Modular Moonshine III. 11 October 1994, corrected 18 Nov 1997

Richard E. Borcherds,^{∗}

D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 2SB, England.

Mathematics department, University of California at Berkeley, CA 94720-3840, U. S. A.

reb@dpmms.cam.ac.uk

www.dpmms.cam.ac.uk/˜ reb

In this paper we complete the proof of Ryba’s modular moonshine conjectures [R] that was started in [B-R]. We do this by applying Hodge theory to the cohomology of the monster Lie algebra over the ring of p-adic integers in order to calculate the Tate cohomology groups of elements of the monster acting on the monster vertex algebra.

Contents.

1. Introduction.

Notation.

2. Representations of Z_{p}[G].

3. The Z[1/2]-form of the monster Lie algebra.

4. The modular moonshine conjectures for p ≥ 13.

5. Calculation of some cohomology groups.

6. Elements of type 3B, 5B, 7B, and 13B.

7. Open problems and conjectures.

1. Introduction.

This paper is a continuation of the earlier paper [B-R] so we will only briefly recall results that are discussed there in more detail. In [B-R] we constructed a modular superalgebra for each element of prime order in the monster, and worked out the structure of this superalgebra for some elements in the monster. In this paper we work out the structure of this superalgebra for the remaining elements of prime order.

Ryba conjectured [R] that for each element of the monster of prime order p of type pA there is a vertex algebra ^{g}V defined over the finite field F_{p} and acted on by the centralizer C_{M}(g) of g in the monster group M, with the property that the graded Brauer character Tr(h|^{g}V) = ∑_{n} Tr(h|^{g}V_{n})q^{n} is equal to the Hauptmodul Tr(gh|V) = ∑_{n} Tr(g|V_{n})q^{n} (where V is the graded vertex algebra acted on by the monster constructed by Frenkel, Lepowsky and Meurman [FLM]). In [B-R] the vertex superalgebra ^{g}V was defined for any element g ∈ M of odd prime order to be the sum of the Tate cohomology groups Ĥ^{0}(g, V [1/2]) ⊕ Ĥ^{1}(g, V [1/2]) for a suitable Z[1/2] form V [1/2] of V, and it was shown that Tr(h|^{g}V) = Tr(h|Ĥ^{0}(g, V [1/2]))− Tr(h|Ĥ^{1}(g, V [1/2])) was equal to the Hauptmodul of gh ∈ M. Hence to prove the modular moonshine conjecture for an element g of type pA it is enough to prove that Ĥ^{1}(g, V [1/2]) = 0. In [B-R] this was shown by explicit calculation for the elements of type pA for p ≤ 11, using the fact that these elements commute with an element of type 2B. For p ≥ 13 this method does not work as these elements do not commute with an element of type 2B. The first main theorem of this paper is theorem 4.1 which states that if g is an element of prime order p ≥ 13 not of type 13B then Ĥ^{1}(g, V [1/2]) = 0 (assuming a certain condition about the monster Lie algebra, whose proof should appear later). Hence Ryba’s conjectures are proved for all elements of M of type pA. (Actually this is not quite correct, because the proof for p = 2 in [B-R] assumes an unproved technical hypothesis.) The proof we give fails for exactly the cases already proved in [B-R].

We can also ask what happens for the other elements of order less than 13. The cases of elements of types 2B or 3C are also treated in [B-R]: for type 3C, Ĥ^{1}(g, V [1/2]) is again zero, and for type 2B, Ĥ^{1}(g, V [1/2]) is zero in even degree and Ĥ^{0}(g, V [1/2]) is zero in odd degree. This leaves the cases where g is of type 3B, 5B, 7B, or 13B, when the Tate cohomology groups Ĥ^{0}(g, V [1/2]) and Ĥ^{1}(g, V [1/2]) are usually both nonzero in each degree. In these cases the structure of the cohomology groups is determined by our second main result, theorem 6.1, which states that the unique element σ of order 2 in the center of

∗ Supported by NSF grant DMS-9401186 and by a Royal Society research professorship.

C_{M}(g)/O_{p}(C_{M}(g)) acts as 1 on Ĥ^{0}(g, V [1/2]) and as −1 on Ĥ^{1}(g, V [1/2]). This determines the modular characters of both cohomology groups because Tr(h|Ĥ^{0}(g, V [1/2]) + Tr(h|Ĥ^{1}(g, V [1/2])) is then given by Tr(hσ|Ĥ^{0}(g, V [1/2])) − Tr(hσ|Ĥ^{1}(g, V [1/2])) which is the Hauptmodul of the element ghσ ∈ M. Hence both Tr(h|Ĥ^{0}(g, V [1/2])) and Tr(h|Ĥ^{1}(g, V [1/2])) can be written explicitly as linear combinations of two Hauptmoduls for any p-regular element h ∈ C_{M}(g). We get two new modular vertex superalgebras acted on by double covers of sporadic groups when g has type 3B or 5B: a vertex superalgebra over F_{3} acted on by 2.Suz, and a vertex superalgebra over F_{5} acted on by 2.HJ. For elements of types 7B and 13B the vertex superalgebras we get are acted on by the double covers 2.A_{7} and 2.A_{4} of alternating groups.

In particular if the integral form V of the monster vertex algebra discussed in [B-R] exists and has the properties conjectured there then all the cohomology groups Ĥ^{i}(g, V) are now known for all elements g of prime order in M.

We now discuss the proofs of the two main theorems. We first quickly dispose of the second main theorem: if an element g ∈ M of odd order commutes with an element of type 2B then it is easy to work out its action on the Z[1/2]-form of the monster vertex algebra and hence the Tate cohomology groups Ĥ^{∗}(g, V [1/2]) can be calculated by brute force, which is what we do in sections 5 and 6 for the elements of types 3B, 5B, 7B, and 13B.

The elements g of prime order p ≥ 13 do not commute with any elements of type 2B (except when g has type 13B or 23AB) so we cannot use the method above. Instead we adapt the proof of the moonshine conjectures in [B]. By using a Z_{p}-form of the monster Lie algebra rather than a Q-form we can find some complicated relations between the coefficients of the series ∑_{n} Tr(h|Ĥ^{0}(g, V [1/2]_{n}))q^{n} and ∑_{n} Tr(h|Ĥ^{1}(g, V [1/2]_{n}))q^{n}. The reason we get more information by using Z_{p}-forms rather than Q_{p}-forms is that the group ring Z_{p}[Z/pZ] has 3 indecomposable modules which are free over Z_{p}, while the group ring Q_{p}[Z/pZ] only has 2 indecomposable modules which are free over Q_{p}, and this extra indecomposable module provides the extra information. The relations between the coefficients we get are similar to the relations defining completely replicable functions, except that some of the relations defining completely replicable functions (about “(2p−1)/p^{2}” of them) are missing. If p is large (≥ 13) we show that these equations have only a finite number of solutions. We then use a computer to find all the solutions, and see that the only solutions imply that Ĥ^{1}(g, V [1/2]) = 0 for all g of prime order at least 13 other than those of type 13B. This proof could probably be made to work for some smaller values of p such as 7 and 11, but the difficulty of showing that there are no extra solutions for the equations of the coefficients increases rapidly as p decreases, because for smaller p there are more equations missing.

