Umbral moonshine and the Niemeier lattices

In this paper, we relate umbral moonshine to the Niemeier lattices - the 23 even unimodular positive-definite lattices of rank 24 with non-trivial root systems. To each Niemeier lattice, we attach a finite group by considering a naturally defined quotient of the lattice automorphism group, and for each conjugacy class of each of these groups, we identify a vector-valued mock modular form whose components coincide with mock theta functions of Ramanujan in many cases. This leads to the umbral moonshine conjecture, stating that an infinite-dimensional module is assigned to each of the Niemeier lattices in such a way that the associated graded trace functions are mock modular forms of a distinguished nature. These constructions and conjectures extend those of our earlier paper and in particular include the Mathieu moonshine observed by Eguchi, Ooguri and Tachikawa as a special case. Our analysis also highlights a correspondence between genus zero groups and Niemeier lattices. As a part of this relation, we recognise the Coxeter numbers of Niemeier root systems with a type A component as exactly those levels for which the corresponding classical modular curve has genus zero. 11F22; 11F37; 11F46; 11F50; 20C34; 20C35


Introduction
In this paper we relate umbral moonshine to the Niemeier lattices. This relation associates one case of umbral moonshine to each of the 23 Niemeier lattices and in particular constitutes an extension of our previous work [1], incorporating 17 new instances. Moreover, this prescription displays interesting connections to genus zero groups (subgroups Γ < SL 2 (R) that define a genus zero quotient of the upper-half plane) and extended Dynkin diagrams via McKay's correspondence.
To explain this moonshine relation, let us first recall what Niemeier lattices are. In 1973 Niemeier classified the even unimodular positive-definite lattices of rank 24 [2]. There are 24 of them, including the so-called Leech lattice discovered almost a decade earlier in the context of optimal lattice sphere packing in 24 dimensions [3]. Niemeier proved that the Leech lattice is the unique even, unimodular and positive-definite lattice of rank 24 with no root vectors (lattice vectors with norm-square 2), while the other 23 even unimodular rank 24-dimensional lattices all have root systems of the full rank 24. Moreover, these 23 lattices are uniquely labelled by their root systems, which are in turn uniquely specified by the following two conditions: first, they are unions of simply-laced root systems with the same Coxeter numbers; second, the total rank is 24.
We will refer to these 23 root systems as the Niemeier root systems and the 23 corresponding lattices as the Niemeier lattices. In this paper we associate a finite group and a set of vectorvalued mock modular forms to each of these 23  where we write q = e 2πiτ . Following Thompson's idea [4] that J(τ ) should be the graded dimension of an infinite-dimensional module for M this observation was later expanded into the full monstrous moonshine conjecture by Conway and Norton [5], conjecturing that the graded character T g (τ ) attached to the monster module and g ∈ M should be a principal modulus for a certain genus zero group Γ g < SL 2 (R). (When a discrete group Γ < SL 2 (R) has the property that Γ\H is isomorphic to the Riemann sphere minus finitely many points, there exists a holomorphic function f on H that generates the field of Γ-invariant functions on H. Such a function f is called a principal modulus, or Hauptmodul, for Γ.) We refer to [6] or the introduction of [1] for a more detailed account of monstrous moonshine.
In 2010 the study of a new type of moonshine was triggered by an observation of Eguchi-Ooguri-Tachikawa, which constituted an analogue of McKay's observation in monstrous moonshine. In the work of Eguchi-Taormina and Eguchi-Ooguri-Taormina-Yang in the 1980's [7][8][9], these authors encountered a q-series H (2) (τ ) = 2 q −1/8 (−1 + 45 q + 231 q 2 + 770 q 3 + 2277 q 4 + · · · ) (1. 3) in the decomposition of the elliptic genus of a K3 surface into irreducible characters of the N = 4 superconformal algebra. It was later understood by Eguchi-Hikami [10] that the above q-series is a mock modular form. See §3 for the definition of mock modular forms. Subsequently the coincidence between the numbers 45, 231, 770, 2277,. . . and the dimensions of irreducible representations of M 24 was pointed out in [11]. This observation was later extended into a Mathieu moonshine conjecture in [12][13][14][15] by providing the corresponding twisted characters, the mock modular forms H (2) g , and was moreover related in a more general context to the K3-compactification of superstring theory in [12]. Very recently, the existence of an infinitedimensional M 24 -module underlying the mock modular form (1.3) and those constructed in [12][13][14][15] was shown by T. Gannon [16], although the nature of this M 24 -module remains mysterious.
Meanwhile, it was found that Mathieu moonshine is but one example of a more general phenomenon, umbral moonshine. In [1] we associated a finite group G ( ) and a vector-valued mock modular form H Despite the discovery of this more general framework of umbral moonshine, encompassing Mathieu moonshine as a special case, which moreover displays various beautiful properties, many questions remained unanswered. For example: why these specific umbral groups G ( ) ? Why are they labelled by divisors of the number 12? What is the structure underlying all these instances of moonshine?
In the present paper we provide partial answers to the above questions. We present evidence that there exists an instance of umbral moonshine naturally associated to each of the 23 Niemeier lattices. As a Niemeier lattice is uniquely determined by its root system X, in the main text we shall use X (or equivalently the corresponding lambency; see Tables 2-3) to label the instances of umbral moonshine. In particular, we construct in each instance an umbral group G X as the quotient of the automorphism group of the corresponding Niemeier lattice L X by the normal subgroup generated by refections in root vectors. This property gives a uniform construction as well as a concrete understanding of the umbral groups.
Similarly, we provide a prescription that attaches to each of the 23 Niemeier lattices a distinguished vector-valued modular form-the umbral mock modular form H X -which conjecturally encodes the dimensions of the homogeneous subspaces of the corresponding umbral module. The Niemeier lattice uniquely specifies the shadow of the mock modular form through a map which associates a unary theta series of a specific type to each of the irreducible simply-laced ADE root systems, as well as unions of such root systems where all the irreducible components have the same Coxeter number. As will be explained in §3.3, this map bears a strong resemblance to the ADE classification by Cappelli-Itzykson-Zuber of modular invariant combinations of the characters of theÂ 1 affine Lie algebra [24]. When applied to the Niemeier root systems, we dictate the resulting unary theta series to be the shadow of the corresponding umbral mock modular form. Together with the natural requirement that H X satisfies an optimal growth condition the specification of the shadow uniquely fixes the desired umbral form (cf. Theorem

and Corollary 4.2).
By associating a case of umbral moonshine to each Niemeier lattice we extend our earlier work on umbral moonshine to include 17 more instances. In fact, the 6 instances discussed in the earlier paper, labelled by the 6 divisors of 12, correspond to pure A-type Niemeier root systems containing only A-type irreducible components. There are 8 pure A-type Niemeier root systems, one for each divisor − 1 of 24, and they are given simply as the union of 24 −1 copies of A −1 . This new proposal relating Niemeier lattices and umbral moonshine can be regarded as a completion of our earlier work [1], in that it includes Niemeier root systems with D-or/and E-components and sheds important light on the underlying structure of umbral moonshine.
More properties of umbral moonshine reveal themselves as new instances are included and as the structure of umbral moonshine is examined in light of the connection to Niemeier lattices.
Recall that in [1] we observed a connection between the (extended) Dynkin diagrams and some of the groups G ( ) via McKay's correspondence for subgroups of SU (2). In the present paper we observe that the same holds for many of the new instances of umbral moonshine, and the result presents itself as a natural sequence of extended Dynkin diagrams with decreasing rank, starting withÊ 8 and ending withÂ 1 . Moreover, we observe an interesting relation between umbral moonshine and the genus zero groups Γ < SL 2 (R) through the shadows of the former and the principal moduli for the latter. As will be discussed in §2.3 and §4.2, our construction attaches a principal modulus for a genus zero group to each Niemeier lattice. In particular, we recognise the Coxeter numbers of the root systems with an A-type component as exactly those levels for which the corresponding classical modular curve has genus zero.
The outline of this paper is as follows. In §2 we give some background on Niemeier lattices, define the umbral finite groups, and discuss the mysterious relation to extended ADE diagrams and genus zero quotients of the upper-half plane. In §3 we introduce various automorphic objects that play a rôle in umbral moonshine, including (mock) modular forms and Jacobi forms of the weak, meromorphic, and mock type. For later use we also introduce the Eichler-Zagier (or Atkin-Lehner) map on Jacobi forms, and an ADE classification of such maps. In §4 we focus on the umbral mock modular forms, which are conjecturally the generating functions of the dimensions of the homogeneous subspaces of the umbral modules. In §4-5 we give explicit formulas for these umbral mock modular forms as well as most of the umbral McKay-Thompson series. This is achieved partially with the help of multiplicative relations, relating McKay-Thompson series in different instances of umbral moonshine corresponding to Niemeier lattices with one Coxeter number being the multiple of the nother. In §6 we present the main results of the paper, which are the umbral moonshine conjectures relating the umbral groups and umbral mock modular forms, and a counterpart for umbral moonshine of the genus zero property of monstrous moonshine. We also observe certain discriminant properties relating the exponents of the powers of q in the mock modular forms and the imaginary quadratic number fields over which the homogeneous submodules of the umbral modules are defined, extending the discriminant properties observed in [1]. Finally, we present some conclusions and discussions in §7.
To provide the data and evidence in support of our conjectures, this paper also contains four appendices. In Appendix A we describe some modular forms and Jacobi forms which are utilised in the paper. In Appendix B we present tables of irreducible characters as well as the characters of certain naturally defined (signed) permutation representations of the 23 umbral groups. In Appendix C we provide the first few dozen coefficients of all the umbral McKay-Thompson series. In Appendix D, using the tables in Appendix B and C, we explicitly present decompositions into irreducible representations for the first 10 or so homogeneous subspaces of S X g The vector-valued cusp form conjectured to be the shadow of H X g , for g ∈ G X and X a Niemeier root system (cf. § §5. 1,6.2). Ω X g The 2m × 2m matrix attached to g ∈ G X for X a Niemeier root system with Coxeter number m (cf. §5.1).
K X The umbral module attached to the Niemeier root system X. A conjectural G X -module with graded-super-characters given by the H X g (cf. §6.1). n g The order of the image of an element g ∈ G X in the quotient groupḠ X (cf. §6.2).
h g The unique positive integer such that n g h g is the product of the shortest and longest cycle lengths in the cycle shapeΠ X g for g ∈ G X and X a Niemeier root system (cf. §6.2). Π X g The cycle shape attached to g ∈ G X via the permutation representation of G X with twisted Euler characterχ X (cf. § §2.4,6.2,B.2). Similarly forΠ X A g ,Π X D g , &c. ν X g The multiplier system of H X g (cf. §6.2).

