Umbral Moonshine and Linear Codes

John F. R. Duncan∗

7 June 2013

Abstract

We exhibit each of the finite groups of umbral moonshine as a distinguished subgroup of the automorphism group of a distinguished linear code, each code being defined over a different quotient of the ring of integers. These code constructions entail permutation representations which we use to give a description of the multiplier systems of the vector-valued mock modular forms attached to the conjugacy classes of the umbral groups by umbral moonshine.

1 Introduction

In 2010 Eguchi–Ooguri–Tachikawa made a remarkable observation [1] relating the elliptic genus

of a K3 surface to the largest Mathieu group M24 via a decomposition of the former into a linear combination of characters of irreducible representations of the small N = 4 superconformal algebra. The elliptic genus is a topological invariant and for any K3 surface it is given by the weak Jacobi form

 2  2  2! θ2(τ, z) θ3(τ, z) θ4(τ, z) ZK3(τ, z) = 8 + + (1.1) θ2(τ, 0) θ3(τ, 0) θ4(τ, 0)

of weight 0 and index 1. The θi here denote Jacobi theta functions (cf. (B.3)). The decompo- sition into N = 4 characters leads to an expression

2 ∞ ! θ1(τ, z) X Z (τ, z) = 24 µ(τ, z) + q−1/8 − 2 + t qn
(1.2) K3 η(τ)3 n n=1

for some tn ∈ Z (cf. [2]) where θ1(τ, z) and µ(τ, z) are defined in (B.3-B.4). By inspection, the

first five tn are given by t1 = 90, t2 = 462, t3 = 1540, t4 = 4554, and t5 = 11592; the observation

∗Department of Mathematics, Case Western Reserve University, Cleveland, OH 44106, U.S.A. E-mail: [email protected]

1 Umbral Moonshine and Linear Codes 2

of [1] is that each of these tn is twice the dimension of an irreducible representation of M24 (cf. [3]). Experience with monstrous moonshine [4, 5, 6], for example, leads us to conjecture that every (2) tn may be interpreted as the dimension of an M24-module Kn−1/8, and that, assuming such a module structure to be known, we may obtain interesting functions by replacing tn = tr| (2) 1 Kn−1/8 (2) with tr| (2) g for non-identity elements g ∈ M24. If we define H (τ) by requiring Kn−1/8

3 2 (2) ZK3(τ, z)η(τ) = θ1(τ, z) (aµ(τ, z) + H (τ)), (1.3) then a = 24 and ! (2) −1/8 X n H (τ) = q −2 + tnq (1.4) n>0 is a slight modification of the generating function of the tn. The inclusion of the term −2 and the factor q−1/8 = e−2πiτ/8 has the effect of improving the modularity: H(2)(τ) is a (weak) −3 3 mock modular form for SL2(Z) with multiplier  (cf. (B.2)), weight 1/2, and shadow 24η (cf. (B.1)), meaning that if we define the completion Hˆ (2)(τ) of the holomorphic function H(2)(τ) by setting Z ∞ Hˆ (2)(τ) = H(2)(τ) + 24 (4i)−1/2 (z + τ)−1/2η(−z¯)3dz, (1.5) −τ¯

ˆ (2) then H (τ) transforms as a modular form of weight 1/2 on SL2(Z) with multiplier system conjugate to that of η(τ)3, so that we have

(γ)−3Hˆ (2)(γτ)j(γ, τ)1/2 = Hˆ (2)(τ)

−1 for γ ∈ SL2(Z) where j(γ, τ) = (cτ + d) when (c, d) is the lower row of γ. (2) The McKay–Thompson series Hg for g ∈ M24 is now defined—assuming knowledge of the (2) L (2) M24-module structure on K = n Kn−1/8—by setting

∞ (2) −1/8 X n−1/8 Hg (τ) = −2q + tr|Kn−1/8 (g)q (1.6) n=1

2πiτ (2) (2) where q = e(τ) = e . Actually the functions Hg are more accessible than M24-module K