Y. Martin [Ma] has recently found a conceptual proof that any completely replicable function is a modular function. Cummins and Gannon [CG] have greatly generalized this result using different methods, and showed that the functions are Hauptmoduls. It seems possible that their methods could be extended to replace the computer calculations in section 4, although Cummins has told me that this would probably require adding in some “missing” relations corresponding to the case (p, mn) ≠ 0 in proposition 3.4.

We summarize the results of this paper and of [B-R] about the vertex superalgebras ^{g}V for all elements g of prime order p in the monster. If g is of type pA, or pB for p > 13, or 3C, then ^{g}V is just a vertex algebra and not a superalgebra. If p ≥ 13 this is proved in this paper by an argument which works for any self dual form of the monster vertex algebra but which relies on computer calculations and on a so far unpublished argument about integral forms of the monster Lie algebra. For p < 13 (or p = 23) this is proved in [B-R] by calculating explicitly for a certain Z[1/2] form of the monster vertex algebra; this avoids computer calculations but only works for one particular form of the monster vertex algebra. Also, if p = 2 the calculation depends on an as yet unproved technical assumption about the Dong-Mason-Montague construction of the monster vertex algebra from an element of type 3B ([DM] or [M]). If g is of type pB for 2 ≤ p ≤ 13 then the super part of ^{g}V does not vanish. If p = 2 then ^{g}V is calculated explicitly in [B-R] (again using the unproved technical assumption) and it turns out that its ordinary part vanishes in odd degrees and its super part vanishes in even degrees. If 3 ≤ p ≤ 13 then ^{g}V is calculated in this paper, and we find that there is an element in C_{M}(g) acting as −1 on the super part and as 1 on the ordinary part of ^{g}V. In all cases we have explicitly described the modular characters of C_{M}(g) acting on the ordinary or super part of any degree piece of ^{g}V in terms of the coefficients of certain Hauptmoduls.

Notation.

[A] is the element of K represented by the module A.

A, B, C G-modules. A_{n} The alternating group on n symbols. Aut The automorphism group of something. c_{g}^{+}(n) The n’th coefficient of the Hauptmodul of g ∈ M, equal to Tr(1|^{g}V_{n}) = dim(Ĥ^{0}(g, V [1/2]_{n})) −

dim(Ĥ^{1}(g, V [1/2]_{n})). c_{g}^{−}(n) dim(Ĥ^{0}(g, V [1/2]_{n})) + dim(Ĥ^{1}(g, V [1/2]_{n})). c_{m,n} Defined in proposition 4.3.

C The complex numbers. C_{M}(g) The centralizer of g in the group M.

Co_{1} Conway’s largest sporadic simple group. E The positive subalgebra of the monster Lie algebra. F The negative subalgebra of the monster Lie algebra. F_{p} The finite field with p elements. f A homomorphism from K to Q defined in lemma 2.3. g An element of G, usually of order p. g_{i} The element g^{i} of G, used when it is necessary to distinguish the multiplication in the group ring from

some other multiplication. ⟨g⟩ The group generated by g. G A group, often cyclic of prime order p and generated by g.

h(A), h_{n} See section 5.

H The Cartan subalgebra of the monster Lie algebra. Ĥ^{i}(G, A) A Tate cohomology group of the finite group G with coefficients in the G-module A. Ĥ^{i}(g, A) means Ĥ^{i}(⟨g⟩, A), where ⟨g⟩ is the cyclic group generated by g. Ĥ^{∗}(g, A) The sum of the Tate cohomology groups Ĥ^{0}(g, A) and Ĥ^{1}(g, A), considered as a super module.

HJ The Hall-Janko sporadic simple group (sometimes denoted J_{2}).

I The indecomposable module of dimension p − 1 over Z_{p}[G], isomorphic to the kernel of the natural map

from Z_{p}[G] to Z_{p} and to the quotient Z_{p}[G]/N_{G}Z_{p}. Im The image of a map. K A ring which is a free Q-module with a basis of 3 elements [Z_{p}] = 1, [Z_{p}[G]], and [I]. Ker The kernel of a map. Λ, Λ̂ The Leech lattice and a double cover of the Leech lattice.

Λ^{n}(A) The n’th exterior power of A. Λ^{∗}(A) The exterior algebra ⊕_{n}Λ^{n}(A) of A.

L An even lattice. m The monster Lie algebra. M The monster simple group. N_{G} The element ∑_{g∈G} g of Z_{p}[G].

O_{p}(G) The largest normal p-subgroup of the finite group G.

p A prime, usually the order of g. Q, Q_{p} The rational numbers and the field of p-adic numbers.

ρ The Weyl vector of the monster Lie algebra. R The real numbers. R_{p} A finite extension of the p-adic integers. R(i) Defined just before lemma 3.2.

σ An automorphism of type 2B in the monster or the automorphism −1 of the Leech lattice. S_{n} A symmetric group.

S^{n}(A) The n’th symmetric power of A. S^{∗}(A) The symmetric algebra ⊕_{n}S^{n}(A) of A.

Suz Suzuki’s sporadic simple group. Tr Tr(g|A) is the usual trace of g on a module A if A is a module over a ring of characteristic 0, and the

Brauer trace if A is a module over a field of finite characteristic. V [1/n] A Z[1/n]-form of the monster vertex algebra.

V_{Λ} The integral form of the vertex algebra of Λ̂.

V_{n} The degree n piece of V. V^{n} An eigenspace of some group acting on V. gV A modular vertex algebra or superalgebra given by Ĥ^{∗}(g, V [1/n]) for some n coprime to |g|. Z The integers. Z_{p} The ring of p-adic integers. ω A cube root of 1 or a conformal vector. Ω The Laplace operator on Λ^{∗}(E).

2. Representations of Z_{p}[G].

We give some auxiliary results about modules over Z_{p}[G] where G is a cyclic group generated by g of prime order p, and in particular calculate the exterior and symmetric algebras of all indecomposable modules. All modules will be free over Z_{p} and will be either finitely generated or Z graded with finitely generated pieces of each degree. All tensor products will be taken over Z_{p}.

Recall from [B-R section 2] that there are 3 indecomposable modules over Z_{p}[G], which are Z_{p}, the group ring Z_{p}[G], and the module I of Z_{p}[G] that is isomorphic to the kernel of the natural map from Z_{p}[G] to Z_{p} and to the quotient Z_{p}[G]/N_{G}Z_{p} (where N_{G} = 1 + g + g^{2} + ⋯ + g^{p−1}). Their Tate cohomology groups are given by Ĥ^{0}(g, Z_{p}) = Z/pZ, Ĥ^{1}(g, Z_{p}) = 0, Ĥ^{0}(g, Z_{p}[G]) = 0, Ĥ^{1}(g, Z_{p}[G]) = 0, Ĥ^{0}(g, I) = 0, and Ĥ^{1}(g, I) = Z/pZ.