Root Systems
In this subsection we give a brief summary of simply-laced root systems and their corresponding Dynkin diagrams. Standard references for this material include [25] and [26].
Let V be a finite-dimensional vector space of rank r over R equipped with an inner product ·, · . For v ∈ V define the hyperplane H v to be the set of elements of V orthogonal to v and the reflection in the hyperplane H v to be the linear map r v : V → V defined by A finite subset X ⊂ V of non-zero vectors is a rank r crystallographic root system if • r α (X) ∈ X for all α ∈ X, • X ∩ Rα = {α, −α} for all α ∈ X, • 2 α, β / α, α ∈ Z for all α, β ∈ X.
Given a root system X we say that X is irreducible provided that it can not be partitioned into proper subsets X = X 1 ∪ X 2 with α 1 , α 2 = 0 for all α 1 ∈ X 1 and α 2 ∈ X 2 . If X is an irreducible root system then there are at most two values for the length α, α 1/2 that occur. If all roots have the same length then the irreducible root system is called simply-laced.
It is possible to choose a subset of roots in X that form a basis of V . We define a subset Φ = {f 1 , f 2 , · · · , f r } ⊂ X to be a set of simple roots provided that • Φ is a basis for V , • each root α ∈ X can be written as a linear combination of the f i with integer coefficients and with either all n i ≤ 0 or all n i ≥ 0.
Given a choice of simple roots we define the positive roots of X to be those α for which all n i ≥ 0 in (2.2). The negative roots are those for which all n i ≤ 0. We also define the height of α as in (2.2) by setting To each irreducible root system we can associate a connected Dynkin diagram as follows.
We associate a node to each simple root. The nodes associated to two distinct simple roots f i , f j are then either not connected if f i , f j = 0 or connected by N ij lines with The Dynkin diagrams associated to simply-laced irreducible root systems all have N ij = {0, 1} and are of type A n , D n , E 6 , E 7 , E 8 as shown in Figure 1. Here the subscript indicates the rank of the associated root system, and in the figure we choose a specific enumeration of simple roots for later use in §2.4.
The height function defines a Z-gradation on the set of roots. Every irreducible root system has a unique root θ of largest height with respect to a given set of simple roots Φ = {f i } with an expansion where the a i are a set of integers known as the Coxeter labels of the root system or Dynkin diagram. If we append the negative of this highest root (the lowest root) to the simple roots of the simply-laced root system, we obtain the extended Dynkin diagrams of typeÂ n ,D n ,Ê 6 ,Ê 7 ,Ê 8 .
These are shown in Figure 2, where we indicate the lowest root with a filled in circle and the simple roots with empty circles. An equivalent definition of the Coxeter number may be given in terms of the Weyl group of X. The Weyl group W X is the group generated by the reflections r α for α ∈ X. The product w = r 1 r 2 . . . r r of reflections r i := r fi in simple roots f i ∈ Φ is called a Coxeter element of W X and is uniquely determined up to conjugacy in W X , meaning that different choices of simple roots and different orderings of the simple roots chosen lead to conjugate elements of W X . The Coxeter number m = m(X) is then the order of any Coxeter element of X.
We obtain a finer invariant of X by considering the eigenvalues of a Coxeter element of W X . Say u 1 , . . . , u r are the Coxeter exponents of X if a Coxeter element w has eigenvalues e 2πiu1/m , . . . , e 2πiur/m (counting multiplicity). This data is conveniently recorded using the notion of Frame shape, whereby a formal product i n ki i (with n i , k i ∈ Z and n i > 0) serves as a shorthand for the rational polynomial i (x ni − 1) ki . For each Coxeter element there is a Frame shape π X -the Coxeter Frame shape of X-such that the corresponding polynomial function coincides with the characteristic polynomial r i=1 (x − e 2πiui/m ) of w. These Frame shapes will play a prominent rôle in what follows. They are given along with the corresponding Coxeter  Table 1.

Lattices
A lattice is a free Z-module L say equipped with a symmetric bilinear form · , · . We say that a lattice L is positive-definite if · , · induces a positive-definite inner product on the vector space L R = L ⊗ Z R. Since L is a free Z-module the natural map L → L R is an embedding and we may identify L with its image in L R . Say that L is integral if we have λ, µ ∈ Z for all λ, µ ∈ L and say that L is even if we have λ, λ ∈ 2Z for each λ ∈ Λ. (An even lattice is necessarily integral.) The dual of L is the lattice L * ⊂ L R defined by setting L * = {λ ∈ L R | λ, L ⊂ Z}. (2.7) Clearly, if L is integral then L * contains L. In the case that L * coincides with (the image of) L (in L R ) we say that L is unimodular. For an even lattice L we call L 2 = {λ ∈ L | λ, λ = 2} the set of roots of L. Frame shapes Table 1: Coxeter numbers, exponents, and Frame shapes The Leech lattice is the unique (up to isomorphism) even, unimodular, positive-definite lattice of rank 24 with no roots [27], and is named for its discoverer, John Leech [3,28]. Shortly after Leech's work, the unimodular even positive-definite lattices of rank 24 were classified by Niemeier [2]; we refer to those with non-empty root sets as the Niemeier lattices. There are exactly 23 Niemeier lattices up to isomorphism, and if L is such a lattice then its isomorphism type is determined by its root set L 2 , which is a union of irreducible simply-laced root systems (cf. §2.1). Say a root system X is a Niemeier root system if it occurs as L 2 for some Niemeier lattice L. The Niemeier root systems are precisely the 23 root systems satisfying the two conditions that first, they are unions of simply-laced root systems with the same Coxeter numbers, and second, the total rank is 24. Explicitly, they are A 24 1 , A 12 (2.10). We will call them the A-type, D-type, and the E-type Niemeier root systems, respectively. We say that a Niemeier root system X has Coxeter number m if m is the Coxeter number of any simple component of X.
Since all the simple components of a Niemeier root system have the same Coxeter number all the type A components appearing have the same rank, and similarly for components of type D and E. So we can write m−1 for some non-negative integer d X A (or X A = ∅), and m the Coxeter number of X, and similarly for X D and X E . For example, if X = A 2 7 D 2 5 then m = 8, X A = A 2 7 , X D = D 2 5 , d X A = d X D = 2 and X E = ∅. (2.12) Before finishing this subsection we will comment on the relation between the Niemeier lattices and the Leech lattice. The covering radius of the Leech lattice is √ 2 according to [29], meaning that R = √ 2 is the minimal positive R such that the 24-dimensional vector space Λ R = Λ ⊗ Z R is covered by placing a closed ball of radius R at each point of Λ, (2.13) A point x ∈ Λ R that realizes the maximum value √ 2 = inf λ∈Λ x − λ is called a deep hole of Λ. Let x ∈ Λ R be a deep hole and let V x be the set of vertices of x, (2.14) It is shown in [29] that if λ, λ ∈ V x with λ = λ then λ − λ 2 ∈ {4, 6, 8}. Following [29] define the hole diagram attached to x by joining vertices λ, λ ∈ V x with a single edge if λ − λ 2 = 6, and by joining them with a double edge if λ − λ 2 = 8. The vertices λ and λ are disjoined in case λ−λ 2 = 4. Then the diagram so obtained is the extended Dynkin diagram corresponding to a Niemeier root system, and all Niemeier root systems arise in this way [29]. Conversely, from each Niemeier lattice one can obtain a different "holy" construction of the Leech lattice [30].

Genus Zero Groups
In this section we attach a genus zero subgroup of SL 2 (R) to each of the 23 Niemeier root systems.
If Γ is a discrete subgroup of SL 2 (R) that is commensurable with the modular group SL 2 (Z) then its natural action on the boundaryR = R ∪ {i∞} of the upper half plane H restricts tô The orbits of Γ onQ are called the cusps of Γ, and the quotient space is naturally a compact Riemann surface (cf. e.g. [31, §1.5]). We adopt the common practice of saying that Γ has genus zero in case X Γ is a genus zero surface.
For n a positive integer the Hecke congruence group of level n, denoted Γ 0 (n), is defined by (2. 16) Say e is an exact divisor of n, and write e n, if e|n and (e, n/e) = 1. According to [5] the normaliser N (Γ 0 (n)) of Γ 0 (n) in SL 2 (R) is commensurable with SL 2 (Z) and admits the description where h is the largest divisor of 24 such that h 2 divides n. So if e n then we obtain a coset W n (e) for Γ 0 (n) in its normaliser by setting Observe that the product of any two elements of W n (e) lies in W n (1) = Γ 0 (n). More generally, the operation e * f = ef /(e, f ) 2 equips the set of exact divisors of n with a group structure isomorphic to that obtained by multiplication of Atkin-Lehner involutions, W n (e)W n (f ) = W n (e * f ). So for S a subgroup of the group of exact divisors of n we may define a group Γ 0 (n) + S, containing and normalizing Γ 0 (n), by taking the union of the Atkin-Lehner cosets W n (e) for e ∈ S. It is traditional [5] to simplify this notation by writing Γ 0 (n) + e, f, . . . in place of Γ 0 (n) + {1, e, f, . . .}.
Define the Coxeter Frame shape π X of an arbitrary root system X to be the product of Coxeter Frame shapes of the irreducible components of X. Next, for a Frame shape π = i n ki i define the associated eta product η π by setting 24) and observe that if X is simply laced and π X is the Coxeter Frame shape of X then where r denotes the rank of X and d X is the number of irreducible components. We may also consider the lambda sum λ π attached to a Frame shape π, which is the function where λ n (τ ) is defined in (A.9). Observe that if π = i n ki i is such that i k i = 0 then definite lattice of rank 24 then the eta product of the Coxeter Frame shape of X is a principal modulus for a genus zero subgroup of SL 2 (R). It would be desirable to have a conceptual proof of this fact.
The relation between the T X and umbral moonshine will be discussed in §4.2, where the weight two Eisenstein forms will play a prominent rôle. (Cf. Table 6.) We have i k i = 0 when π X = i n ki i for every Niemeier root system X, so the functions f X and T X are related by It is interesting to note that all of the Γ X , except for X = A 24 , appear in monstrous moonshine as genus zero groups for whom monstrous McKay-Thompson series serve as principal moduli. Indeed, all of the Frame shapes π X , except for X = A 24 , are Frame shapes of elements of Conway's group Co 0 , the automorphism group of the Leech lattice (cf. [5, §7]). We observe that for the cases that π X is the Frame shape of an element in Co 0 the corresponding centralizer in Co 0 typically contains a subgroup isomorphic to G X .
We include the ATLAS names [33] (see also [5]) for the monstrous conjugacy classes corresponding to the groups Γ X via monstrous moonshine in the rows labelled Γ X in Table 2. Extending the notation utilised in [1] we assign lambencies -now symbols rather than integers-to each Niemeier system X according to the prescription of Table 2. The lambencies then serve to name the groups Γ X also, according to the convention that n corresponds to Γ 0 (n), and 12 + 4 corresponds to Γ 0 (12)+4, &c. It will be convenient in what follows to sometimes use ( ) in place of X, writing G ( ) , H ( ) , &c., to label the finite groups and mock modular forms associated to the corresponding Niemeier root system.