(for which no concrete construction is yet known) since one only needs to know tr|Kn−1/8 g for (2) a few values of n if one assumes the function Hg to have good modular properties. Concrete (2) proposals made in [7, 8, 9, 10] entail the prediction that Hg should be a certain concretely −3 defined (cf. [2]) mock modular form of weight 1/2 for Γ0(ng) with shadow proportional to  Umbral Moonshine and Linear Codes 3

(2) where ng is the order of g ∈ M24, and the existence of a compatible M24-module K has now been established by Gannon [11]. In [12] it was shown that the observation of Eguchi–Ooguri–Tachikawa belongs to a family of relationships—umbral moonshine—between finite groups G(`) and vector-valued mock modular (`) (`) (`) (`) forms Hg = (Hg,1,...,Hg,`−1) for g ∈ G , that support the existence of infinite-dimensional bi-graded G(`)-modules

(`) M M (`) K = Kr,n−r2/4`, (1.7) 0

(`) −1/4` X n−r2/4` Hg,r(τ) = −2δr,1q + tr|K(`) (g)q . (1.8) r,n−r2/4` n∈Z r2−4n`<0

The cases of umbral moonshine presented in [12] are indexed by the positive integers ` such that ` − 1 divides 12. In this note we give constructions of the umbral groups G(`) as automorphisms of linear codes over rings Z/`, and we show, as an application, how to use the resulting permutation representations to describe the multiplier systems of the umbral McKay– (`) Thompson series Hg .

Table 1: The groups of umbral moonshine.

` 2 3 4 5 7 13 (`) G M24 2.M12 2.AGL3(2) GL2(5)/2 SL2(3) Z/4Z

It is striking that the codes arising all appear in the article [13] written in connection with the Leech lattice. One consequence is that each of the umbral groups G(`) may be regarded as a subgroup of the Conway group Co0, this being the automorphism group of the Leech lattice. Another consequence is the suggestion that there might be analogous cases of umbral moonshine for the remaining 17 codes (or equivalently, Niemeier root systems) appearing in [13]. Confirmation of this suggestion will be demonstrated in a forthcoming article [14]. Umbral Moonshine and Linear Codes 4

2 Codes

In this section we review the notion of linear code over a ring Z/m and define a distinguished linear code G(`) over Z/` for each ` such that ` − 1 divides 12. n A (linear) code over Z/m of length n is a Z/m-submodule of (Z/m) . Let {ei | i ∈ Ω} denote the standard basis for (Z/m)n, the index set Ω having cardinality n. We equip (Z/m)n 0 P 0 P with a Z/m-valued Z/m-bilinear form by setting (C,C ) = i∈Ω cici in case C = i∈Ω ciei and 0 P 0 n ⊥ n C = i∈Ω ciei, and given S ⊂ (Z/m) we define S = {D ∈ (Z/m) | (C,D) = 0, ∀C ∈ S}. We say that a code C < (Z/m)n is self-orthogonal in case C ⊂ C⊥ and we say that C is self-dual if it is maximally self-orthogonal, meaning that C = C⊥. Given a code C of length n over Z/m we n define Aut(C) to be the subgroup of GLn(Z/m) that stabilizes the subspace C < (Z/m) , and we define Aut±(C) to be the subgroup of Aut(C) consisting of signed coordinate permutations, meaning that

± Aut (C) = {γ ∈ GLn(Z/m) | γ(C) ⊂ C and γ(B) ⊂ B} (2.1)