Lemma 2.1. The tensor products of these modules are given as follows:

Z_{p} ⊗ X = X (for any X)
Z_{p}[G] ⊗ X = Z_{p}[G]^{dim(X)} (the sum of dim(X) copies of Z_{p}[G])
I ⊗ I = Z_{p}[G]^{p−2} ⊕ Z_{p}.

Proof. The case Z_{p} ⊗X is trivial. If X has a basis x_{1}, …, x_{n} then for any fixed i the elements g^{j} ⊗g^{j}(x_{i}) (0 ≤ j < p) form a basis for a submodule of Z_{p}[G] ⊗ X isomorphic to Z_{p}[G], and Z_{p}[G] ⊗ X is the direct sum of these submodules for all i, which proves the result about Z_{p}[G] ⊗ X. For the case I ⊗ I we look at the cohomology sequence of the short exact sequence

0 → I ⊗ I → I ⊗ Z_{p}[G] → I ⊗ Z_{p} → 0

to see that Ĥ^{0}(I ⊗ I) = Z/pZ and Ĥ^{1}(I ⊗ I) = 0, which implies that I ⊗ I must be the sum of Z_{p} and some copies of Z_{p}[G]. The number of copies can be worked out by looking at the dimensions of both sides. This proves lemma 2.1.

Corollary 2.2. If A and B are Z_{p}[G]-modules then Ĥ^{∗}(g, A ⊗ B) = Ĥ^{∗}(g, A) ⊗ Ĥ^{∗}(g, B).

Proof. Check each of the 9 cases when A and B are indecomposable modules using lemma 2.1. This proves corollary 2.2.

We form the ring K which is a free Q-module with a basis of 3 elements [Z_{p}] = 1, [Z_{p}[G]], and [I] corresponding to the indecomposable modules of Z_{p}[G], whose addition and multiplication are those induced by tensor products and direct sums of modules. If A is a Z_{p}[G]-module then we write [A] for the element of K represented by A. Notice that this is not the same as the (tensor product with Q of the) Grothendieck ring of finitely generated projective Z_{p}[G] modules because we do not assume the relations given by nonsplit short exact sequences. The Grothendieck ring (tensored with Q) is a quotient of our ring K by the ideal generated by [Z_{p}] + [I] − [Z_{p}[G]].

Lemma 2.3. There are exactly 3 ring homomorphisms dim, Tr(g|.), and f from K to Q (which correspond to the 3 elements of the spectrum of K). They are given by

1. dim([Z_{p}]) = 1, dim([Z_{p}[G]]) = p, dim([I]) = p − 1.

2. Tr(g|[Z_{p}]) = 1, Tr(g|[Z_{p}[G]]) = 0, Tr(g|[I]) = −1.

3. f([Z_{p}]) = 1, f([Z_{p}[G]]) = 0, f([I]) = 1.

Proof. From corollary 2.2 we know that the products in the ring K are given by [Z_{p}[G]][Z_{p}[G]] = p[Z_{p}[G]], [Z_{p}[G]][I] = (p − 1)[Z_{p}[G]], and [I][I] = (p − 2)[Z_{p}[G]] + 1. Hence if x ∈ Q and y ∈ Q are the images of [Z_{p}[G]] and I under some homomorphism they must satisfy x^{2} = px, xy = (p − 1)x, and y^{2} = (p − 2)x + 1. The only solutions to these equations are (x, y) = (p, p − 1), (0, 1), or (0, −1). This proves lemma 2.3.

The first two homomorphisms in lemma 2.3 are the homomorphisms which factor through the Grothendieck ring and will not be very interesting to us. We can use them in place of f in the arguments in the rest of the paper, but this only leads to relations between the coefficients of ∑_{n} dim(V_{n})q^{n} and ∑_{n} Tr(g|V_{n})q^{n}, which are essentially the relations already used in [B] to determine these functions. It is the third homomorphism f which detects the integral structure on V [1/2] which will give us enough extra information to prove theorem 4.1.

In general, taking the trace of any fixed element of a group gives a homomorphism from a representation ring of a group to some field, and quite often (e.g., for the complex representation ring of a group) this gives a 1:1 correspondence between homomorphisms and (certain sorts of) conjugacy classes of the group. Lemma 2.3 gives an example of a homomorphism f not constructed in this way.

A permutation module is a module for G with a basis acted on by G. We recall from [B-R] that a module A is a permutation module if and only if Ĥ^{1}(g, A) = 0, which in turn is equivalent to saying that A is a sum of copies of Z_{p} and Z_{p}[G].

Lemma 2.4. If A is a permutation module over any ring for any finite group g then so are the symmetric powers S^{n}(A). If all elements of G act on a basis of A as products of cycles of odd length then the exterior powers Λ^{n}(A) are also permutation modules.

Proof. For symmetric powers this is proved in [B-R]. For the proof for exterior powers, let a_{1}, …, a_{n} be a basis for A acted on by G. Then G acts on the set of all elements of the form ±a_{i_{1}} ∧ ⋯ ∧ a_{i_{n}}, which consists of all elements of a basis together with their negatives. If the action of G on the basis of A has the property that all elements of G are products of cycles of odd length, then there is no element of G that maps any element of the form ±a_{i_{1}} ∧ ⋯ ∧ a_{i_{n}} to its negative. This means that we can find a G-invariant subset of these elements which forms a basis for Λ^{n}(A). This proves lemma 2.4.

Lemma 2.5. The exterior powers of indecomposable modules are given by

Λ^{0}(Z_{p}) = Z_{p}
Λ^{1}(Z_{p}) = Z_{p}
Λ^{n}(Z_{p}) = 0 if n ≠ 0, 1
Λ^{0}(Z_{p}[G]) = Z_{p}
Λ^{p}(Z_{p}[G]) = Z_{p} if p is odd, and I if p = 2
Λ^{n}(Z_{p}[G]) = Z_{p}[G]^{\binom{p−1}{n}} if 1 ≤ n ≤ p − 1
Λ^{n}(Z_{p}[G]) = 0 if n > p
Λ^{n}(I) = Z_{p} ⊕ Z_{p}[G]^{(\binom{p−1}{n}−1)/p} if n is even and 0 ≤ n < p
Λ^{n}(I) = I ⊕ Z_{p}[G]^{(\binom{p−1}{n}+1−p)/p} if n is odd and 0 ≤ n < p
Λ^{n}(I) = 0 if n ≥ p

Proof. For the exterior powers of Z_{p}[G] with p odd we apply lemma 2.4 to see that all exterior powers are sums of Z_{p} and Z_{p}[G], and the exterior powers can then be identified by checking the dimension and the trace of g. The cohomology groups of the exterior powers of I can then be worked out by induction using the exact sequence

0 → Λ^{n}(I) → Λ^{n}(Z_{p}[G]) → Λ^{n−1}(I) → 0.