Umbral Groups
Given a Niemeier root system X we may consider the automorphism group of the associated Niemeier lattice L X . The reflections in roots of L X generate a normal subgroup of the full automorphism group of L X -the Weyl group of X-which we denote W X . We define G X to be the corresponding quotient, The particular groups G X arising in this way are displayed in Table 3. Observe 1 that the group G ( ) of [1] appears here as G X for X the unique root system with a component A −1 . In fact, the G ( ) of [1] are exactly those G X for which X is of the form X = A d −1 with even d. It will develop that, for every Niemeier root system X, the representation theory of G X is intimately related to a set of vector-valued mock modular forms H X g , to be introduced in §4-5.
As mentioned in §2.3, it will often be useful to use the lambencies to label the groups and mock modular forms associated to a given Niemeier root system. To this end we define G (n) = G X in case Γ 0 (n) has genus zero and X is the unique A-type Niemeier root system with 1 We are grateful to George Glauberman for first alerting us to this fact.
Coxeter number n (cf. (2.8)). We define G (2n+n) = G X when Γ 0 (2n) + n has genus zero and X is the unique D-type Niemeier root system with Coxeter number 2n (cf. (2.9)). We write G (12+4) for G X when X = E 4 6 and we write G (30+6, 10,15) for G X when X = E 3 8 . Observe that the subgroupŴ X < Aut(L X ) consisting of automorphisms of L X that stabilize the irreducible components of X is also normal in Aut(L X ). DefineḠ X to be the corresponding quotient,Ḡ so thatḠ X is precisely the group of permutations of the irreducible components of X induced by automorphisms of L X , and is a quotient of G X (viz., the quotient byŴ X /W X ) since W X <Ŵ X . It turns out thatŴ X /W X has order 2 when X A = ∅ or X = X E = E 4 6 , has order 3 when X = X D = D 6 4 , and is trivial otherwise. Remark 2.3. In terms of the notation of [30] The groups G X andḠ X come naturally equipped with various permutation representations.
To see this choose a set Φ of simple roots for L X , meaning a set which is the union of sets of simple roots for each irreducible root sublattice of L X . Then Φ constitutes a basis for the 24-dimensional space L X R , and for each g ∈ G X there is a unique element in the preimage of g under Aut(L X ) → G X that belongs to the subgroup Aut(L X , Φ), consisting of automorphisms of L X that stabilize Φ as a set (i.e. act as permutations of the irreducible root subsystems followed by permutations-corresponding to Dynkin diagram automorphisms-of simple roots within irreducible root subsystems). Thus we obtain a section G X → Aut(L X ) of the projection Aut(L X ) → G X whose image is Aut(L X , Φ), and composition with the natural contains the roots in Φ belonging to type A components of X, and similarly for Φ D and Φ E .
since Aut(L X ) cannot mix roots that belong to non-isomorphic root systems, so we obtain and interpretχ X D andχ X E similarly. Observe thatχ X (and hence alsoχ X A ,χ X D andχ X E ) are independent of the choice of Φ. We setχ X A = 0 in case Φ A is empty, and similarly forχ X D andχ X E . We haveχ (2.32) The charactersχ X A , &c., are naturally decomposed further as follows. Suppose that X A = ∅.
(2.11)) and we may write where the superscript indicates the A m−1 component to which the simple root f i j belongs, and the inner products between the f i j for varying j are as described by the labeling in Figure 1 (so that f i j , f i k is −1 or 0 according as the nodes labelled f j and f k are joined by an edge or not). Then for fixed j the vectors We denote the corresponding character g →χ X A g since the isomorphism type of the representation is evidently independent of the choice of j. Observe thatχ X A is generally not a faithful character since permutations of Φ A arising from diagram automorphisms, exchanging f i with different j in the range 0 < j < m/2 furnishing isomorphic (signed permutation) representations; we denote the corresponding character g → χ X A g . Since the f i j are linearly independent we can conclude that by counting the possibilities for j in each case.
If Φ D is non-empty then m is even and where, similar to the above, the superscript indicates the D m/2+1 component to which the simple root f i j belongs, and the inner products between the f i j for varying j are as described in Figure 1. Suppose first that m > 6. Then m/2 + 1 > 4 and the only non-trivial diagram automorphism of D m/2+1 has order 2 and interchanges f i m/2 and f i m/2+1 . So we find that for 1 ≤ j < m/2 the sets we denote the character of this (i.e. any one of the these) permutation representation(s) by g →χ X D g . We define χ X D to be the (signed permutation) character of the representation spanned by the vectors when X D = ∅ and m > 6. In case m = 6 the group of diagram automorphisms of D m/2+1 = D 4 is a copy of S 3 , acting transitively on the sets {f i 1 , f i 3 , f i 4 } (for fixed i), so we defineχ X D to be the character attached to the (permutation) representation of G X spanned by and defineχ X D to be the character of the representation spanned by the vectors Evidentlỹ in case m = 6. In preparation for §5.1, where the characters defined here will be used to specify certain vector-valued cusp forms of weight 3/2, we define and, as above, the superscripts enumerate simple components of X E and the subscripts indicate inner products for simple vectors within a component as per Figure 1. Defineχ X E to be the character of G X attached to the permutation representation spanned by the set {f i 3 | 1 ≤ i ≤ d E } (for example). In case n = 6 write χ X E for the character of G X attached to the representation when n = 6, the invariant subspace with character 2χ X E being spanned by the vectors f i , the twisted Euler characters attached to G X . They are given explicitly in the tables of §B.2. As mentioned above, we will use them to attach a vector-valued cusp form S X g to each g ∈ G X for X a Niemeier root system in §5.1.

McKay Correspondence
The McKay correspondence [34] relates finite subgroups of SU(2) to the extended Dynkin diagrams of ADE type by associating irreducible representations of the finite groups to nodes of the corresponding diagrams. A beautiful explanation for this can be given in terms of resolutions of simple singularities C 2 /G for G < SU(2) [35,36]. In §3.5 of [1] we observed a curious connection between the umbral groups G ( ) and certain finite subgroups D ( ) < SU (2) In [1] it was observed that a Dynkin diagram of rank 11 − may be attached to each G ( ) for ∈ {3, 4, 5, 7} in the following manner. If p = (25 − )/( − 1) then p is a prime and there is a unique (up to conjugacy) subgroupL ( ) <Ḡ ( ) such thatL ( ) is isomorphic to PSL 2 (p) and acts transitively in the degree 24 NowL ( ) has a unique (up to isomorphism) subgroupD ( ) of index p inL ( ) -a fact which is peculiar to the particular p arising-andD ( ) is a finite subgroup of SO(3) whose preimage D ( ) in SU(2) realises the extended diagram ∆ ( ) corresponding (cf. Figure 2) to a Dynkin diagram of rank 11 − via McKay's correspondence. In the present setting, with groups G ( ) defined for all such that Γ 0 ( ) has genus zero, and in particular for 3 ≤ ≤ 10, it is possible to extend this correspondence as follows.
Since ( To aid in the reading of Table 4 we note here the exceptional isomorphisms In [1] we used the common abbreviation L n (q) for PSL n (q).

Automorphic Forms
In this section we discuss modular objects that play a rôle in the moonshine relation between mock modular forms and finite groups that is the main focus of this paper.
In what follows we take τ in the upper half-plane H and z ∈ C, and adopt the shorthand notation e(x) = e 2πix . We also define q = e(τ ) and y = e(z) and write for the natural action of SL 2 (R) on H and let We also write for the action of SL 2 (Z) on H × C.

Mock Modular Forms
We briefly recall modular forms, mock modular forms, and their vector-valued generalisations.
Let Γ be a discrete subgroup of the group SL 2 (R) that is commensurable with the modular group SL 2 (Z). For w ∈ 1 2 Z say that a function ψ : Γ → C is a multiplier system for Γ with weight w if for all γ 1 , γ 2 ∈ Γ. Given such a multiplier system ψ for Γ we may define the (ψ, w)-action of Γ on the space O(H) of holomorphic functions on the upper half-plane by setting for f ∈ O(H) and γ ∈ Γ. We then say that f ∈ O(H) is an (unrestricted) modular form with multiplier ψ and weight w for Γ in the case that f is invariant for this action; i.e. f | ψ,w γ = f for all γ ∈ Γ. We say that an unrestricted modular form f for Γ with multiplier ψ and weight w is a weak modular form in case f has at most exponential growth at the cusps of Γ. We say that f is a modular form if (f |ψ ,w σ)(τ ) remains bounded as (τ ) → ∞ for any σ ∈ SL 2 (Z), and we say f is a cusp form if (f |ψ ,w σ)(τ ) → 0 as (τ ) → ∞ for any σ ∈ SL 2 (Z).
Suppose that ψ is a multiplier system for Γ with weight w, and g is a modular form for Γ with the inverse multiplier systemψ : γ → ψ(γ) and dual weight 2 − w. Then we may use g to twist the (ψ, w)-action of Γ on O(H) by setting With this definition, we say that f ∈ O(H) is an (unrestricted) mock modular form with multiplier ψ, weight w and shadow g for Γ if f is invariant for this action; i.e. f | ψ,w,g γ = f for all γ ∈ Γ. We say that an unrestricted mock modular form f for Γ with multiplier ψ, weight w and shadow g is a weak mock modular form in case f has at most exponential growth at the cusps of Γ. We say that f is a mock modular form if (f |ψ ,w σ)(τ ) remains bounded as (τ ) → ∞ for any In this paper we will consider the generalisation of the above definition to vector-valued (weak) mock modular forms with n components, where the multiplier ψ : Γ → GL n (C) is a (projective) representation of Γ. From the definition (3.6), it is not hard to see that the multiplier ψ of a (vector-valued) mock modular form is necessarily the inverse of that of its shadow. To avoid clutter, we omit the adjective "weak" in the rest of the paper when there is no room for confusion.
Following Zwegers [37] and Zagier [38] we define a mock theta function to be a q-series h = n a n q n such that for some λ ∈ Q the assignment τ → q λ h| q=e(τ ) defines a mock modular form of weight 1/2 whose shadow is a unary (i.e. attached to a quadratic form in one variable) theta series of weight 3/2. In §6 we conjecture that specific sets of mock theta functions appear as McKay-Thompson series associated to infinite-dimensional modules for the groups G X (cf. §2.4), where X is a Niemeier root system.

Jacobi Forms
We first discuss Jacobi forms following [39]. For every pair of integers k and m, we define the m-action of the group Z 2 and the (k, m)-action of the group SL 2 (Z) on the space of holomorphic where γ ∈ SL 2 (Z) and λ, µ ∈ Z. We say a holomorphic function φ : H × C → C is an (unrestricted) Jacobi form of weight k and index m for the Jacobi group SL 2 (Z) Z 2 if it is invariant under the above actions, φ = φ| k,m γ and φ = φ| m (λ, µ), for all γ ∈ SL 2 (Z) and for all In what follows we refer to the transformations (3.7) and (3.8) as the elliptic and modular transformations, respectively.
The invariance of φ(τ, z) under τ → τ + 1 and z → z + 1 implies a Fourier expansion φ(τ, z) = n,r∈Z c(n, r)q n y r (3.9) and the elliptic transformation can be used to show that c(n, r) depends only on the discriminant D = r 2 − 4mn and on r mod 2m. In other words, we have c(n, r) = Cr(r 2 − 4mn) wherẽ r ∈ Z/2mZ and r =r mod 2m, for some appropriate function D → Cr(D). An unrestricted Jacobi form is called a weak Jacobi form, a (strong) Jacobi form, or a Jacobi cusp form when the Fourier coefficients satisfy c(n, r) = 0 whenever n < 0, Cr(D) = 0 whenever D > 0, or Cr(D) = 0 whenever D ≥ 0, respectively. In a slight departure from the notation in [39] we denote the space of weak Jacobi forms of weight k and index m by J k,m .
In what follows we will need two further generalisations of the above definitions. The first is relatively straightforward and replaces SL 2 (Z) by a finite index subgroup Γ ⊂ SL 2 (Z) in the modular transformation law. (One has to consider Fourier expansions (3.9) for each cusp of Γ.) The second is more subtle and leads to meromorphic Jacobi forms which obey the modular and elliptic transformation laws but are such that the functions z → φ(τ, z) are allowed to have poles lying at values of z ∈ C corresponding to torsion points of the elliptic curve C/(Zτ + Z).
The elliptic transformation (3.7) implies (cf. [39]) that a (weak) Jacobi form of weight k and index m admits an expansion in terms of the index m theta functions, Recall that the vector-valued function θ m = (θ m,r ) satisfies where S ,T are the 2m × 2m unitary matrices with entries T rr = e r 2 4m δ r,r . (3.14)