± where B denotes the set {±ei | i ∈ Ω}. Observe that Aut(C) and Aut (C) coincide when m ∈ {2, 3, 4}, but are generally different otherwise. We now identify a distinguished linear code G(`) over Z/` for each ` such that ` − 1 divides 12. The code G(`) will have length 24/(` − 1) and the construction we give will be either a rephrasing or direct reproduction of a construction given (much earlier) in [13]. In particular, it will develop that G(2) is the extended binary Golay code and G(3) is the extended ternary Golay code. The remaining G(`) are visible, in a certain sense, inside the Leech lattice (cf. [13]) and may be regarded as natural analogues of the extended binary and ternary Golay codes defined over larger quotients of the ring of integers. (2) 24 To define G equip (Z/2) with the standard basis {ei} and index this basis with the set Ω(2) = {∞} ∪ Z/23. Let N be the subset of Ω(2) consisting of the elements of Z/23 that are not squares in Z/23, so that N = {5, 7, 10, 11, 14, 15, 17, 19, 20, 21, 22}, and define

X 24 Ci = e∞ + en+i ∈ (Z/2) (2.2) n∈N

24 for i ∈ Z/23. Then the subspace of (Z/2) generated by the set {Ci | i ∈ Z/23} is a self-dual linear code over Z/2 of length 24 which we denote G(2). In fact, G(2) is a copy of the extended binary Golay code (see various chapters in [15] for more details) and the automorphism group (2) (2) of G is isomorphic to the largest Mathieu group, M24. The group G defined in [12] is also Umbral Moonshine and Linear Codes 5

± (2) (2) (2) isomorphic to M24 so we have Aut (G ) = Aut(G ) ' G . (3) 12 To define G equip (Z/3) with the standard basis {ei} and take the index set to be Ω(3) = {∞} ∪ Z/11. The set of non-squares in Z/11 is N = {2, 6, 7, 8, 10}. Let Q be the 12 complement of N in Z/11, so that Q = {0, 1, 3, 4, 5, 9}, and define Ci ∈ (Z/3) for i ∈ Z/11 by setting

X X 12 Ci = 2e∞ + 2en+i + en+i ∈ (Z/3) . (2.3) n∈N n∈Q

(3) Then the code G generated by the Ci is a copy of the extended ternary Golay code (cf. (3) [15]) and the automorphism group of G is isomorphic to a group 2.M12, being the unique

(up to isomorphism) non-trivial double cover of the Mathieu group M12 (cf. [3]). Again we ± (3) (3) (3) have Aut (G ) = Aut(G ) and the group G defined in [12] is also isomorphic to 2.M12, so Aut±(G(3)) ' G(3). 8 For ` = 4 equip (Z/4) with the standard basis, indexed by Ω(4) = {∞} ∪ Z/7, let N denote 8 the set {3, 5, 6} of non-squares in Z/7, and define Ci ∈ (Z/4) for i ∈ Z/7 by setting

X 8 Ci = 3e∞ + 2ei + en+i ∈ (Z/4) . (2.4) n∈N

(4) 8 (4) Define G to be the Z/4-submodule of (Z/4) generated by the set {Ci | i ∈ Z/7}. Then G is a copy of the octacode [16] (see also [17, §3.2]). The automorphism group of G(4) has a central subgroup of order 2, generated by the symmetry C = (ci) 7→ (−ci), and modulo this central subgroup we obtain the affine general linear group of degree 3 over a field with 2 elements, which 4 is the same as the stabilizer in GL4(2) of a line in (Z/2) . Comparing with the definition of G(4) given in [12] we find that Aut±(G(4)) = Aut(G(4)) ' G(4). Now consider the case that ` = 5. Index the standard basis of (Z/5)6 with the set Ω(5) = {∞} ∪ {0, 1, 2, 3, 4} and define

6 Ci = e∞ + e1+i + 4e2+i + 4e3+i + e4+i ∈ (Z/5) (2.5)

(5) 6 ± (5) for i ∈ Z/5. Then the Ci generate a self-dual code G < (Z/5) . We see that Aut (G ) is a (5) proper subgroup of Aut(G ) since the latter contains the central element ei 7→ 2ei, for example, ± (5) which does not preserve B = {±ei}. The group Aut (G ) is a double cover of S5, regarded as a permutation group on 6 points via the isomorphism S5 ' PGL2(5). The particular double cover arising is perhaps a little unfamiliar in that it does not contain the Schur double cover of