The cohomology groups determine how many copies of Z_{p} and I occur in the decomposition into indecomposable modules, so we can work out the exterior powers of I from this. The remaining cases of lemma 2.5 are trivial, so this proves lemma 2.5.

These exterior powers induce operators on the ring K which we also denote by Λ^{n}. For example, we can use these operators to define Adams operations ψ^{n} on the ring K by

∑_{n∈Z}(−1)^{n}Λ^{n}([A])q^{n} = exp(− ∑_{n∈Z} ψ^{n}([A])q^{n}/n).

Corollary 2.6. If p is odd then

∑_{n∈Z}(−1)^{n}f([Λ^{n}(Z_{p})])q^{n} = 1 − q
∑_{n∈Z}(−1)^{n}f([Λ^{n}(Z_{p}[G])])q^{n} = 1 − q^{p}
∑_{n∈Z}(−1)^{n}f([Λ^{n}(I)])q^{n} = (1 + q^{p})/(1 + q)

Proof. This follows immediately from lemma 2.5 and lemma 2.3.

Lemma 2.7. The symmetric powers of indecomposable modules are given by

S^{n}(Z_{p}) = Z_{p} if n ≥ 0
S^{n}(Z_{p}[G]) = Z_{p}[G]^{\binom{p+n−1}{n}/p} if (p, n) = 1
S^{n}(Z_{p}[G]) = Z_{p}[G]^{(\binom{p+n−1}{n}−1)/p} ⊕ Z_{p} if p|n
S^{n}(I) = Z_{p} ⊕ Z_{p}[G]^{(\binom{p+n−2}{n}−1)/p} if p|n
S^{n}(I) = I ⊕ Z_{p}[G]^{(\binom{p+n−2}{n}+1−p)/p} if n ≡ 1 mod p
S^{n}(I) = Z_{p}[G]^{(\binom{p+n−2}{n})/p} if n ≢ 0, 1 mod p

Proof. For symmetric powers of Z_{p} this is trivial and for symmetric powers of Z_{p}[G] it can be proved in the same way as in lemma 2.5. If p = 2 then the case of I is easy to do as I is just one dimensional. For the case of I when p is odd we let N_{G} be the element 1 + g + ⋯ + g^{p−1} ∈ Z_{p}[G] and consider the quotient of the symmetric algebra S^{∗}(Z_{p}[G]) by the ideal (N_{G}). This is the universal commutative algebra generated by Z_{p}[G]/N_{G}Z_{p}, so it is S^{∗}(Z_{p}[G]/N_{G}Z_{p}) which is isomorphic to S^{∗}(I) because Z_{p}[G]/N_{G}Z_{p} is isomorphic to I. Therefore there is an exact sequence

0 → S^{n−1}(Z_{p}[G]) → S^{n}(Z_{p}[G]) → S^{n}(I) → 0.

Using the known cohomology groups of the symmetric powers of Z_{p}[G] and the exact sequence of cohomology groups of this exact sequence gives the cohomology groups of S^{n}(I) and hence the number of times that Z_{p} and I occur in the decomposition of S^{n}(I). This proves lemma 2.7.

Corollary 2.8. The cohomology rings of the symmetric algebras of indecomposable modules are give as follows.

Ĥ^{∗}(S^{∗}(Z_{p})) is a polynomial algebra with one ordinary generator of degree 1. Ĥ^{∗}(S^{∗}(Z_{p}[G])) is a polynomial algebra with one ordinary generator of degree p. Ĥ^{∗}(S^{∗}(I)) is the tensor product of a polynomial algebra with one ordinary generator of degree p and an exterior algebra with one super generator of degree 1 if p is odd. (If p is even then Ĥ^{∗}(S^{∗}(I)) is a polynomial algebra on one super generator of degree 1, but we do not need this case.)

Proof. This follows from the calculation of the symmetric powers in lemma 2.7. The product structure on the symmetric algebras can be worked out using the formula d(ab) = (da)b ± a(db) relating the cup product with the map d from Ĥ^{i}(C) to Ĥ^{i+1}(A) associated to the exact sequence 0 → A → B → C → 0. This proves corollary 2.8.

Lemma 2.9. Suppose that A_{−1} = 0, A_{0}, …, A_{n}, A_{n+1} = 0 is a sequence of modules and we have maps d from A_{i} to A_{i+1} and d^{∗} from A_{i+1} to A_{i} for all i such that d^{2} = 0, d^{∗2} = 0, and dd^{∗} + d^{∗}d = k for some fixed integer k coprime to p. Then

[A_{0}] − [A_{1}] + [A_{2}] − ⋯ + (−1)^{n}[A_{n}] = 0.

Proof. The fact that k is a unit in Z_{p} implies that A_{i} is the direct sum dd^{∗}(A_{i}) ⊕ d^{∗}d(A_{i}) and that d is an isomorphism from d^{∗}d(A_{i}) to dd^{∗}(A_{i+1}). The lemma follows from this because the sequence splits as the sum of these isomorphisms. (Notice that the relation between the [A_{i}]’s does not follow just from the exactness of d; for example, look at the short exact sequence 0 → I → Z_{p}[G] → Z_{p} → 0. It is easy to check that in this case there is a map d^{∗} with d^{∗2} = 0 and dd^{∗} + d^{∗}d = p, so the condition that (k, p) = 1 is also necessary.) This proves lemma 2.9.

Lemma 2.10. Suppose that A is a finitely generated free module over Z or Z_{p} with a symmetric self dual form (,). Then this induces a symmetric self dual form (defined in the proof) on any exterior power Λ^{n}(A).

Proof. We define the bilinear form on Λ^{n}(A) by

(a_{1} ∧ ⋯ ∧ a_{n}, b_{1} ∧ ⋯ ∧ b_{n}) = ∑_{σ∈S_{n}} χ(σ)(a_{1}, b_{σ(1)}) ⋯ (a_{n}, b_{σ(n)})

where χ(σ) is 1 or −1 depending on whether σ is an even or odd permutation. This obviously defines a symmetric bilinear form on Λ^{n}(A) with values in Z or Z_{p}, and we have to check it is unimodular. The determinant of Λ^{n}(A) is equal to a det(A)^{b} for some constants a and b depending on n, but not on A. We can work out a by letting A be the lattice with a base of orthogonal elements a_{1}, a_{2}, … with norms ±1, when we see that Λ^{n}(A) also has an orthogonal base of elements a_{i_{1}} ∧ a_{i_{2}} ⋯ with i_{1} < i_{2} < ⋯ with norms ±1, so both A and Λ^{n}(A) have discriminant 1 and so a = 1. Hence if A is self dual then Λ^{n}(A) has discriminant 1 and must also be self dual. This proves lemma 2.10. (It is easy to check that the constant b is equal to dim(Λ^{n}(A))/ dim(A) but we do not need this.)

