-functions of quadratic twists of the abelian variety \( J_0(11) \) (which turns out to be an elliptic curve because \( X_0(11) \) has genus one). On the other hand the coefficients of \( \mathcal{H} \) and \( \mathcal{H}_{11} \) may be expressed in terms of class numbers of imaginary-quadratic number fields by construction (3.8)–(3.9), (3.11)–(3.12). It follows that the integer-coefficient property of (3.10) {“prevents”} the \( L \)-function of a twist of \( J_0(11) \) from vanishing if a corresponding linear combination of imaginary-quadratic class numbers is not divisible by \( n_{11}=5 \). This, together with Kolyvagin’s work [e19] on Euler systems, allows to prove the following.
(Call \( D\in\mathbb{Z} \) a fundamental discriminant if \( D \) is the discriminant of \( \mathbb{Q}(\sqrt{D}) \). Equivalently, \( D \) is odd and squarefree, or \( D=4d \) for some odd and squarefree \( d \).)
Just as Ogg’s torsion conjecture (3.1) is valid for all primes, the story (3.13)–(3.16) we have just told for \( p=11 \) generalizes to all primes too. To explain this define \begin{equation}\label{eqn:np^{\prime}} \tag{3.17} n_p^{\prime}:=\operatorname{num}\left(\frac{p+1}{6}\right), \quad d_p^{\prime}:=\operatorname{den}\left(\frac{p+1}{6}\right), \quad d_p:=\operatorname{den}\left(\frac{p-1}{12}\right) \end{equation} (cf. (3.1), and write \( \operatorname{den}(\alpha) \) for the denominator of \( \alpha\in \mathbb{Q} \)), and define \begin{equation}\label{eqn:msHZZpZZ1} \tag{3.18} \mathcal{H}^{\mathbb{Z}/p\mathbb{Z}}_{1}:=12n_p^{\prime}\mathcal{H}. \end{equation} Supersingular elliptic curves make a pleasing reentrance now (cf. Proposition 1.1), and not just in the characteristics \( p \) for which \( X_0(p)+p \) has genus zero: In [e56] the endomorphism rings of the supersingular elliptic curves in characteristic \( p \) are used to prove the existence of an integer-coefficient cusp form \( \varphi_p \) such that \begin{equation}\label{eqn:msHZZpZZp} \tag{3.19} \mathcal{H}^{\mathbb{Z}/p\mathbb{Z}}_{p} :=\frac{1}{n_p}\left(\frac{d_pd_p^{\prime}}{6}p\mathcal{H}_p-d_pn_p^{\prime}\mathcal{H}\right) -\frac{p}{n_p}\varphi_p \end{equation} is an integer-coefficient holomorphic mock modular form of weight \( \frac32 \) for \( \Gamma_0(4p) \). (See Lemma 4.1.4 of [e56].) These forms (3.18)–(3.19) are congruent modulo \( p \) by construction, so if we define \( \mathcal{H}^{\mathbb{Z}/p\mathbb{Z}}_g := \mathcal{H}^{\mathbb{Z}/p\mathbb{Z}}_{o(g)} \) for \( g\in \mathbb{Z}/p\mathbb{Z} \) then there exists a virtual graded \( \mathbb{Z}/p\mathbb{Z} \)-module \begin{equation}\label{eqn:WZZpZZ} \tag{3.20} W^{\mathbb{Z}/p\mathbb{Z}}=\bigoplus_{D < 0}W^{\mathbb{Z}/p\mathbb{Z}}_D \end{equation} (cf. (3.4)) such that \begin{equation}\label{eqn:ZZpZZ} \tag{3.21} C^{\mathbb{Z}/p\mathbb{Z}}_g(D)=\operatorname{tr}(g\,|\,W^{\mathbb{Z}/p\mathbb{Z}}_D) \end{equation} for each \( D < 0 \) (cf. (3.5)), where \begin{equation} \tag{3.22} \mathcal{H}^{\mathbb{Z}/p\mathbb{Z}}_g = -n_p^{\prime} +\sum_{D < 0} C_g^{\mathbb{Z}/p\mathbb{Z}}(D)q^{|D|} \end{equation} (cf. (3.3)). We recover \( \mathcal{H}^{2.M_{12}}_{1A} \) (see (3.10)) by taking \( p=11 \) in (3.18), and we recover \( \mathcal{H}^{2.M_{12}}_{11A} \) (see (3.13)) by taking \( p=11 \) and making a suitable choice of \( \varphi_{p} \) in (3.19). We recover the restriction of \( W^{2.M_{12}} \) (cf. (3.4)) to any cyclic subgroup of \( 2.M_{12} \) of order 11 by taking \( p=11 \) (and making a suitable choice of \( \varphi_{p} \)) in (3.20). The involvement of \( \varphi_{p} \) in (3.19) allows to prove the following, which specializes to Proposition 3.1 when \( p=11 \).