The Eichler-Zagier Operators and an ADE Classification
We now turn to a discussion of the Eichler-Zagier operators on Jacobi forms and establish an where 1 = n 1 < n 2 < · · · < m = n σ0(m) are the divisors of m. It is easy to check that these matrices automatically satisfy (3.17).
as is evident from the definition (3.18) of Ω m (n) as well as the reflection property (3.14) of the componentsh m,r ofh m .
In fact, as we will now show, for a given vector-valued modular formh m = (h m,r ) and any given divisor n of m, the new Jacobi formh T m · Ω m (n) · θ m can be obtained from the original onẽ h T m · θ m via a natural operator-the so-called Eichler-Zagier operator [39]-on Jacobi forms.
Given positive integers n, m such that n|m, we define an Eichler-Zagier operator W m (n) acting on a function f : e m a 2 n 2 τ + 2 a n z + ab n 2 f τ, z + a n τ + b n . (3.20) It is easy to see that the operator W m (n) commutes with the index m elliptic transformation for all µ, λ ∈ Z and in particular preserves the invariance under elliptic transformations. Moreover, one can easily check that the modular invariance (3.8) is also preserved. As a result W m (n) maps a(n unrestricted) Jacobi form of weight k and index m to another (unrestricted) Jacobi form of the same weight and index. Moreover, when n m this operation is an involution on the space of strong Jacobi forms. This involution is sometimes referred to as an Atkin-Lehner involution for Jacobi forms due to its intimate relation to Atkin-Lehner involutions for modular forms [42,43]. We will explain and utilise some aspects of this relation in §4.2.
For later use, we define the more general operator W = ni|m c i W m (n i ) by setting The relation between the Eichler-Zagier operators W and the transformation on Jacobi forms where Ω is a linear combination of the matrices Ω m (n) in (3.18), can be seen via the action of the former on the theta functions θ m,r . Notice that In terms of the 2m-component vector θ m = (θ m,r ), we have In other words, the Jacobi forms we discussed above in terms of the matrices Ω m (n) are simply the images of the original Jacobi forms under the corresponding Eichler-Zagier operators. This property makes it obvious thath T m · Ω m (n) · θ m is also a Jacobi form since W m (n) preserves the transformation under the Jacobi group. This relation will be important in the discussion in §4.
Apart from the modularity (Jacobi form) condition, it is also natural to impose a certain positivity condition. As we will see, this additional condition leads to an ADE classification of the matrices Ω. To explain this positivity condition, first recall that all the entries of the matrices Ω m (n) for any divisor n of m are non-negative integers and it might seem that any positivity condition would be redundant. However, we have also seen that the description of the theta-coefficientsh m,r of a weight k, index m Jacobi form as a vector with 2m components has some redundancy since different components are related to each other byh m,r = (−1) kh m,−r (cf. (3.14)). For the purpose of the present paper we will from now on consider only the case of odd k, where there are at most m − 1 independent components in (h m,r ). In this case, using the property (3.17) we can rewrite the Jacobi formh T m · Ω · θ m as As a result, it is natural to consider the 2m×2m matrices Ω =  with a natural normalisation Evidently, they are in one-to-one correspondence with the non-negative integer combinations of (h m,r ) and (θ m,r − θ m,−r ), with r, r = 1, 2, . . . , m − 1, with the coefficient of the termh m,1 θ m,1 equal to 1.
It turns out that this problem has been studied by Cappelli-Itzykson-Zuber [24]. They found a beautiful ADE classification of such 2m × 2m matrices Ω (see Proposition 2 of [24]) and we present these matrices in Table 5, denoting by Ω X the matrix corresponding to the irreducible simply-laced root system X. The motivation of [24] was very different from ours: these authors were interested in classifying the modular invariant combinations of chiral and anti-chiral characters of the affine Lie algebraÂ 1 (the SU(2) current algebra). However, as the modular transformation of theÂ 1 characters at level m − 2 is the same up to scalar factors as that of the index m theta functions θ m,r , the relevant matrices are also the commutants of the same S and T matrices satisfying (3.16).
The relation between the Eichler-Zagier operators and the Ω m (n) matrices discussed earlier makes it straightforward to extend the above ADE classification to an ADE classification of Eichler-Zagier operators. Combining the results of the above discussion, we arrive at the following theorem.
if Ω coincides with a matrix Ω X corresponding to an irreducible simply-laced root system X with Coxeter number m via Table 5.
Conversely, any 2m×2m matrix Ω for which the above statement is true necessarily coincides with a matrix Ω X corresponding to an irreducible simply-laced root system X with Coxeter number m. Moreover, the resulting (unrestricted) Jacobi form is the image of the original Jacobi formh T m · θ m under the Eichler-Zagier operator W X defined by replacing Ω m (n) in Ω X with W m (n) (cf. (3.32)).
Next, the positivity and integrality conditions on c r,r are equivalent to those on the entries Ω r,r − Ω r,−r given in (3.28-3.29). The linear combinations of Ω m (n i ) satisfying (3.28-3.29) were shown in [24] to correspond to ADE root systems via Table 5. Finally, the equality (3.32) follows from the equality (3.25).
The relation between Ω X and the ADE root system X lies in the following two facts. First, Ω X is a 2m × 2m matrix where m is the Coxeter number of X. Moreover, Ω X r,r − Ω X r,−r = α X r for r = 1, . . . , m − 1 coincides with the multiplicity of r as a Coxeter exponent of X (cf. Table   1). Note the striking similarity between the expression for Ω X and the denominator of the Coxeter Frame shape π X (cf. §2.1). For instance, Ω X = Ω m (1) for X = A m−1 is nothing but the 2m × 2m identity matrix.
More generally, for a union X = i X i of simply-laced root systems with the same Coxeter number, we let Then we have the relation among different (odd) weight k and index m Jacobi forms corresponding to the same vectorvalued modular formh m = (h m,r ). Note that the operator W X is in general no longer an involution and often not even invertible. The Eichler-Zagier operators W X corresponding to Niemeier root systems (cf. §2.2) will play a central rôle in §5.

From Meromorphic Jacobi Forms to Mock Modular Forms
In subsection §3.1 we have seen the definition of mock modular form and its vector-valued generalisation. One of the natural places where such vector-valued mock modular forms occur is in the theta expansion of meromorphic Jacobi forms. To be more precise, following [37] and [40] we will establish a uniform way to separate a meromorphic Jacobi form ψ into its polar and finite parts The finite part will turn out to be a mock Jacobi form, admitting a theta expansion as in . For a given pole z = z s of a weight 1 index m meromorphic Jacobi form ψ, we will consider the image ψ P zs under Av m of a suitably chosen meromorphic function F zs (y) that has a pole at z = z s , such that ψ − ψ P zs is regular at all z ∈ z s + Z + τ Z.
In the remaining part of this subsection we will first review (following [37,40]) this construction of mock modular forms in more detail, and then extend the discussion of the Eichler-Zagier operators to meromorphic Jacobi forms and study how they act on the polar and the finite part separately. This will allow us to establish an ADE classification of mock Jacobi forms of a specific type in the next section and constitutes a crucial element in the construction of the umbral mock modular forms H X .
A Simple Pole at z = 0 To start with, consider a meromorphic Jacobi form ψ(τ, z) of weight 1 and index m, with a simple pole at z = 0 and no other poles. Define the polar part of ψ to be where χ(τ )/πi is the residue of ψ(τ, z) at z = 0. For the applications in the present paper we need only consider the case that χ(τ ) = χ is a constant. With this definition, one can easily check that ψ F = ψ − ψ P is indeed a holomorphic function with no poles in z.
Note that where we define the generalised Appell-Lerch sum for j ∈ 1 2 Z. The function µ m,0 enjoys the following relation to the modular group SL 2 (Z). Define the completion of µ m,0 (τ, z) by settinĝ where S m,r (τ ) denotes the unary theta series thenμ m transforms like a Jacobi form of weight 1 and index m for SL 2 (Z) but is clearly no longer holomorphic whenever m > 1. From the above definition and the transformation (3.13) of the theta functions, we see that S m = (S m,r ) is a (vector-valued) weight 3/2 cusp form for Returning to our weight 1 index m Jacobi form ψ, assumed to have a simple pole at z = 0, it is now straightforward to see that the finite part ψ F = ψ − ψ P , a holomorphic function on H × C, has a completion given bŷ that transforms like a Jacobi form of weight 1 and index m for SL 2 (Z). As such, ψ F is an example of a mock Jacobi form (cf. [40, §7.2]). Since both ψ and ψ P are invariant under the index m elliptic transformation, so is the finite part ψ F . This fact guarantees a theta expansion of ψ F analogous to that of a (weak) Jacobi form (3.10) where h = (h r ) is a weight 1/2 vector-valued holomorphic function on H whose completion transforms as a weight 1/2 vector-valued modular form with 2m components. As such, we conclude that h = (h r ) is a vector-valued mock modular form for the modular group SL 2 (Z) with shadow χ S m = (χ S m,r ).
Note that S 1 vanishes identically. This is a reflection of the fact that µ 1,0 , in contrast to the µ m,0 for m > 1, coincides with its completion, and is thus (already) a meromorphic Jacobi form, of weight 1 and index 1. By construction it has simple poles at z ∈ Zτ + Z and nowhere else, and we also have the explicit formula (3.43) (See §A.2 for θ 1 (τ, z).) The function µ 1,0 is further distinguished by being a meromorphic Jacobi form with vanishing finite part; a "Cheshire cat" in the language of [40, §8.5]. It will play a distinguished rôle in §4, where it will serve as a device for producing meromorphic Jacobi forms of weight 1 from (weak, holomorphic) Jacobi forms of weight 0.

Simple Poles at n-Torsion Points
Next we would like to consider the more general situation in which we have a weight 1 index m meromorphic Jacobi form ψ with simple poles at more general torsion points z ∈ Qτ + Q.
We introduce the row vector with two elements s = (α β) to label the pole at z s = ατ + β and write y s = e(z s ). For the purpose of this paper we will restrict our attention to the n-torsion points satisfying nz ∈ Zτ + Z, where n is a divisor of the index m. Focus on a pole located at say z s = ατ + β with α, β ∈ 1 n Z. Again following [40], we require the corresponding polar term to be given by the formula ψ P zs (τ, z) = πi Res z=zs (ψ(τ, z)) Av m y y s −2mα y/y s + 1 generalising (3.35). One can easily check that ψ − ψ P zs has no pole at z ∈ z s + Zτ + Z. As before, the above polar part is invariant under the elliptic transformation by construction.
To discuss its variance under the modular group, first notice that the transformation (τ, z) → γ(τ, z) maps the pole at z s to a different pole according to s → sγ. As a result, to obtain a mock Jacobi form for SL 2 (Z) from a meromorphic Jacobi form with poles at n-torsion points z s (where n is the smallest integer such that z s ∈ τ n Z + 1 n Z), we should consider meromorphic Jacobi forms that have poles at all the n-torsion points. Moreover, the modular transformation of ψ dictates that the residues of the poles satisfy D s (γτ ) = D sγ (τ ), where we have defined, after [40], D s (τ ) = e(mαz s ) Res z=zs (ψ(τ, z)). (3.45) More specifically, we would like to consider the situation where ψ satisfies Res z=− a n τ − b n ψ(τ, z) = χe(−ma(aτ + b)/n 2 )/nπi, for a, b = 0, 1, . . . , n − 1, (3.46) corresponding to the simplest case where the function D s (τ ) is just a constant. Without loss of generality we will also assume for the moment that ψ has no other poles, as the more general situation can be obtained by taking linear combinations. In this case, using (3.44) it is not hard to see that the polar parts contributed by the poles at these n-torsion points are given by the images under the Eichler-Zagier operator W m (n) (cf. (3.20)) of the polar term contributed by the simple pole at the origin, so that From (3.38) and the fact that the Eichler-Zagier operators preserve the Jacobi transformations, we immediately see how considering Jacobi forms with simple poles at torsion points leads us to vector-valued mock modular forms with more general shadows. In this case, from (3.25), (3.38) and (3.47) it is straightforward to see that the completion of the polar part again transforms like a Jacobi form of weight 1 and index m for SL 2 (Z).
Following the same argument as before, we conclude that the theta-coefficients of the finite define a vector-valued mock modular form h = (h r ), whose completion is given bŷ and whose shadow is hence given by a vector of unary theta series whose r-th component equals where r ∈ Z/2mZ. In particular, this means that the vector-valued mock modular forms arising from meromorphic Jacobi forms in this way are closely related to mock theta functions, as their shadows are always given by unary theta series.
Finally, we also note that the Eichler-Zagier operators and the operations of extracting polar and finite parts are commutative in the following sense. Proof. Denote the set of poles of ψ in the unit cell by S, and focus on the pole of ψ at z * = −ã/ñτ −b/ñ ∈ S. From the action of W m (n) we see that ψ|W m (n) has poles at all z ∈ z * + 1 n Z + τ n Z. Focussing on the pole at z = z s = z * − (aτ /n + b/n), from (3.20) we get Res z=zs (ψ|W m (n))(τ, z) = 1 n e( 2mazs n )e( ma n 2 (aτ + b))Res z=z * (ψ(τ, z)), (3.53) which leads to e( 2mazs n )e( ma n 2 (aτ + b)) k∈Z q mk 2 y 2mk (q k y/y s ) 2m( a n +ã n ) q k y/y s + 1 q k y/y s − 1 where y s denotes y s = e(z s ) = e(−(ã/ñ + a/n)τ − (b/ñ + b/n)) in the second line.
By direct comparison using (3.20) and (3.44), this is exactly ψ P |W m (n) and this finishes the proof.
Since all the operations involved are linear, we also have the following corollary. Moreover, if we denote the theta-coefficients of ψ F by h ψ = ((h ψ ) r ) and its shadow by S ψ = ((S ψ ) r ) with r ∈ Z/2mZ, then the theta-coefficients of (ψ|W) F form a vector-valued mock modular form h ψ|W satisfying

56)
with shadow given by where As a result, the relations (3.32) between different Jacobi forms of the same index can be applied separately to the polar and the finite part. In the present paper we will mostly be concerned with the application of the above to the case thatñ = 1. For later use it will be useful to note the following property. Proof. From the property (3.47) of ψ P and the elliptic transformation of µ m,0 , it follows that ψ P (τ, z) = −ψ P (τ, −z) and therefore ψ F (τ, z) = −ψ F (τ, −z). As such, the vector-valued mock Together with S m,r = −S m,−r , the lemma follows from the action of W m (n) (3.25) on θ m = (θ m,r ) and the definition (3.18) of the matrix Ω m (n).