The analogue of this lemma for symmetric powers is false. The proof breaks down because the element a_{i_{1}}a_{i_{2}} ⋯ with i_{1} ≤ i_{2} ≤ ⋯ does not have norm ±1 if some of the i_{j}’s are equal. In section 5 we will construct a graded algebra h(A), such that the homogeneous components of h(A) have self dual symmetric bilinear forms if A does, and h(A) ⊗ Q_{p} is a sum of tensor products of symmetric powers of A.

The following corollary is purely for entertainment and is not used elsewhere in the paper.

Corollary 2.11. There is a 28-dimensional self dual positive definite lattice with no roots, acted on by the orthogonal group O_{8}^{+}(F_{2}).

Proof. By lemma 2.10 the lattice Λ^{2}(E_{8}) is self dual, and is obviously acted on by the automorphism group of the E_{8} lattice modulo −1, which is just O_{8}^{+}(F_{2}). It is easy to check that this lattice has no roots either by calculation or by observing that there is no suitable root lattice acted on by O_{8}^{+}(F_{2}). This proves lemma 2.11.

The lattice Λ^{2}(E_{8}) has theta function

∑_{λ∈Λ^{2}(E_{8})} q^{λ^{2}/2} = 1 + 2240q^{3/2} + 98280q^{4/2} + 1790208q^{5/2} + 19138560q^{6/2} + · · · .

The vectors of norm 3 are just the 2240 vectors generating one of the 1-dimensional sublattices of the form Λ^{2}(L) where L is one of the 1120 sublattices of E_{8} isomorphic to A_{2}.

3. The Z[1/2]-form of the monster Lie algebra.

We find some relations between the dimensions of the Tate cohomology groups Ĥ^{i}(g, V [1/2]) by studying the Lie algebra cohomology of the positive subalgebra of a Z_{p}-form of the monster Lie algebra. In the next section we will use these relations to compute the cohomology groups in the case when g has order at least 13. We recall that V [1/2] is the self dual Z[1/2]-form of Frenkel, Lepowsky, and Meurman’s monster vertex algebra [FLM] described in [B-R] (but the arguments in this section work for any self dual Z_{p}-form of the monster vertex algebra).

This section should really be about the integral form of the monster Lie algebra, but this has not been constructed yet because of the lack of a construction of a self dual integral form of the monster vertex algebra. However, as we are only looking at the action of elements of odd order p in the monster and tensoring everything with Z_{p}, a Z[1/2]-form is just as good as an integral form.

We briefly recall some properties of the monster Lie algebra m from [B]. This is a II_{1,1}-graded Lie algebra acted on by the monster, where II_{1,1} is the 2-dimensional even self dual Lorentzian lattice whose element (m, n) has norm −2mn. The piece of degree (m, n) of m is isomorphic to V_{mn} if (m, n) ≠ (0, 0) and to R^{2} if (m, n) = (0, 0). The Lie algebra m is the sum of 3 subalgebras E, F, and H, where the Cartan subalgebra H is the piece of degree (0, 0) of m, E is the sum of all pieces of degree (m, n) of m for m > 0, and F is the sum of all pieces of degree (m, n) of m for m < 0.

The Z[1/2] form on V induces a self dual Z[1/2]-form and hence a Z_{p}-form on m. From now on we will write m[1/2] for this Z[1/2]-form. In the preprint version of this paper I implicitly assumed that the degree (m, n) piece of m was self dual and isomorphic as a module over the monster to V_{mn}. At the last moment I realized that this is not at all clear. The problem is that although the no-ghost theorem gives an isomorphism of both spaces tensored with Q, and both spaces have Z[1/2]-forms, there is no obvious reason why this isomorphism should map one Z[1/2] form to the other. (In fact for the fake monster Lie algebra the corresponding isomorphism does not preserve the integral forms.) Fortunately in this paper we only need the following weaker statement:

Assumption. If m < p then the degree (m, n) piece of m ⊗ Z_{p} is self dual under the natural bilinear form and isomorphic to V_{mn} ⊗ Z_{p} as a Z_{p} module acted on by the monster.

I believe I have a proof of this; if all goes well it will appear in a paper provisionally titled “The fake monster formal group”. Until this paper appears all the statements in sections 3 and 4 and the modular moonshine conjectures for elements of type pA (p ≥ 13) should have this assumption added as a hypothesis.

I do not know if the condition m < p in the assumption above is necessary; if it is not, then this condition (usually added in parentheses) can be missed out of some of the following lemmas. It seems very likely that self duality still holds without it, but I am not so sure about the action of the monster being the same if both m and n are divisible by p.

The monster Lie algebra has a Weyl vector ρ, i.e., a vector in the root lattice II_{1,1} such that (ρ, α) = −(α, α)/2 for every simple root α. This follows from [B, theorem 7.2] which states that this simple roots are the vectors (1, n) for n ≥ 0 or n = −1, with multiplicities c(n). Hence the Weyl vector is ρ = (1, 0).

We recall some facts about Hodge theory applied to Lie algebra cohomology. The self dual symmetric bilinear form on the monster Lie algebra identifies E with the dual space of F, and using the Cartan involution which maps F to E we see that we have a symmetric bilinear form on the Lie algebra E (which is the restriction of the contravariant form of the monster Lie algebra to E) and in particular we can identify E with its dual. The Lie algebra cohomology is the cohomology of an operator d which maps each space Λ^{i}(E) to Λ^{i+1}(E). The adjoint of this operator is denoted by d^{∗}, and we define the Laplace operator Ω by Ω = dd^{∗} + d^{∗}d. As for any generalized Kac-Moody algebra with a Weyl vector, the action of the Laplace operator Ω = dd^{∗} + d^{∗}d on the degree α = (m, n) ∈ II_{1,1} piece of Λ^{∗}(E) is given by multiplication by (α, α + 2ρ)/2 = (m − 1)n. The Lie algebra cohomology group H^{i}(E) of E can be identified with the zero eigenspace of Ω on Λ^{i}(E). The Laplace operator restricted to Λ^{1}(E) = E is half the partial Casimir operator Ω_{0} defined in [K, section 2.5]; the extra factor of 1/2 is necessary to make things work well over rings not containing 1/2, but this is not important in this paper as we work over the ring of p-adic integers for odd p which always contains 1/2.

Proposition 3.1. The operators d and d^{∗} act on the Z[1/2]-form Λ^{∗}(E[1/2]) ⊂ Λ^{∗}(E) of the exterior algebra (in all degrees (m, n) with m < p).