A5 as a subgroup; it can be realized explicitly as the quotient of GL2(5) by its unique central Umbral Moonshine and Linear Codes 6

subgroup of order 2. (Note that the centre of GL2(5) is cyclic of order 4.) Upon comparison with [12] we conclude that Aut±(G(5)) ' G(5). For ` = 7 we take Ω(7) = {∞} ∪ Z/3 as an index set for the standard basis of (Z/7)4 and we 4 define Ci ∈ (Z/7) by setting

4 Ci = e∞ + 2ei + e1+i + 6e2+i ∈ (Z/7) (2.6)

(7) for i ∈ Z/3. We define G to be the Z/7-submodule generated by the Ci for i ∈ Z/3 and (7) ± (7) observe that G is a self-dual code over Z/7 with Aut (G ) a double cover of PSL2(3) ' A4. ± (7) (7) In fact the double cover arising is SL2(3) and we have Aut (G ) ' G . The remaining code is G(13) which has length 2 = 24/(13 − 1) and which we may take to 2 (13) be generated by e∞ + 5e0 ∈ (Z/13) . (In this case we set Ω = {∞} ∪ Z/1.) The code G(13) is self-dual (since 12 + 52 ≡ 0 (mod 13)) and Aut±(G(13)) is cyclic of order 4, generated ± (13) (13) explicitly by (c∞, c0) 7→ (c0, −c∞). Once again we find Aut (G ) ' G and we conclude that Aut±(G(`)) ' G(`) for all ` (such that ` − 1 divides 12).

3 Automorphy

We now take G(`) = Aut±(G(`)) for ` ∈ {2, 3, 4, 5, 7, 13} and give an explanation of how these constructions may be used to describe the automorphy of the vector-valued mock modular forms (`) (`) Hg attached (in [12]) to the elements of G via umbral moonshine. Observe that G(`) is a code of length 24/(` − 1) over Z/` for each `. Thus we obtain a permutation representation of degree 24 for G(`) by considering its action on the set

(`) {cei | i ∈ Ω , c ∈ Z/`, c 6= 0} (3.1)

(`) of non-zero multiplies of the basis vectors ei. Write Π˜ g for the cycle shape attached to g ∈ G arising from this permutation representation and write g 7→ χ˜g for the corresponding character (`) 12 (`) of G . Then, for example, Π˜ g = 2 if g is the central involution in G and ` 6= 4. (For each ` (`) the group G contains the transformation ei 7→ −ei, which is the unique central involution of (`) (4) 8 8 G if ` > 2.) In the case that ` = 4 and g is the central involution of G we have Π˜ g = 1 2 . Define a second permutation representation of degree 24/(` − 1) for G(`) by considering the (`) (`) action of G on the sets Ei = {cei | c ∈ Z/`, c 6= 0} for i ∈ Ω , which constitute a system of (`) imprimitivity for the degree 24 permutation representation of G just defined. Write Π¯ g for the corresponding cycle shapes and g 7→ χ¯g for the character, and observe that this permutation Umbral Moonshine and Linear Codes 7

representation is generally not faithful, for the central involution ei 7→ −ei acts trivially. We write g 7→ g¯ for the natural map from G(`) to its quotient G¯(`) by the central subgroup generated (`) (`) (2) by ei 7→ −ei. (We have G ' G¯ when ` = 2 since ei = −ei for i ∈ Ω .) Observe that the smaller permutation representationχ ¯ is an irreducible constituent of the larger oneχ ˜. Indeed, the latter contains b`/2c copies of the former, and b(` − 1)/2c copies of a faithful representation of degree 24/(` − 1), whose character we denote g 7→ χg, which is just that which you obtain by taking the matrices representing the action of G(`) = Aut±(G(`)) as elements of GLn(Z/`)—these matrices having exactly one non zero entry ±1 in each row and column—and regarding them as elements of GLn(C). (Here n = 24/(` − 1).)