The Umbral Mock Modular Forms
Following the general discussion of the relevant automorphic objects in the previous section, in this section we will start specifying concretely the vector-valued mock modular forms which encode, according to our conjecture, the graded dimensions of certain infinite-dimensional modules for the umbral groups defined in §2. We will specify the shadows of these functions-the umbral mock modular forms-in §4.1. Subsequently, in §4.2 we will show how these shadows distinguish the Niemeier root systems through a relation to genus zero groups and their principal moduli.
Afterwards we will provide explicit expressions for the umbral forms of the A-type Niemeier root systems by specifying a set of weight 0 weak Jacobi forms. The umbral forms of D-and E-type Niemeier root systems will be specified in the next section.

The Umbral Shadows
In §3.4 we have seen how the theta expansion of the finite part of a meromorphic Jacobi form gives rise to a vector-valued mock modular form, and how different configurations of poles lead to different shadows. The shadows of the mock modular forms obtained in this way are always given by unary theta series. In this subsection we will see how the ADE classification discussed in section §3.3 leads to particular cases of the above construction. Moreover, by combining the ADE classification and the construction of mock modular forms from meromorphic Jacobi forms discussed in §3.4, we will associate a specific shadow S X , or equivalently a pole structure of the corresponding meromorphic Jacobi form ψ X , to each of the Niemeier root systems X.
Consider a meromorphic Jacobi form ψ with weight 1 and index m. Recall from (3.47) that the contribution to its polar part ψ P from the simple poles at the n-torsion points with residues satisfying (3.46), is given by Clearly, one may consider a linear combination of expressions as in (4.1). Consider a weight 1 index m meromorphic Jacobi form ψ with poles at n 1 -,. . . ,n κ -torsion points where n j |m, and where each n j contributes c j e(−ma(aτ + b)/n 2 j )/n j πi to the residue of the pole located at − a nj τ − b nj . From the above discussion it follows that its polar part is given by By taking a linear combination of expressions as in (3.48) we see that its completion, given bŷ where Ω = κ i=1 c i Ω m (n i ), transforms like a Jacobi form of weight 1 and index m for SL 2 (Z). Immediately we conclude that the theta-coefficients of the finite part ψ F = ψ − ψ P constitute a vector-valued mock modular form with (the r-th component of) the shadow given by (Ω r,r − Ω r,−r )S m,r , r = 1, . . . , m − 1. X with Coxeter number m we associate a 2m-vector-valued cusp form S X , of weight 3/2 for SL 2 (Z), with r-th component given by where m denotes the Coxeter number of X and the matrix Ω X is defined as in Table 5. For instance, we have For the D-series root systems with even rank we have and in the case that n is odd we have (4.10)  where for simplicity we have only specified S En r for r ∈ {1, · · · , m−1} ⊂ Z/2mZ. The remaining components are determined by the rule S En −r = −S En r . Following (3.31), more generally for a union X = i X i of simply-laced root systems with the same Coxeter number we have S X = i S Xi . With this definition, S X is given by the matrix From the above discussion we see that the cusp form S X arises naturally as the shadow of a vector-valued mock modular form obtained from the theta expansion of a meromorphic Jacobi , whose polar part is given by (4.14) It is not hard to see that such meromorphic Jacobi forms exist for any simply- for a more detailed discussion, but note that µ 1,0 is denoted there also by Ψ 1,1 .) The Corollary 3.3 then shows that its image −µ 1,0 φ|W X under the corresponding Eichler-Zagier operator is a weight 1 index m meromorphic Jacobi with polar part coinciding with (4.14).
Besides specifying the poles of the meromorphic Jacobi form, in what follows we will also require an optimal growth condition (equivalent to the optimal growth condition formulated in [40], with its name derived from the fact that it guarantees the slowest possible growth of the coefficients of the corresponding meromorphic Jacobi form). It turns out that there does not always exist a meromorphic Jacobi form with polar part given by (4.14) that moreover satisfies this optimal growth condition for an arbitrary simply-laced root system X, but when it does exist it is unique.
Theorem 4.1. Let X be a simply-laced root system with all irreducible components having the same Coxeter number m. There exists at most one weight 1 index m meromorphic Jacobi form ψ X satisfying the following two conditions. First, its polar part (ψ X ) P is given by (4.14).
Second, its finite part satisfies the optimal growth condition as τ → i∞ for all r ∈ Z/2mZ.
Proof. If there are two distinct meromorphic Jacobi forms satisfying the above conditions, then their difference is necessarily a weight 1 index m weak Jacobi form of optimal growth in the sense of [40]. But Theorem 9.7 of [40] is exactly the statement that no such weak Jacobi form exists.
Corollary 4.2. Let X be a simply-laced root system with all irreducible components having the same Coxeter number m. Then there exists at most one vector-valued mock modular form h X for SL 2 (Z) with shadow S X that satisfies the optimal growth condition (4.16).
Proof. Let h X be a vector-valued mock modular form satisfying the above conditions. Consider ψ = (ψ X ) P + r h X r θ m,r with (ψ X ) P given by (4.14). From the fact that the multiplier of a mock modular form is the inverse of that of its shadow and from the discussion in §3.4, ψ is a weight 1 index m meromorphic Jacobi form satisfying the conditions of Theorem 4.1. It then follows from Theorem 4.1 that such h X is unique if it exists.
So far our discussion has been very general, applicable to any simply-laced root system with all irreducible components having the same Coxeter number. In the next subsection we will see how the cusp forms S X with X given by one of the 23 Niemeier root systems play a distinguished rôle. The S X for X a Niemeier root system are called the umbral shadows, We will demonstrate the existence of meromorphic Jacobi forms ψ X satisfying the conditions of Theorem 4.1 for X an A-type Niemeier root system in Proposition 4.6, and for D-and Etype Niemeier root systems in Proposition 5.2. The resulting vector-valued functions defined by the theta-coefficients of the finite parts of these ψ X are the umbral mock modular forms, to be denoted H X = (H X r ). The first few dozen Fourier coefficients of the umbral mock modular forms are tabulated in Appendix C. For some of the Niemeier root systems X, the meromorphic Jacobi forms ψ X are closely related to some of the meromorphic Jacobi forms analysed in [40, §9.5]; for other Niemeier root systems X, the corresponding shadows S X fall outside the range of analysis in [40] and this is why we find mock Jacobi forms of optimal growth for values of m other than those appearing in the work of Dabholkar-Murthy-Zagier.

From Niemeier Lattices to Principal Moduli
In the last subsection we have seen how ADE root systems have an intimate relation to meromorphic Jacobi forms. More precisely, to a simply-laced root system with all irreducible components having the same Coxeter number m, we associate a pole structure for meromorphic Jacobi forms of weight 1 and index m. Equivalently, we associate a weight 3/2 vector-valued cusp form to every such root system, which plays the rôle of the shadow of the mock modular form arising from the meromorphic Jacobi form via the relation discussed in §3. 4. In this subsection we see how the shadows S X attached to Niemeier root systems are distinguished, and in particular, how they are related to the genus zero groups Γ X of §2.3.
Recall [43] that a skew-holomorphic Jacobi form of weight 2 and index m is a smooth functioñ φ(τ, z) on H × C which is periodic in both τ and z with period 1, transforms under the Stransformation asφ and has a Fourier expansioñ where Cφ(∆, r) = 0 for ∆ < 0. We denote the space of such functions by J + 2,m . Recall that an integer ∆ is called a fundamental discriminant if ∆ = 1 or ∆ is the discriminant of a quadratic number field. Following Skoruppa [43] (see also [44]), given a pair (∆ 0 , r 0 ) where ∆ 0 is a positive fundamental discriminant that is a square modulo 4m and ∆ 0 = r 2 0 mod 4m, we may associate a weight 2 modular form for Γ 0 (m) to eachφ ∈ J + 2,m , where ∆0 a denotes the Jacobi symbol and cφ(∆ 0 , r 0 ) denotes a suitably chosen constant term.
From the discussion in §3.3, it is not difficult to see that given a simply-laced root system X with each of its irreducible components having the same Coxeter number m, we may consider of weight 2 and index m.
Applying S ∆0,r0 with the simplest choice (∆ 0 , r 0 ) = (1, 1) to the skew-holomorphic Jacobi form σ X , and using the fact that the Jacobi symbol 1 a = 1 for all positive integers a, we arrive at a weight 2 form on Γ 0 (m) where r is the rank of the root system X and α X r is the multiplicity of the multiplicity of r as a Coxeter exponent of X, which also coincides with the "diagonal" coefficient Ω X r,r − Ω X r,−r of S m,r in the r-th component of the vector-valued cusp form S X = (S X r ) (cf. Table 1 and §3.3). For X = A m−1 , one can easily see from the fact that α X r = 1 for all r ∈ {1, 2, . . . , m − 1} that the associated weight-two form is nothing but the following Eisenstein form at level m (cf.
One can compute the function f X in a similar way for the D-and E-series and arrive at the result in Table 6. From this table one observes that the weight two form f X has a close relation to the Coxeter Frame shape π X , and hence also to the matrix Ω X according to Table 5. We now discuss this further.
From Ω X where X has Coxeter number m, we can obtain a map on the space spanned by the weight two modular forms {λ n (τ ), n|m} of level m by replacing each Ω m (m 1 ) in Ω X with the operator w m (m 1 ) which acts on Eisenstein forms according to where m 1 and m 2 are assumed to divide m, and m 1 * Then the weight two form corresponding to a simply-laced root system X with Coxeter number m is nothing but where π X is the Coxeter Frame shape of X (cf. §2.1) and λ π , for π an arbitrary Frame shape, cm de ∈ SL 2 (R) (cf. (3.5)). On the other hand, from (3.25) we see that the skew-holomorphic form σ X can be obtained as the image of the Eichler-Zagier operator Taken together, at a given m and for a given union X of simply-laced root systems with Coxeter number m we have the equality where we have written W X = i W m (e i ) explicitly in terms of its different components. At the same time, for the cases that all e i ||m, we also have For these cases, we note that the equality f Am−1 |W m (e i ) = S 1,1 σ Am−1 |W m (e i ) can be viewed as a consequence of the relation between the Eichler-Zagier operators and the Atkin-Lehner involutions observed in [42,43]. Due to this relation, the Eichler-Zagier operators that define involutions are sometimes referred to as Atkin-Lehner involutions on Jacobi forms in the literature.
We conclude this section by observing that if X is a Niemeier root system then where σ X is the skew-holomorphic Jacobi form defined by S X in (4.21) and T X is the principal modulus for Γ X defined in §2. 3. In this way we obtain a direct connection between the umbral shadows S X and the genus zero groups Γ X attached to Niemeier root systems in §2.3.