(See [e33] for generalities on twists of commutative algebraic groups.)
The reader will notice that Proposition 3.1 doesn’t use any more of the \( 2.M_{12} \)-module structure on \( W^{2.M_{12}} \) than its restriction to a cyclic subgroup of order 11. This restricted module structure, in turn, depends mainly on the congruence \begin{equation} \tag{3.23} 11\mathcal{H}_{11}-12\mathcal{H} \equiv \varphi_{11} \bmod 5 \end{equation} (cf. (3.13)), but the same can be said of Proposition 3.1. This suggests that it might not be necessary to mention modules for finite groups in order to establish said result. Sure enough, the conclusion of Proposition 3.1 was established much earlier in [e18], without any reference to graded virtual modules for finite groups.
So is there arithmetic meaning to the \( 2.M_{12} \)-module structure on \( W^{2.M_{12}} \)? In [e55] a positive answer to this question is formulated, for the restriction, \begin{equation}\label{eqn:WM11} \tag{3.24} W^{M_{11}}=\bigoplus_{D < 0}W^{M_{11}}_D, \end{equation} of \( W^{2.M_{12}} \) to a copy of the smallest sporadic simple group, \( M_{11} \), which may be described as the stabilizer of a point for the 5-transitive action of \( M_{12} \) on 12 points, \begin{equation} \tag{3.25} \# M_{11} = 2^4\cdot3^2\cdot5\cdot11 \end{equation} (cf. (2.1)). (In fact there are two conjugacy classes of subgroups of \( 2.M_{12} \) isomorphic to \( M_{11} \): One maps to the stabilizer of a point under the natural map \( 2.M_{12}\to M_{12} \) (cf. (2.15)), but the other maps to a transitive subgroup of \( S_{12} \). To define \( W^{M_{11}} \) (3.24) we should restrict \( W^{2.M_{12}} \) to a transitive copy of \( M_{11} \).)
It develops that \( M_{11} \) has elements of order 8, and for such an element the associated McKay–Thompson series on \( W^{M_{11}} \), which we denote \( \mathcal{H}^{M_{11}}_{8AB} \), takes the form \begin{equation} \tag{3.26} \mathcal{H}^{M_{11}}_{8AB}(\tau) = -2\theta(4\tau)(2\theta(64\tau)-\theta(16\tau))^2 - \varphi_{8|4} \end{equation} (cf. (3.13)), where \( \theta(\tau):=\sum_n q^{n^2} \) and \( \varphi_{8|4} \) is a certain cusp form of weight \( \frac32 \) for \( \Gamma_0(32) \) (with a character of order 4). Now \( \varphi_{8|4} \) is related to the quadratic twists \begin{equation}\label{eqn:y2x3D2x} \tag{3.27} y^2 = x^3-D^2x \end{equation} of the elliptic curve over \( \mathbb{Q} \) defined by (1.13) in the same way as \( \varphi_{11} \) is related to the quadratic twists (3.16) of \( J_0(11) \). These twists (3.27) are in turn related to the “congruent number problem” of antiquity: It develops that \( |D| \) is a congruent number — that is, the area of a right triangle with rational side lengths — if and only if the elliptic curve defined by (3.27) has infinitely many rational points. (See [e4] for the early history of the congruent number problem, see [e42] for a more recent expository account, and see [e13] for an explanation of the connection to elliptic curves.)
The relationship between \( \varphi_{8|4} \), the elliptic curves of (3.27) and the congruent number problem plays a key role in the proof of the following result.
The stabilizer of a point for the 5-transitive action of \( M_{24} \) on 24 points, denoted \( M_{23} \), is also a sporadic simple group, \begin{equation} \tag{3.28} \# M_{23} = 2^7\cdot3^2\cdot5\cdot7\cdot11\cdot23 \end{equation} (cf. (2.2)). In forthcoming joint work with Cheng and Mertens we show that the \( \mathbb{Z}/p\mathbb{Z} \)-module structure on \( W^{\mathbb{Z}/p\mathbb{Z}} \) (cf. (3.20)) extends to a \( M_{23} \)-module structure when \( p=23 \). The existence or otherwise of richer module structures on the \( W^{\mathbb{Z}/p\mathbb{Z}} \) for general \( p \) is a focus for future research.