Proof. The main point is that all homogeneous pieces of the Z[1/2]-form of the exterior algebra are self dual under the contravariant bilinear form. This follows from lemma 2.10 and the fact that each homogeneous component is a finite sum of tensor products of exterior powers of the homogeneous components of E[1/2]. The result now follows from the fact that if d is a homomorphism of finite dimensional free Z[1/2]-modules from A to B and A and B have self dual symmetric inner products then the adjoint d^{∗} from B to A is well defined. This proves proposition 3.1.

We let R(0) be the ring of formal Laurent series K[[r, q]][1/r, 1/q] with coefficients in K, and we let R(1) be the space of power series in R(0) only involving monomials r^{m}q^{n} with p|mn, and we let R(2) be the subring of R(0) of power series only involving powers of r^{p} and q^{p}. So we have the inclusions R(2) ⊂ R(1) ⊂ R(0) and R(1) is a module over R(2) (but is not a ring itself). (We can also define R(0), R(1), and R(2) for arbitrary generalized Kac-Moody algebras, in particular for the fake monster Lie algebra: R(0) is a completion of the group ring of the root lattice, R(1) is a subspace corresponding to the elements α of the root lattice with p|(α, α)/2, and R(2) is the subring corresponding to p times the root lattice.) If A is a Z[1/2]-module then we write [A] for [A ⊗_{Z[1/2]} Z_{p}].

Lemma 3.2. If (p, (m − 1)n) = 1 (and m < p) then the piece of degree (m, n) of

[Λ^{∗}(E[1/2])] = [Λ^{0}(E[1/2])] − [Λ^{1}(E[1/2])] + [Λ^{2}(E[1/2])] − ⋯

vanishes. In other words

r^{−1}Λ^{∗}( ∑_{m>0,n∈Z} r^{m}q^{n}[V_{mn}])

lies in R(1).

Proof. The Laplace operator Ω = dd^{∗} +d^{∗}d acts as multiplication by (m−1)n on the degree (m, n) piece of Λ^{∗}(E[1/2]), so the result follows from lemma 2.9 if we let A_{i} be the degree (m, n) piece of Λ^{i}(E[1/2]). This proves lemma 3.2.

Lemma 3.3. If A is a Z_{p}[G]-module with Tr(g|A) = c^{+}, f([A]) = c^{−}, (and m < p) then

f(Λ^{∗}(r^{m}q^{n}A)) = (1 − r^{m}q^{n})^{(c^{+}+c^{−})/2}(1 + r^{m}q^{n})^{(c^{+}−c^{−})/2}U

where U is a unit of the ring R(2).

Proof. It is sufficient to prove this for an indecomposable module A because both sides are multiplicative. We just check each of the three possibilities for A using lemma 2.6.

If A = Z_{p} then c^{+} = c^{−} = 1 and f(Λ^{∗}(r^{m}q^{n}[Z_{p}])) = 1 − r^{m}q^{n}, so lemma 3.3 is true in this case.

If A = Z_{p}[G] then c^{+} = c^{−} = 0 and f(Λ^{∗}(r^{m}q^{n}[Z_{p}[G]])) = 1 − r^{mp}q^{np} which is a unit in R(2), so lemma 3.3 is true in this case.

If A = I then c^{+} = −1, c^{−} = 1 and

f(Λ^{∗}(r^{m}q^{n}[I])) = (1 + r^{mp}q^{np})/(1 + r^{m}q^{n}) = U(1 + r^{m}q^{n})^{−1},

so lemma 3.3 is also true in this case. This proves lemma 3.3.

We define integers c_{g}^{+}(n) and c_{g}^{−}(n) by

c_{g}^{+}(n) = Tr(g|V_{n}) = dim(Ĥ^{0}(g, V [1/2]_{n})) − dim(Ĥ^{1}(g, V [1/2]_{n}))
c_{g}^{−}(n) = f(V [1/2]_{n}) = dim(Ĥ^{0}(g, V [1/2]_{n})) + dim(Ĥ^{1}(g, V [1/2]_{n})).

We use the numbers c_{g}^{+}(n) and c_{g}^{−}(n) rather than the apparently simpler numbers dim(Ĥ^{0}(g, V [1/2]_{n})) and dim(Ĥ^{1}(g, V [1/2]_{n})) because they turn out to be coefficients of Hauptmoduls, and because c_{g}^{+}(n) is already known.

Proposition 3.4. If (p, mn) = 1 (and m < p) then the coefficient of r^{m}q^{n} in

r^{−1} ∏_{m>0,n∈Z}(1 − r^{m}q^{n})^{(c_{g}^{−}(mn)+c_{g}^{+}(mn))/2}(1 + r^{m}q^{n})^{(c_{g}^{+}(mn)−c_{g}^{−}(mn))/2}

vanishes. In other words this power series lies in R(1).

Proof. We apply the homomorphism f to the expression in lemma 3.2 and use lemma 3.3. We find that the expression in this proposition is equal to an element of R(1) times a unit in R(2), and is therefore still an element of R(1). Therefore its coefficients of r^{m}q^{n} vanish unless p|mn. This proves proposition 3.4.

Proposition 3.4 can be generalized in the obvious way to any generalized Kac-Moody algebra which has a self dual Z_{p}-form, a Weyl vector, and an integral root lattice.

We can also ask whether or not the coefficients of r^{m}q^{n} in proposition 3.4 vanish when p|mn. Some numerical calculations suggest that they usually do not.

4. The modular moonshine conjectures for p ≥ 13.

In this section we prove the following theorem, which completes the proof of the modular moonshine conjectures of [R section 6] (apart from a small technicality in the case p = 2.)

Theorem 4.1. If g ∈ M is an element of prime order p ≥ 17 or an element of type 13A then Ĥ^{1}(g, V [1/2]) = 0.

This implies that ^{g}V = Ĥ^{0}(g, V [1/2]) = Ĥ^{∗}(g, V [1/2]) is a vertex algebra whose modular character is given by Hauptmoduls, and whose homogeneous components have the characters of [R definition 2].

The proof of this theorem will occupy the rest of this section. We have to show that the numbers c_{g}^{−}(n) of proposition 3.4 are equal to the numbers c_{g}^{+}(n), because the difference is twice the dimension of Ĥ^{1}(g, V [1/2]_{n}). We start by summarizing what we know about these numbers.

Lemma 4.2.

1. The numbers c_{g}^{−}(n) and c_{g}^{+}(n) satisfy the relations given in proposition 3.4.

2. The numbers c_{g}^{−}(n) are integers, with c_{g}^{−}(n) ≡ c_{g}^{+}(n) mod 2.

3. c_{g}^{−}(n) ≥ |c_{g}^{+}(n)| and (p − 2)c_{g}^{+}(n) + pc_{g}^{−}(n) ≤ 2c_{1}^{+}(n), (where c_{1}^{+}(n) is the coefficient of q^{n} in the elliptic modular function).