From Weight Zero Jacobi Forms to Umbral Mock Modular Forms
The goal of this subsection is to construct the umbral mock modular form H X which (conjecturally) encodes the graded dimension of the umbral module K X (cf. §6.1) for every A-type Niemeier root system X. In §4.1 we have seen how to associate an umbral shadow S X to a Niemeier root system X. Equivalently, we can associate a pole structure, which together with the optimal growth condition (4.16) determines a(t most one) meromorphic weight 1 index m Jacobi form ψ X according to Theorem 4.1. In this subsection we will explicitly construct meromorphic Jacobi forms ψ X satisfying the conditions of Theorem 4.1 for X an A-type root system via certain weight 0 Jacobi forms φ X (cf. Proposition 4.6). After obtaining these ψ X , the procedure discussed in §3.4 then immediately leads to the umbral forms H X . It is also possible to specify the D-and E-type Niemeier root systems in a similar way. However, the discussion would become somewhat less illuminating and we will instead specify these in an arguably more elegant way in §5.
Our strategy in the present subsection is the following. As mentioned in §4.1, if we take a weight 0 index m − 1 weak Jacobi form φ with φ(τ, 0) = 1, then −µ 1,0 φ|W X is a weight 1 index m meromorphic Jacobi whose pole structure is automatically of the desired form (cf. (3.43)).
Namely, it always leads to a vector-valued mock modular form whose shadow is given by S X . In this subsection we will see how to select (uniquely) a weight 0 form φ X such that the resulting weight 1 Jacobi form satisfies the optimal growth condition (4.16).
For the simplest cases this optimal growth condition can be rephrased in terms of weight 0 Jacobi forms using the language of characters of the N = 4 superconformal algebra. Recall from [7,8] that this algebra contains subalgebras isomorphic to the affine Lie algebraÂ 1 as well as the Virasoro algebra, and in a unitary representation the former of these acts with level m − 1, for some integer m > 1, and the latter with central charge c = 6(m − 1). The unitary h,j are labelled by the two "quantum numbers" h and j which are the eigenvalues of L 0 and 1 2 J 3 0 , respectively, when acting on the highest weight state. (We adopt a normalisation of the SU(2) current J 3 such that the zero mode J 3 0 has integer eigenvalues.) In the Ramond sector of the superconformal algebra there are two types of highest weight representations: the short (or BPS, supersymmetric) ones with h = m−1 4 and j ∈ {0, 1 2 , · · · , m−1 2 }, and the long (or non-BPS, non-supersymmetric) ones with h > m−1 in the short and long cases, respectively, [8], where the function µ m,j (τ, z) is defined as in (3.37).
Lambencies 2, 3, 4, 5, 7, 13 It turns out that for the pure A-type Niemeier root systems given as the union of 24/( − 1) copies of A −1 for ( − 1)|12, the relevant criterion for φ ( ) is that of an extremal Jacobi form [1, §2.5]. The idea of an extremal Jacobi form can be viewed as a generalisation of the concept of an extremal Virasoro character, a notion that was introduced in [45] and discussed in [46] in the context of pure AdS 3 gravity.
With the above definitions, following [1], for m a positive integer and φ a weak Jacobi form for some a h,j ∈ C. Note the restriction on n in the last summation in (4.35). Write J ext 0,m−1 for the subspace of J 0,m−1 consisting of extremal weak Jacobi forms.
We recall here that the extremal condition has a very natural interpretation in terms of the mock modular forms h = (h r ) of weight 1/2 via the relation More precisely, the extremal condition is equivalent to the condition that as τ → i∞, which clearly implies the optimal growth condition q 1/4m h r (τ ) = O(1) of Theorem 4.1.
In [1] we proved that dim arising from the extremal condition (4.35) will determine the unique weight 1 meromorphic Jacobi forms ψ X satisfying the conditions of Theorem 4.1 for (4.31) where m is the Coxeter number of X.

Lambencies 9, 25
In order to include the other two pure A-type Niemeier root systems (X = A 3 8 and X = A 24 ), it is sufficient to weaken the extremal condition slightly and consider weight 0 and index m − 1 Jacobi forms admitting an expression for some a h,j ∈ C. Notice that we have weakened the bound from r 2 − 4mn < 0 to r 2 − 4mn ≤ 0 in the last summand. This is directly related to the fact that for m = 9, 25 there exists 0 < r < m such that r 2 = 0 mod 4m.
There exists at most one solution to (4.38) up to rescaling (look ahead to Lemma 4.3), and inspection reveals that at m = 9, 25 we have the non-zero solutions ϕ (9) 1 and ϕ (25) 1 , in the notation of §A.4. These Jacobi forms will determine, by way of (4.31), the unique weight 1 meromorphic Jacobi forms ψ X satisfying the conditions of Theorem 4.1 for X ∈ {A 3 8 , A 24 }.
Lambencies 6, 10 In order to capture X = A 4 5 D 4 and X = A 2 9 D 6 at m = 6 and m = 10 (the cases with m given by the product of two distinct primes), we will relax the extremal condition further and consider weight 0 and index m − 1 Jacobi forms admitting an expression for some a h,j ∈ C. For this more general condition we also have a uniqueness property.  Proof. If φ 1 and φ 2 are two weight 0 Jacobi forms that can be written as in (4.39), there exists a linear combination φ of φ 1 and φ 2 satisfying Equivalently, ψ = µ 1,0 φ is a weight 1 index m weak Jacobi form with Fourier expansion ψ(τ, z) = n, c(n, )q n y , where c(n, ) = 0 for 2 − 4mn > 1. But such a weight 1 index m weak Jacobi form does not exist according to Theorem 9.7 of [40], based on the fact that there is no (strong) Jacobi form of weight 1 and index m for any positive integer m, as shown earlier by Skoruppa [47].
We therefore conclude that φ = 0, and φ 1 and φ 2 are linearly dependent. This more general condition (4.39) singles out the weight 0 Jacobi forms ϕ (6) 1 and ϕ (10) 1 (cf. §A.4) at m ∈ {6, 10} in addition to those already mentioned, for which m is prime or the square of a prime. As above, these weight 0 forms will determine, by way of (4.31), the unique weight 1 meromorphic Jacobi forms ψ X satisfying the conditions of Theorem 4.1 for X ∈ {A 4 5 D 6 , A 2 9 D 6 }. We are left with the A-type Niemeier root systems with Coxeter numbers that are not square-free and not squares of primes: they are X = A 2 7 D 2 5 , A 11 D 7 E 6 , A 15 D 9 , A 17 E 7 with m = 8, 12, 16, 18 respectively. To discuss these cases, let us first point out a subtlety in our procedure for determining the weight 1 meromorphic Jacobi form ψ X from a weight 0 meromorphic Jacobi form φ X using (4.31). Although the resulting weight 1 form ψ X is unique following Theorem 4.1, in general the corresponding weight 0 form φ X is not. In other words, there could be more than one φ X satisfying (4.31) for a given ψ X . For the A-type cases with m ∈ {2, 3, 4, 5, 6, 7, 9, 10, 13, 25} discussed above, there is no such ambiguity since the matrix Ω X corresponding to the Eichler-Zagier operator W X is invertible. On the other hand, corresponding to m ∈ {8, 12, 16, 18}. At the same time, in these cases there exists a unique d > 1 such that d 2 is a proper divisor of m and correspondingly, there is an interesting feature in the space of Jacobi forms. This is due to the fact (cf. [40, §4.4]) that if ψ k,m/d 2 (τ, z) transforms as a weight k index m/d 2 Jacobi form then ψ k,m/d 2 (τ, dz) transforms as a weight k index m Jacobi form. It turns out that the umbral forms discussed above, in particular those with lambency = 2, 3, 4, help to determine the weight 0 forms φ X at lambency = 8, 12, 16, 18 by requiring the "square relation" where d is the unique integer such that d 2 is a proper divisor of different from 1, and ψ ( /d 2 ) is the weight 1 meromorphic Jacobi form with index /d 2 that we have constructed above via (4.31). This extra condition (4.42) eliminates the kernel of the Eichler-Zagier operator W X and renders our choice for φ X unique. Notice that, following Table 3 we have used the lambency to denote the Niemeier root system X, and the former simply coincides with the Coxeter number for the A-type cases discussed in this subsection.
To specify this particular choice of φ X , let us impose the following condition. For a nonsquare-free m which is not a square of a prime, we consider the weight 0 index m − 1 Jacobi forms φ, such that the finite part of the weight 1 index m Jacobi form −µ 1,0 φ|(1 is a mock Jacobi form with expansion n, c(n, )q n y and c(n, r) = 0 whenever r 2 − 4mn > 1, i.e.
From the above discussion we arrive at the following uniqueness property for such Jacobi forms.
At m ∈ {8, 12, 16, 18}, applying the above Proposition and choosing with φ (m) as in Table 7 gives the weight 0 forms we need in order to specify the remaining umbral mock modular forms H X for X of A-type.
The explicit expressions for φ X for all A-type Niemeier root systems X are given in Table 7, where the basis we use for weight 0 weak Jacobi forms is summarised in §A.4 and ϕ 3 ) Given φ X and using the Eichler-Zagier operator W X defined in §3.3, the formula (4.31) gives the weight 1 meromorphic Jacobi form ψ X . From there, using the method described in §3.4 we can separate it into the polar and the finite part ψ X = (ψ X ) P + (ψ X ) F in a canonical way. As can be verified by inspection, the choices of φ X specified in this subsection determine solutions to the hypotheses of Theorem 4.1.
Proposition 4.6. Let X be one of the 14 Niemeier root systems with an A-type component (cf. (2.8)) and let φ X be as specified in Table 7. Then the meromorphic Jacobi form ψ X determined by (4.31) is the unique such function satisfying the conditions of Theorem 4.1. Write then H X = (H X r ) is the unique vector-valued mock modular form with shadow S X satisfying the optimal growth condition (cf. Corollary 4.2).
As a result, the weight 0 weak Jacobi forms constructed in this subsection define the umbral mock modular forms H X for each of the A-type Niemeier root systems (cf. (2.8)). The first few dozen coefficients of the components H X r are given in Appendix C. It is a reflection of the close relationship between the notions of optimal growth formulated here and in [40] that the umbral forms H X attached to A-type Niemeier root systems are closely related to the mock modular forms of weight 1 that appear in §A.2 of [40].

The Umbral McKay-Thompson Series
The purpose of this section is to discuss the umbral McKay-Thompson series H X g conjecturally defining the graded character of the group G X attached to the umbral module K X (cf. §6.1).

Shadows
In §2.4 we described the umbral groups G X and attached twisted Euler charactersχ X A , χ X A , χ X D , χ X D , &c., to the A-, D-, and E-components of each Niemeier root system. In this section we will explain how to use these characters to define a function S X g , for each g ∈ G X , which turns out to be the shadow of the vector-valued mock modular form H X g . Let X be a Niemeier root system and suppose that m is the Coxeter number of X. Then given g ∈ G X we define 2m × 2m matrices Ω X A g , Ω X D g , and Ω X E g , with entries indexed by Z/2mZ × Z/2mZ, as follows. We define Ω X A g by setting where P s m is the diagonal matrix (of size 2m × 2m, with entries indexed by Z/2mZ × Z/2mZ) with r-th diagonal equal to 1 or 0 according as r = s mod 2 or not, In (5.2) we write δ i,j(n) for the function that is 1 when i = j (mod n) and 0 otherwise. Note there are no type A components in the Niemeier root system X.
If X D = ∅ then m is even. For m > 6 we define Ω X D g by setting whilst for m = 6-an exceptional case due to triality for D 4 -we define Ω = Ω X D g so that The matrices Ω X E g are defined by setting Now for X a Niemeier root system we set Ω X g = Ω X A g +Ω X D g +Ω X E g , and we define S X g by setting This generalises the construction (4.13). We conjecture (cf. §6.2) that the vector-valued function S X g is the shadow of the mock modular form H X g attached to g ∈ G X . We will specify (most of) the H X g explicitly in the remainder of §5.
Remark 5.1. The matrices Ω X , corresponding to the case where [g] is the identity class, admit an ADE classification as explained in §3.3. It is natural to ask what the criteria are that characterise these matrices Ω X g , attached as above to elements g ∈ G X via the twisted Euler characters defined in §2.4.