χ˜g = b`/2cχ¯g + b(` − 1)/2cχg (3.2)

(`) (`) It is now easy to describe the shadow of the vector-valued mock modular form Hg = (Hg,r), (`) (`) (`) (`) for it is given by Sg = (Sg,r) where Sg,r =χ ¯gS`,r for r odd, and Sg,r = χgS`,r for r even, where Sm,r denotes the unary theta series

X (2km+r)2/4m Sm,r(τ) = (2km + r)q . (3.3) k∈Z

Note that Sm = (Sm,r) is a vector-valued cusp form of weight 3/2 for the modular group SL2(Z).

n1 nk Given a cycle shape Π = m1 ··· mk with ni > 0 for 1 ≤ i ≤ k and m1 < m2 < ··· < mk (`) call mk the largest factor of Π and call m1 the smallest factor. For each g ∈ G define ng to be the largest factor of Π¯ g (this turns out to be the same as the order ofg ¯) and define Ng to be the product of the smallest and largest factors of Π˜ g. The significance of these values for (`) (`) the automorphy of Hg is that ng is the level of Hg —i.e., the smallest positive integer such (`) that the vector-valued mock modular form Hg is a mock modular form for Γ0(ng)—and Ng (`) is the smallest positive integer such that the multiplier system for Hg coincides with that of

S` = (S`,r) when restricted to Γ0(Ng). (`) (`) The expression for Sg just given determines the multiplier system of Hg —since the mul- tiplier system of a mock modular form is the inverse of the multiplier system of its shadow (cf. (`) [18])—in the case thatχ ¯g and χg are both non-zero. The multiplier system of Hg may be described as follows in the case thatχ ¯gχg = 0. Define v(`) = ` + 2 in case ` is odd (i.e. ` ∈ {3, 5, 7, 13}), and set v(`) = ` − 1 when ` is even (`) (i.e. ` ∈ {2, 4}). Let σ denote the inverse of the multiplier system of the cusp form S` = (S`,r) Umbral Moonshine and Linear Codes 8

(`) and define ψn|h :Γ0(n) → GL`−1(C) for positive integers n and h by setting

      (`) a b (`) cd (`) a b ψ   = e −v σ   (3.4) n|h c d nh c d in case h divides n, and otherwise

      (`) a b (`) cd (n, h) (`) a b c(d+1)/n c/n ψ   = e −v σ   J K (3.5) n|h c d nh n c d where J is the diagonal matrix J = diag(1, −1, 1, ··· ) with alternating ±1 along the diagonal, and K is the “reverse shuffle” permutation matrix corresponding to the permutation

(1, ` − 1)(2, ` − 2)(3, ` − 3) ··· (3.6)

`−1 (`) (`) of the standard basis {e1, . . . , e`−1} of C . Now the multiplier system of Hg is given by ψn|h for n = ng and h = hg = Ng/ng whenχ ¯gχg = 0. Note that the factor (n, h)/n can usually be ignored in practice, for there is just one case in whichχ ¯gχg = 0 and h = hg does not divide n = ng and (n, h) 6= n; viz., the case that ` = 3 (3) 1 1 1 1 and g ∈ G satisfies Π¯ g = 2 10 and Π˜ g = 4 20 .