4. The numbers c_{g}^{+}(n) are the coefficients of the Hauptmodul of the element g ∈ M.

Proof. These properties follow easily from proposition 3.4 together with the fact that Z_{p} ⊗ V_{n} has dimension c_{1}^{+}(n) and is the sum of (c_{g}^{−}(n) + c_{g}^{+}(n))/2 copies of Z_{p}, (c_{g}^{−}(n) − c_{g}^{+}(n))/2 copies of I, and some copies of Z_{p}[G]. This proves lemma 4.2.

We will prove theorem 4.1 by showing that if g satisfies the conditions of theorem 4.1 then the conditions 1 to 4 in lemma 4.2 imply that c_{g}^{−}(n) = c_{g}^{+}(n). We find a finite set of possible solutions of conditions 1, 2, and 4 of lemma 4.2, and find that for each p only one of these satisfies the condition 3. (There are often several solutions not satisfying condition 3.) The proof is just a long messy calculation and the reader should not waste time looking at it.

We first put the relations into a more convenient form.

Proposition 4.3. There are integers c_{m,n} defined for m, n > 0 (and m < p) with the following properties. 1.

− log (1 + ∑_{m>0,n>0} c_{m,n}r^{m}q^{n}) ≡ ∑_{m>0,n>0} ∑_{d|(m,n)} \frac{c_{g}^{(−)^{d}}(mn/d^{2})}{d}r^{m}q^{n} mod r^{p}

2. If (p, mn) = 1 then c_{m+1,n} = c_{m,n+1}.

Proof. This follows from proposition 3.4 if we multiply both sides by (r − q)/rq and then take the logarithm of both sides. This proves proposition 4.3.

Lemma 4.4. If p > 13 then the values of c_{g}^{−}(n) for n ≤ 21, c_{g}^{−}(2n) for 2n ≤ 32, and c_{g}^{−}(36) and c_{g}^{−}(45) are given by polynomials in the numbers c_{g}^{−}(i) for i = 1, 2, 4, 5 with coefficients in Z_{3}.

Proof. We can evaluate the elements c(n), n ≤ 21 and c(2n), n ≤ 16 using the argument for cases 1 and 2 of lemma 4.7 below. We can evaluate c(36) by looking at the coefficient of r^{4}q^{9} in proposition 4.3 and using the fact that we know c_{m,n} for m ≤ 3, n ≤ 8. Similarly we can evaluate c(45) by looking at the coefficient of r^{5}q^{9}. Notice that the term (∑_{m,n} c_{m,n}r^{m}q^{n})^{3}/3 has 3’s only in the denominators of coefficients of r^{m}q^{n} when 3|m and 3|n, so we do not get problems from this for the coefficients of r^{4}q^{9} and r^{5}q^{9} (but we do get problems from this if we try to work out c_{g}^{−}(27) using the same method). This proves lemma 4.4.

If p = 13 we run into trouble when we try to determine c_{g}^{−}(26); this is why the argument in lemma 4.4 does not work in this case. Also the argument breaks down if we try to work out c(27), because when we look at the coefficient of r^{3}q^{9} we get an extra term c_{g}^{−}(3)/3 which does not have coefficients in Z_{3}. This is why we use the coefficients c_{g}^{−}(1), c_{g}^{−}(2), c_{g}^{−}(4), and c_{g}^{−}(5) rather than c_{g}^{−}(1), c_{g}^{−}(2), c_{g}^{−}(3), and c_{g}^{−}(5).

Lemma 4.4 is the reason why our arguments do not work for p < 13 (and do not work so smoothly for p = 13). As p gets smaller we have fewer relations to work with and it gets more difficult to determine all coefficients in terms of the c_{g}^{−}(m)’s for small m. It is probably possible to extend lemma 4.4 to cover some smaller primes with a lot of effort. Fortunately it is not necessary to do the cases p < 13 because these cases have already been done by explicit calculations in [B-R].

Lemma 4.5. If c_{g}^{−}(1), c_{g}^{−}(2), c_{g}^{−}(4), and c_{g}^{−}(5) are known mod 3^{n} for some n ≥ 1, and p > 13, and (c_{g}^{−}(1), 3) = 1 if p = 13, then these values of c_{g}^{−}(m) are determined mod 3^{n+1}.

Proof. By lemma 4.4 we see that we can determine the values mod 3^{n} of c_{g}^{−}(n) for n ≤ 16, c_{g}^{−}(2n) for 2n ≤ 32, and c_{g}^{−}(36) and c_{g}^{−}(45).

But now if we look at the coefficient of r^{3}q^{3m} of 3.4 with m = 1, 2, 4, or 5, we see that c_{g}^{−}(9m) + c_{g}^{+}(2m)/2 + c_{g}^{−}(m)/3 = c_{g}^{−}(m)^{3}/3+ (some polynomial in known c_{g}^{−}(i)’s with coefficients that are 3-adic integers). But c_{g}^{−}(m)^{3}/3 mod 3^{n} depends only on c_{g}^{−}(m) mod 3^{n} if n ≥ 1, so we can determine c_{g}^{−}(m)/3 mod 3^{n} and hence c_{g}^{−}(m) mod 3^{n+1}. This proves lemma 4.5.

This lemma is the reason that we use 3-adic rather than 2-adic approximation. If we try to prove the lemma above for 2^{n+1} instead of 3^{n+1}, all we find is that we can determine the c_{g}^{+}(m)’s mod 2^{n+1}, which is useless because we already know these numbers.

Proposition 4.6. If the numbers c_{g}^{−}(n) satisfy the conditions 1, 2, and 4 of lemma 4.2 then the numbers c_{g}^{−}(1), c_{g}^{−}(2), c_{g}^{−}(4), and c_{g}^{−}(5) are congruent mod 3^{29} to the coefficients of q, q^{2}, q^{4}, and q^{5} of one of the following power series. (The second column is a genus zero group whose Hauptmodul appears to be the function with coefficients c_{g}^{−}(n). This has not been checked rigorously as it is not necessary for the proof of theorem 4.1.)