Prime Lambencies
In this subsection we review the mock modular forms H X g = (H X g,r ) conjecturally encoding the graded characters of the umbral module K X (cf. Conjecture 6.1) of the umbral group G X for the five Niemeier root systems X with prime Coxeter numbers. Explicitly, these are the root systems g,r ) with r = 1, 2, . . . , − 1, using the notation given in Table 3. In the next subsection we will see that they determine many of the umbral McKay-Thompson series attached to the other Niemeier root systems with non-prime Coxeter numbers. The discussion of this subsection follows that of [1]. The McKay-Thompson series for = 2, X = A 24 1 were first computed in [12][13][14][15].
In order to give explicit formulas for the mock modular forms H ( ) where is the umbral form specified in §4.3 corresponding to the identity class 1A and whose Fourier coefficients are given in Appendix C with H ( ) 1A,r . We also let and where the characters χ X A g andχ where S ,r is again the unary theta series given in (3.39). For > 3 we also specify further weight two modular forms by setting As explained in detail in [1], specifying F  (5.13) and the order of g and g are either the same or related by a factor of 2 or 1/2. For such paired classes we have In particular, H ( ) g,even = 0 for the self-paired classes. For = 7, 13, this serves to constrain the function H ( ) g and supports the claims regarding their modular properties discussed in §6.2. We refer the readers to §4 of [1] for explicit expressions for the weight two forms F ( ) g and F ( ),2 g . Note that, from the discussion in §5.1, the relation (5.13) implies that the shadows attached to such paired classes satisfy Therefore, the paired relation (5.14) can be viewed as a consequence of (the validity of) Conjecture 6.5.

Multiplicative Relations
In this subsection we will describe a web of relations, the multiplicative relations, among the First we will discuss the horizontal relations, summarised in Note that the relation between the shadows non-trivially with S X g ,r , S X g,r = 0. This fact, together with the more general multiplicative relations (5.21), can again be viewed as the consequence of the conjectured uniqueness of such mock modular forms (Conjecture 6.5). See Table 8 for the list of such horizontal relations. In particular, for the identity element a relation H X r = r c r ,r H X r holds for some c r ,r ∈ Z for all the five pairs (X , X) of Niemeier root systems with the same Coxeter numbers.
Note that, together with the discussion in §4.3 and Proposition 3.2, this implies more specifically that the umbral mock modular forms H X r can be obtained as the theta-coefficients of the finite part of the meromorphic weight 1 Jacobi form for the four pairs (X , X) ∈ {(6, 6 + 3), (10, 10 + 5), (12, 12 + 4), (18, 18 + 9)} with A-type root systems X and with the weight 0 Jacobi forms given in Table 7.
In fact, a linear relation between the shadows attached to different Niemeier root systems can happen more generally and not just among those with the same Coxeter numbers. The first indication that non-trivial relations might exist across different Coxeter numbers is the following property of the building blocks of the umbral shadow. As one can easily check, the unary theta function S m = (S m,r ) defined in (3.39) at a given index m can be re-expressed in terms of those at a higher index as for any positive integer n. The above equality makes it possible to have the relation r c r,r S X g ,r (kτ ) = S X g,r (τ ) (5.20) for some k ∈ Z >0 . We will see that this relation between the umbral shadows does occur for many pairs of Niemeier root systems (X , X) with m(X ) = km(X). Moreover, whenever this relation holds non-trivially with S X g ,r , S X g,r = 0, the corresponding relation among the McKay-Thompson series also holds.
We summarise a minimal set of such relations in Table 9. In Tables 8 and 9, when it is not explicitly specified, the relation holds for all values of r such that all H X g ,r1 and H X g,r2 appearing on both sides of the equation have 1 ≤ r 1 ≤ m(X ) and 1 ≤ r 2 ≤ m(X). These multiplicative relations form an intricate web relating umbral moonshine at different

Mock Theta Functions
In this subsection we record relations between the McKay-Thompson series of umbral moonshine and known mock theta functions. Many of the mock theta functions arising appear either in Ramanujan's last letter to Hardy or in his lost notebook [48]. In what follows we will give explicit expressions for the mock theta series using the q-Pochhammer symbol For lambency 2, two of the functions H g (τ ) are related to Ramanujan's mock theta functions of orders 2 and 8 through (5.23) For = 3 we encounter the following order 3 mock theta functions of Ramanujan: 6D,2 (τ ) = 2q 2/3 ρ(−q), .

Specification
In this subsection we will combine the different types of data on the   For the lambency = 6 + 3, corresponding to X = D 6 4 , the corresponding umbral group has the form G X = 3.Ḡ X (5.61) and the conjugacy classes withχ X D (g) > 0 form pairs satisfying .
We are left to determine the second components H g,2 (τ ), which are given by g,r ) for all [g] ⊂ G (4) . For = 6, the vertical relations in Table 9 to the McKay-Thompson series of lambencies = 2, 3 suffice to specify all H 1A,r in Table 9 and the paired relation (5.59) determines H (18) 3A and H 6A . For = 8, the vertical relations to = 4 recorded in Table 9, together with the paired relation (5.59) and H  Table 9 together with the paired relation 3A,6 (τ ) = H 6A,r and finishes the specification for = 9. For = 10, the vertical relations to = 2 and = 5 recorded in Table 9 specify all H    Table 8 and the vertical relations in Table 9  Proposition 5.2. Let X be one of the D-or E-type Niemeier root systems (cf. (2.9), (2.10)).
The vector-valued mock modular form H X = H X 1A specified above is the unique vector-valued mock modular form with shadow S X satisfying the optimal growth condition (cf. Corollary 4.2).
Together with Proposition 4.6, this proposition establishes our construction of the unique vector-valued mock modular form H X corresponding to all 23 Niemeier root systems X.

Conjectures
In this section we pose the umbral moonshine conjectures connecting the umbral groups G X and the mock modular forms H X g discussed in the previous sections.

Modules
In this section we formulate a conjecture that relates the umbral McKay-Thompson series H X g to an infinite-dimensional G X -module K X .
Recall that a super- is a linear operator preserving the grading then the super-trace of T is given by str V T = tr V0 T − tr V1 T where tr W T denotes the usual trace of T on W . We say that V is purely even (odd) when V = V0 (V1). If V is a G-module, with G-action preserving the Z/2Z-grading, then the function g → str V g is called the super-character of G determined by V .
Conjecture 6.1. Let X be a Niemeier root system and let m be the Coxeter number of X.
There exists a naturally defined Z/2mZ × Q-graded super-module for G X such that the graded super-character attached to an element g ∈ G X coincides with the vector-valued mock modular form

2)
where c X = 1 except for X = A 3 8 , for which c X = 3. Moreover, the homogeneous component K X r,d of K X is purely even if d > 0.
The reason for the exceptional value c X = 3 for X = A 3 8 is the curious fact that there are no integer combinations of irreducible characters of G X that coincide with the coefficients q −D/36 , D = −27λ 2 for some integer λ, of H X g,r (τ ) (cf. Conjecture 6.11). For example, the minimal positive integer c for which g → cH X g,6 is a graded virtual super-character of G X is c = c X = 3. Combining the above conjecture and the paired relations (5.59) and (5.63) of the McKay-Thompson series we arrive at the following. Conjecture 6.2. Let X be a Niemeier root system and set c = #G X /#Ḡ X . Then the Q-graded G X -module K X r is a faithful representation of G X when r ≡ 0 (mod c) and factors throughḠ X otherwise.
As mentioned in §1, for the special case X = A 24 1 , Conjecture 6.1 has been shown to be true by T. Gannon in [16], although the construction of K X is still absent. In this case, we have c = 1 and hence Conjecture 6.2 is automatically true. It should be possible to apply the techniques similar to that in [16] to prove Conjecture 6.1 for other Niemeier root systems X.

Modularity
We have attached a cusp form S X g to each g ∈ G X in §5.1 by utilising the naturally defined permutation representations of G X , and the corresponding twisted Euler characters, that are described in §2.4. We begin this section with an explicit formulation of the conjecture that these cusp forms describe the shadows of the super-characters attached to the conjectural G X -module K X .
In preparation for the statement define n g and h g for g ∈ G X as follows. Take n g to be the order of the image of g inḠ X (cf. §2.4), and set h g = N g /n g where N g denotes the product of the shortest and longest cycle lengths of the permutation which is the image of g under G X → Sym Φ . These values are on display in the tables of §B.2. They may also be read off from the cycle shapesΠ X g andΠ X A g ,Π X D g , &c., attached to the permutation representations constructed in §2.4, for n g is the maximum of the cycle lengths appearing inΠ X A g ,Π X D g and Π X E g , and ifΠ X g = j m1 1 · · · j m k k with j m1 1 < · · · < j m k 1 and m i > 0 then h g = j 1 j k /n g . Conjecture 6.3. The graded super-characters (6.2) for fixed X and g ∈ G X and varying r ∈ Z/2mZ define the components of a vector-valued mock modular form H X g of weight 1/2 on Γ 0 (n g ) with shadow function S X g .
Let ν X g denote the multiplier system of H X g . Since the multiplier system of a mock modular form is the inverse of the multiplier system of its shadow, Conjecture 6.3 completely determines the modular properties of H X g -i.e. the matrix-valued function ν X g -when S X g is non-vanishing. However, it may happen that S X g vanishes identically and H X g is a(n honest) modular form. The following conjecture puts a strong restriction on ν X g even in the case of vanishing shadow.
Conjecture 6.4. The multiplier system ν X g for H X g coincides with the inverse of the multiplier system for S X when restricted to Γ 0 (n g h g ).

Moonshine
We now formulate a conjecture which may be regarded as the analogue of the principal modulus property (often referred to as the genus zero property) of monstrous moonshine.
The monstrous McKay-Thompson series T g , for g an element of the monster, are distinguished in that each one is a principal modulus with pole at infinity for a genus zero group Γ g , meaning that T g is a Γ g -invariant function on the upper-half plane having a simple pole at the infinite cusp of Γ g , but having no poles at any other cusps of Γ g . Equivalently, T g satisfies the for some group Γ g , where T g | 1,0 γ is the function τ → T g (γτ ) by definition (cf. (3.5)). Note that the existence of a non-constant function T g satisfying the conditions (6.3) implies that the group Γ g has genus zero, for such a function necessarily induces an isomorphism from X Γg (cf. (2.15)) to the Riemann sphere. As such, if we assume that both T g and T g satisfy these conditions and both qT g and qT g have the expansion 1 + O(q) near τ → i∞, then T g and T g differ by an additive constant; i.e. T g = T g + C for some C ∈ C. In other words, the space of solutions to (6.3) is 1 or 2 dimensional, according as the genus of X Γg is positive or 0.
Observe the similarity between condition (ii) of (6.3) and the optimal growth condition (4.16). Since q −1 is the minimal polar term possible for a non-constant Γ g -invariant function on the upper-half plane, assuming that the stabiliser of infinity in Γ g is generated by ± ( 1 1 0 1 ), the condition (ii) is an optimal growth condition on modular forms of weight 0; the coefficients of a form with higher order poles will grow more quickly. The condition (iii) naturally extends this to the situation that Γ g has more than one cusp. Accordingly, we now formulate an analogue of (6.3)-and an extension of the optimal growth condition (4.16) to vector-valued mock modular forms of higher level-as follows. Suppose that ν is a (matrix-valued) multiplier system on Γ 0 (n) with weight 1/2, and suppose, for the sake of concreteness, that ν coincides with the inverse of the multiplier system of S X , for some Niemeier root system X, when restricted to Γ 0 (N ) for some N with n|N . Observe that, under these hypotheses, every component H r has a Fourier expansion in powers of q 1/4m where m is the Coxeter number of X, so q −1/4m is the smallest order pole that any component of H may have. Say that a vector-valued function H = (H r ) is a mock modular form of optimal growth for Γ 0 (n) with multiplier ν, weight 1/2 and shadow S if In condition (i) of (6.4) we write | ν,1/2,S for the weight 1/2 action of Γ 0 (n) with multiplier ν and twist by S (cf. (3.6)), on holomorphic vector-valued functions on the upper-half plane.
Recall that ν X g denotes the multiplier system of H X g , and S X g is its shadow. Recall also that n g denotes the order of (the image of) g ∈ G X in the quotient groupḠ X . We now conjecture that the umbral McKay-Thompson series all have optimal growth in the sense of (6.4), and this serves as a direct analogue of the Conway-Norton conjecture of monstrous moonshine, that all the monstrous McKay-Thompson series are principal moduli for genus zero subgroups of SL 2 (R); or equivalently, that they are all functions of optimal growth in the sense of (6.3). Conjecture 6.5. Let X be a Niemeier root system and let g ∈ G X . Then H X g is the unique, up to scale, mock modular form of optimal growth for Γ 0 (n g ) with multiplier ν X g , weight 1/2 and shadow S X g .
Conjecture 6.5 should serve as an important step in obtaining a characterisation of the mock modular forms of umbral moonshine. Note that the above conjecture has been proven in §4.1 (cf. Corollary 4.2) for the identity class of G X for all Niemeier root systems X. Note that in the case of the identity class we have Γ 0 (n) = SL 2 (Z) which has the cusp (representative) at i∞ as the only cusp. As a result, the more general conditions in (6.4) reduce to the condition (4.16) discussed in §4. For X = A 24 1 , this conjecture was proven for all conjugacy classes of G X = M 24 in [51]. See also [52] for related results in this case.