Acknowledgement

The author thanks RIMS and the organizers of the RIMS workshop Research on finite groups and their representations, vertex operator algebras, and algebraic combinatorics, Kyoto, Japan, January 2013, for support and hospitality. The author’s understanding of umbral moonshine, and the content of this note in particular, has developed from many conversations with many people, but the greatest burden has fallen upon his coauthors Miranda Cheng and Jeffrey Harvey, to whom the author is extremely grateful. We are indebted to George Glauberman for the observation that the groups of umbral moonshine are related to deep holes in the Leech lattice. Umbral Moonshine and Linear Codes 9 and g χ , and also the g ¯ Π 6 4 | 6 2 and 6 1 g 2 1 1 6 | 3 ˜ | Π 23 2 1 2 1 , so we just give the ¯ 2 1 χ 1 4 = 3 23 1 6 | 5 | 21 χ 4 1 1 3 = ¯ 1 χ 15 1 21 1 4 5 | 6 | 5 = 2 4 1 ` 3 1 1 1 4 14 4 1 15 = 2 we have ˜ 1 1 4 | 2 | 7 ` 2 1 4 2 , as well as the cycle shapes 1 1 g 1 χ 4 2 12 14 2 4 4 | | 4 12 . In case 2 } , and g 1 13 χ , 12 2 12 7 1 3 4 | 8 | , 6 3 1 5 , 4 1 4 , 2 3 2 , 6 1 12 1 3 | 2 3 | 11 6 2 1 1 ∈ { 2 ` 2 11 2 3 | | 12 10 2 2 Table 2: Characters and cycle shapes at 2 2 8 1 8 1 2 1 10 4 2 | | , for each 1 8 ) 2 1 ` ( 2 1 G 3 ∈ 1 2 1 8 7 | | 24 3 2 0 2 0 0 1 1 0 1 3 24 8 0 6 0 0 4 0 4 2 0 g 1A 2A 2B 3A 3B 4A 4B 4C 5A 6A 6B 1 7 1 1 7AB 8A 10A 11A 12A 12B 14AB 15AB 21AB 23AB g g ] ] g g h h g g | | g g ¯ ¯ ¯ ¯ χ χ [ [ g g Π Π n n for each g h | g n . The conjugacy class names are chosen to coincide with those of [12]. g ¯ A Characters The tables in this section describe the character values ¯ symbols Π Umbral Moonshine and Linear Codes 10 1 1 4 1 | 22 11 − 1 1 2 1 2 1 11 11 2 1 1 1 1 1 8 11 11 | 20 10 1 1 4 2 2 1 1 8 1 8 14 1 1 4 1 1 2 7 1 2 | 7 1 2 2 − 1 1 2 1 2 1 1 1 2 1 4 8 10 8 8 | 1 2 3 1 1 7 4 4 7 7 | 3 1 1 1 1 6 1 2 1 0 0 0 1 2 8 4 1 3 6 | 4 4 1 2 − 1 7 2 1 2 2 | 2 2 1 4 2 1 = 3 = 4 1 1 1 2 ` ` 6 6 2 1 2 8 4 2 8 | 3 3 2 4 2 1 4 2 2 1 2 3 1 1 1 2 4 6 6 | 3 1 2 2 24 6 6 2 2 | 6 12 2 2 6 2 2 0 1 4 6 2 2 5 | 2 0 0 0 1 2 6 10 3 3 2 | − 2 2 2 1 − 2 2 1 2 2 4 1 5 5 2 4 6 2 1 1 1 3 3 3 | 4 6 2 2 4 1 1 2 4 4 2 8 2 4 1 1 3 2 2 | 1 8 4 2 4 1 1 2 4 5 5 4 4 | 2 4 4 2 6 2 2 2 | 4 4 4 4 12 4 | 4 3 6 4 4 4 4 | 4 2 3 3 4 8 | 2 3 3 Table 3: Characters and cycle shapes at Table 4: Characters and cycle shapes at 2 2 3 3 4 | 12 3 0 0 0 0 2 4 3 3 6 2 | 2 3 3 − 1 2 8 2 2 8 0 0 0 0 2 3 6 8 2 | 3 3 8 1 − 3 6 1 1 1 4 1 1 8 | 24 4 3 2 3 3 88000428 2 0021 1 2 | 1 12 1A 2A 2B 4A 4B 2C 3A 6A 6BC 8A 4C 7AB 14AB 1 1 4 − 2 1 4 8 g 2 2 ] g g h g g | 4 8 g ¯ ˜ ¯ χ χ [ g Π Π 1 1 n 8 2 2 6 6 | 2 4 4 2 12 0 4 | 12 12 1 2 − 12 24 1 1 12 1212 0 4 4 3 3 0 0 0 4 2 2 0 1 1 0 2 0 1 1 1A 2A 4A 2B 2C 3A 6A 3B 6B 4B 4C 5A 10A 12A 6C 6D 8AB 8CD 20AB 11AB 22AB 1 1 g ] g g h g g | g ¯ ˜ ¯ χ χ [ g Π Π n Umbral Moonshine and Linear Codes 11