Discriminants
One of the most striking features of umbral moonshine is the apparently intimate relation between the number fields on which the irreducible representations of G X are defined and the discriminants of the vector-valued mock modular form H X . In this subsection we will discuss this "discriminant property", extending the discussion in [1].
First, for a Niemeier root system with Coxeter number m we observe that the discriminants of the components H X r of the mock modular form H X = H X 1A determine some important properties of the representations of G X . Here we say that an integer D is a discriminant of H X if there exists a term q d = q − D 4m with non-vanishing Fourier coefficient in at least one of the components. The following result can be verified explicitly using the tables in § §B,D.
Proposition 6.6. Let X be one of the 23 Niemeier root systems. If n > 1 is an integer satisfying 1. there exists an element of G X of order n, and 2. there exists an integer λ that satisfies at least one of the following conditions and such that D = −nλ 2 is a discriminant of H X . First, (n, λ) = 1, and second, λ 2 is a proper divisor of n, then there exists at least one pair of irreducible representations and * of G X and at least one element g ∈ G X such that tr (g) is not rational but tr (g), tr * (g) ∈ Q( √ −n) (6.5) and n divides ord(g).
The list of integers n satisfying the two conditions of Proposition 6.6 is given in Table 10.
We omit from the table lambencies of Niemeier root systems for which there exists no integer n satisfying these conditions.
From now on we say that an irreducible representation of the umbral group G X is of type n if n is an integer satisfying the two conditions of Proposition 6.6 and the character values of generate the field Q( √ −n). Evidently, irreducible representations of type n come in pairs ( , * ) with tr * (g) the complex conjugate of tr (g) for all g ∈ G X . The list of all irreducible representations of type n is also given in Table 10. are not all real. We can now state the next observation.
Proposition 6.7. For each Niemeier root system X, an irreducible representation of G X has Frobenius-Schur indicator 0 if and only if it is of type n for some n defined in Proposition 6.6.
The Schur index of an irreducible representation of a finite group G is the smallest positive integer s such that there exists a degree s extension k of the field generated by the character values tr (g) for g ∈ G such that can be realised over k. Inspired by Proposition 6.7 we make the following conjecture.
Conjecture 6.8. If is an irreducible representation of G X of type n then the Schur index of is equal to 1.
In other words, we conjecture that the irreducible G X -representations of type n can be realised over Q( √ −n). For X = A 24 1 this speculation is in fact a theorem, since it is known [53] that the Schur indices for M 24 are always 1. For X = A 12 2 it is also known [53] that the Schur indices forḠ (3) = M 12 are also always 1. Moreover, the representations of G (3) 2.Ḡ (3) with characters χ 16 and χ 17 in the notation of Table 12 have been constructed explicitly over Q( in [54]. Finally, Proposition 6.7 constitutes a non-trivial consistency check for Conjecture 6.8 since the Schur index is at least 2 for a representation with Frobenius-Schur indicator equal to −1. (χ 8 , χ 9 ), (χ 10 , χ 11 ), (χ 12 , χ 13 )  Equipped with the preceding discussion we are now ready to state our main observation for the discriminant property of umbral moonshine. Proposition 6.9. Let X be a Niemeier root system with Coxeter number m. Let n be one of the integers in Table 10 and let λ n be the smallest positive integer such that D = −nλ 2 n is a discriminant of H X . Then K X r,−D/4m = n ⊕ * n where n and * n are dual irreducible representations of type n. Conversely, if is an irreducible representation of type n and −D is the smallest positive integer such that K X r,−D/4m has as an irreducible constituent then there exists an integer λ such that D = −nλ 2 .
Extending this we make the following conjecture. Conjecture 6.10. Let X be a Niemeier root system with Coxeter number m. If D is a discriminant of H X which satisfies D = −nλ 2 for some integer λ then the representation K X r,−D/4m has at least one dual pair of irreducible representations of type n arising as irreducible constituents.
Conjecture (6.10) has been verified for the case X = A 24 1 in [55]. We conclude this section with conjectures arising from the observation (cf. §D) that the conjectural G X -module K X r,d is typically isomorphic to several copies of a single representation. We say a G-module V is a doublet if it is isomorphic to the direct sum of two copies of a single representation of G, and interpret the term sextet similarly. Conjecture 6.11. Let X be a Niemeier root system and let m be the Coxeter number of X.
Then the representation K X r,−D/4m is a doublet if and only if D = 0 and D = −nλ 2 for any integer λ and for any n listed in Table 10 corresponding to X. If X = A 3 8 then the representation K X r,−D/4m is a sextet if and only if D = −27λ 2 for some integer λ.
In particular, for the nine Niemeier root systems 3) that the graded super-characters H X g arising from the action of G X on K X are vector-valued mock modular forms with concretely specified shadows S X g , and 3. (Conjecture 6.5) that the umbral McKay-Thompson series H X g are uniquely determined by an optimal growth property which is directly analogous to the genus zero property of monstrous moonshine.
To lend evidence in support of these conjectures we explicitly identify (almost all of) the umbral McKay-Thompson series H X g . In this way, from the 23 Niemeier root systems we obtain 23 instances of umbral moonshine, encompassing all the 6 instances previously discussed in [1] and in particular the case with G X = M 24 first discussed in the context of the K3 elliptic genus [11]. Apart from uncovering 17 new instances, we believe that the relation to Niemeier lattices sheds important light on the underlying structure of umbral moonshine. First, the construction of the umbral group G X is now completely uniform: G X is the outer-automorphism group of the corresponding Niemeier lattice (cf. (2.30)). Second, it provides an explanation for why the 6 instances discussed in [1] are naturally labelled by the divisors of 12: they correspond to Niemeier root systems given by evenly many copies (viz., 24/( − 1)) of an A-type root system A −1 . Third, it also sheds light on the relation between umbral moonshine and meromorphic weight 1 Jacobi forms as well as weight 0 Jacobi forms. For as we have seen in §3.4, the umbral forms H X can be constructed uniformly by taking theta-coefficients of finite parts of certain weight 1 meromorphic Jacobi forms, but in general the relevant meromorphic Jacobi form has simple poles not only at the origin but also at non-trivial torsion points whenever the corresponding root system has a Dor E-type root system as an irreducible component. As a result, in those cases the relation to weight 0 Jacobi forms is less direct as the Eichler-Zagier operator W X of (4.31) is no longer proportional to the identity. In particular, in these cases the umbral mock modular form H X does not arise in a direct way from the decomposition of a weight 0 (weak) Jacobi form into irreducible characters for the N = 4 superconformal algebra.
Recall that the relevant weight 0 Jacobi form in the construction described in §4.3 coincides with the elliptic genus of a K3 surface in the case of the Niemeier root system X = A 24 1 ( = 2, G X = M 24 ). As the relation to weight 0 forms becomes less straightforward in the more general cases, the relation between umbral moonshine and sigma models, or in fact any kind of conformal field theory, also becomes more opaque. An interesting question is therefore the following. What, if any, kind of physical theory or geometry should attach to the more general instances of umbral moonshine?
To add to this puzzle, the Borcherds lift of the K3 elliptic genus is a Siegel modular form which also plays an important rôle in type II as well as heterotic string theory compactified on K3 × T 2 [56][57][58]. As pointed out in [12] and refined in [17], Mathieu moonshine in this context (corresponding here to X = A 24 1 ) leads to predictions regarding Siegel modular forms which have been partially proven in [59]. Furthermore, this Siegel modular form also serves as the square of the denominator function of a generalised Kac-Moody algebra developed by Gritsenko-Nikulin in the context of mirror symmetry for K3 surfaces [60][61][62][63][64]. As discussed in detail in [1, §5.5], many of these relations to string theory and K3 geometry extend to some of the other 5 instances of umbral moonshine discussed in that paper. Since the relation between umbral forms and weight 0 modular forms is modified when D-or E-type root systems are involved, it would be extremely interesting to determine how the above-mentioned relations to string theory, K3 surfaces, and generalised Kac-Moody algebras manifest in the more general cases.
Regarding K3 surfaces, note that Niemeier lattices have a long history of application to this field, and the study of the symmetries of K3 surfaces in particular. See for instance [65].
See [66] for an analysis involving all of the Niemeier lattices. It would be interesting to explore the extent to which recent work [22,23] applying the Niemeier lattice L X to the X = A 24 1 case of umbral moonshine can be extended to other Niemeier root systems in light of [66].
In another direction, the physical context of Mathieu moonshine has been extended recently to K3 compactifications of heterotic string theory with 8 supercharges [67]. As the structure of theories with 8 supercharges is much less rigid than those with 16 supercharges, one might speculate that a suitable generalisation of [67]

A.2 Jacobi Theta Functions
We define the Jacboi theta functions θ i (τ, z) as follows for q = e(τ ) and y = e(z).
Note that there are competing conventions for θ 1 (τ, z) in the literature and our normalisation may differ from another by a factor of −1 (or possibly ±i).

A.3 Higher Level Modular Forms
The congruence subgroups of the modular group SL 2 (Z) that are most relevant for this paper are the Hecke congruence groups A modular form for Γ 0 (N ) is said to have level N . For N a positive integer a modular form of weight 2 for Γ 0 (N ) is given by where σ(k) is the divisor function σ(k) = d|k d. The function λ N is, of course, only non-zero when N > 1.
Observe that a modular form on Γ 0 (N ) is a modular form on Γ 0 (M ) whenever N |M , and for some small N the space of forms of weight 2 is spanned by the λ d (τ ) for d a divisor of N .
A discussion of the ring of weak Jacobi forms of higher level can be found in [68].

B Characters
In §B.1 we give character tables (with power maps and Frobenius-Schur indicators) for each group G X for X a Niemeier root system. These were computed with the aid of the computer algebra package GAP4 [70]. We use the abbreviations a n = √ −n and b n = (−1 + √ −n)/2 in these tables.
The tables in §B.2 furnish cycle shapes and character values-the twisted Euler charactersattached to the representations of the groups G X described in §2.4. Using this data we can obtain explicit expressions for the shadows S X g of the vector-valued mock modular forms H X g according to the prescription of §5.1.

B.2 Euler Characters
The tables in this section describe the twisted Euler characters and associated cycle shapes attached to each group G X in §2.4. According to the prescription of §5.1 the character valuesχ X A g , χ X A g , &c., can be used to describe the shadows of the vector-valued mock modular forms H X g attached to each g ∈ G X by umbral moonshine. We also identify symbols n g |h g which are used in §6.2 to formulate conjectures about the modularity of H X g . By definition n g is the order of the image of g ∈ G X inḠ X and h g = N g /n g where N g denotes the product of shortest and longest cycle lengths appearing in the cycle shapeΠ X g .

C Coefficient Tables
In this section we furnish tables of Fourier coefficients of small degree for the vector-valued mock modular forms H X g attached to elements g ∈ G X . For each Niemeier root system X and each conjugacy class [g] ∈ G X we give a table that displays the coefficients of H X g,r for sufficiently many r that any other component coincides with one of these up to sign. For instance, we always have H X g,r = −H X g,−r , so it suffices to list the H X g,r for 0 < r < m when the Coxeter number of X is m. When X has no A-type components there are further redundancies, so that H X g,2 = H X g,4 = 0 and H X g,1 = H X g,5 , for example, when X = D 6 4 . Recall that H X g is conjectured to coincide with the graded trace function of the umbral module K X for all Niemeier root systems X except for X = A 8 3 , and the relation has an additional factor of 3 when X = A 8 3 (cf. Conjecture 6.1).