Table 5: Characters and cycle shapes at ` = 5 [g] 1A 2A 2B 2C 3A 6A 5A 10A 4AB 4CD 12AB ng|hg 1|1 1|4 2|2 2|1 3|3 3|12 5|1 5|4 2|8 4|1 6|24

χ¯g 662200110 2 0 χg 6 −6 −2 2 0 0 1 −1 0 0 0 6 6 2 2 2 2 2 2 1 1 1 1 3 2 1 1 Π¯ g 1 1 1 2 1 2 3 3 1 5 1 5 2 1 4 6 24 12 12 8 8 2 4 4 4 2 2 6 4 2 4 2 Π˜ g 1 2 2 1 2 3 6 1 5 2 10 4 1 2 4 12

Table 6: Characters and cycle shapes at ` = 7 [g] 1A 2A 4A 3AB 6AB

ng|hg 1 1|4 2|8 3 3|4

χ¯g 4 4 0 1 1 χg 4 -4 0 1 -1 4 4 2 1 1 1 1 Π¯ g 1 1 2 1 3 1 3 24 12 6 6 6 3 3 Π˜ g 1 2 4 1 3 2 6

Table 7: Characters and cycle shapes at ` = 13 [g] 1A 2A 4AB

ng|hg 1 1|4 2|8

χ¯g 2 2 0 χg 2 -2 0 2 2 1 Π¯ g 1 1 2 24 12 6 Π˜ g 1 2 4 Umbral Moonshine and Linear Codes 12

B Special Functions

The Dedekind eta function, denoted η(τ), is a holomorphic function on the upper half-plane defined by the infinite product

Y η(τ) = q1/24 (1 − qn) (B.1) n>0

2πiτ where q = e(τ) = e . It is a modular form of weight 1/2 for the modular group SL2(Z) with multiplier  : SL2(Z) → C so that

η(γτ)(γ) j(γ, τ)1/2 = η(τ) (B.2)

−1 for all γ ∈ SL2(Z), where j(γ, τ) = (cτ + d) in case (c, d) is the lower row of γ. Setting q = e(τ) and y = e(z) we use the following conventions for the four standard Jacobi theta functions.

∞ 1/8 1/2 Y n n −1 n−1 θ1(τ, z) = −iq y (1 − q )(1 − yq )(1 − y q ) n=1 ∞ 1/8 1/2 Y n n −1 n−1 θ2(τ, z) = q y (1 − q )(1 + yq )(1 + y q ) n=1 ∞ (B.3) Y n n−1/2 −1 n−1/2 θ3(τ, z) = (1 − q )(1 + y q )(1 + y q ) n=1 ∞ Y n n−1/2 −1 n−1/2 θ4(τ, z) = (1 − q )(1 − y q )(1 − y q ) n=1

We write µ(τ, z) for the Appell-Lerch sum defined by setting

−iy1/2 X (−1)`ynq`(`+1)/2 µ(τ, z) = ` . (B.4) θ1(τ, z) 1 − yq `∈Z Umbral Moonshine and Linear Codes 13

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