p = 71 Γ_{0}(71)+  q^{−1} +q  +q^{2}  +q^{3}  +q^{4}  +2q^{5} + ⋯
p = 59 Γ_{0}(59)+  q^{−1} +q  +q^{2} +2q^{3} +2q^{4}  +3q^{5} + ⋯
p = 47 Γ_{0}(47)+  q^{−1} +q +2q^{2} +3q^{3} +3q^{4}  +5q^{5} + ⋯
p = 47 Γ_{0}(94)+  q^{−1} +q  +q^{3}  +q^{4}  +q^{5} + ⋯
p = 41 Γ_{0}(41)+  q^{−1} +2q +2q^{2} +3q^{3} +4q^{4}  +7q^{5} + ⋯
p = 41 Γ_{0}(82|2)+  q^{−1}  +q^{3}  +q^{5} + ⋯
p = 31 Γ_{0}(31)+  q^{−1} +3q +3q^{2} +6q^{3} +9q^{4} +13q^{5} + ⋯
p = 31 Γ_{0}(62)+ q^{−1} +q +q^{2} +2q^{3} +q^{4} +3q^{5} + ⋯
p = 29 Γ_{0}(29)+  q^{−1} +3q +4q^{2} +7q^{3} +10q^{4} +17q^{5} + ⋯
p = 29 Γ_{0}(58|2)+ q^{−1} +q +q^{3} +q^{5} + ⋯
p = 23 Γ_{0}(23)+  q^{−1} +4q +7q^{2} +13q^{3} +19q^{4} +33q^{5} + ⋯
p = 23 Γ_{0}(46) + 23  q^{−1}  −q^{2}  +q^{3}  −q^{4}  +q^{5} + ⋯
p = 23 Γ_{0}(46)+  q^{−1} +2q +q^{2} +3q^{3} +3q^{4}  +5q^{5} + ⋯
p = 19 Γ_{0}(19)+  q^{−1} +6q +10q^{2} +21q^{3} +36q^{4} +61q^{5} + ⋯
p = 19 Γ_{0}(38)+  q^{−1} +2q +2q^{2} +5q^{3} +4q^{4}  +9q^{5} + ⋯
p = 19 Γ_{0}(38|2)+  q^{−1} +2q  +q^{3}  +3q^{5} + ⋯
p = 17 Γ_{0}(17)+  q^{−1} +7q +14q^{2} +29q^{3} +50q^{4} +92q^{5} + ⋯
p = 17 Γ_{0}(34)+  q^{−1} +3q +2q^{2} +5q^{3} +6q^{4} +12q^{5} + ⋯
p = 17 Γ_{0}(34|2)+ q^{−1} +q +3q^{3} +4q^{5} + ⋯
p = 13 Γ_{0}(13)+  q^{−1} +12q +28q^{2} +66q^{3} +132q^{4} +258q^{5} + ⋯
p = 13 Γ_{0}(26)+  q^{−1} +4q +4q^{2} +10q^{3} +12q^{4} +26q^{5} + ⋯
p = 13 Γ_{0}(26|2)+  q^{−1} +2q  +4q^{3}  +6q^{5} + ⋯

It is also possible (but unlikely) that there are other solutions for p = 13 for which c_{g}^{−}(1) does not satisfy the inequalities 0 ≤ c_{g}^{−}(1) < 3^{10}.

Proof. For each prime p we use a computer to test all 81 possibilities for c_{g}^{−}(1), c_{g}^{−}(2), c_{g}^{−}(4), and c_{g}^{−}(5) mod 3. If p > 13 then we can calculate the p-adic expansion of all the coefficients c_{g}^{−}(n) recursively using lemma 4.5 above, and we reach a contradiction by looking at coefficients of proposition 4.3 except in the cases above. For p = 13 this does not quite work as we have not shown the values mod 3^{n} determine those mod 3^{n+1}, so we can adopt the crude procedure of just testing all 3^{4} possibilities for the c_{g}^{−}(i)’s mod 3^{n+1} for each solution mod 3^{n} we have found, and checking to see which of them leads to contradictions. There were some cases for p = 13 where this did not lead to a contradiction but did at least lead to the conclusion that c_{g}^{−}(1) < 0 or c_{g}^{−}(1) ≥ 3^{10}. (If c_{g}^{−}(1) is not divisible by 3 then even when p = 13 the numbers c_{g}^{−}(i) mod 3^{n} determine the numbers c_{g}^{−}(n) mod 3^{n+1}. When c_{g}^{−}(1) was divisible by 3, there were several cases where the coefficient c_{g}^{−}(5) was not determined mod 3^{n+1} by the identities used by the computer program. However in all the cases looked at by the computer with 3|c_{g}^{−}(1), the values of c_{g}^{−}(1), c_{g}^{−}(2), c_{g}^{−}(4), and 3c_{g}^{−}(5) mod 3^{n} uniquely determined their values mod 3^{n+1}.) This proves proposition 4.6, at least if one believes the computer calculations. (Anyone who does not like computer calculations in a proof is welcome to redo the calculations by hand.)

Lemma 4.7. If the coefficients c_{g}^{−}(1), c_{g}^{−}(2), c_{g}^{−}(4), and c_{g}^{−}(5) are equal to the coefficients c_{g}^{+}(1), c_{g}^{+}(2), c_{g}^{+}(4), and c_{g}^{+}(5) for g an element of prime order p > 13 or an element of type 13A then all the coefficients are determined by the relations of proposition 3.4.

Proof. This proof is just a long case by case check using induction. We will repeatedly use the fact that the coefficients of r^{m}q^{n} of both sides of proposition 4.3 are equal.

Case 1. c_{g}^{−}(2n) for 2n ≢ 0 mod p. By looking at the coefficient of r^{2}q^{n} and using the fact that c_{2,n} = c_{1,n+1} (as (p, n) = 1) we see that c_{g}^{−}(2n) can be written as a polynomial in c(n + 1) and c(i) for 1 ≤ i < n.

Case 2. c_{g}^{−}(2n+1) for 2n+1 ≢ ±1 mod p. By looking at the coefficient of r^{2}q^{2n} we can express c_{g}^{−}(2n+1) in terms of c_{g}^{−}(4n) and known quantities. But c_{g}^{−}(4n) can be evaluated by looking at the coefficient of r^{4}q^{n} and using the fact that c_{4,n} = c_{3,n+1} = c_{2,n+2} (as 2n + 1 ≢ ±1 mod p).

The recursion formulas given by cases 1 or 2 can be written out explicitly and are given in [B, (9.1)]. In that paper they are sufficient to determine all coefficients because the analogues of the relations in proposition 3.4 hold whenever mn ≠ 0.

The remaining cases of the proof are unusually tedious so we only sketch them briefly.

Case 3. c_{g}^{−}(2n) for 2n ≡ 0 mod p, n odd. We can express this in terms of c_{2,n} = c_{3,n−1}, which we can evaluate by calculating c_{g}^{−}(3(n − 1)) = c_{g}^{−}(2(3(n − 1)/2)) using case 1.

For the rest of the proof it is convenient to evaluate the cluster of elements c_{g}^{−}(2pn − 1), c_{g}^{−}(2pn + 1), and c_{g}^{−}(4pn), simultaneously. (These are the only cases left to do.)

Case 4. 2np ≡ 2 mod 3. We evaluate c_{g}^{−}(2np + 1) in terms of c_{3,(2np+1)/3}. We evaluate c_{g}^{−}(2np − 1) by expression c_{g}^{−}(2np − 1)c(2) in terms of c_{2,2np+1} and evaluating the latter by comparing it with c_{3,2(2np+1)/3}. We evaluate c_{g}^{−}(4np) by expressing c_{3,2np+2} in terms of c_{g}^{−}(2)c_{2,2np} and evaluating the latter by comparing it with c_{2,3np+3}.

Case 5. 2np ≡ 1 mod 3. We evaluate c_{g}^{−}(2np − 1) in terms of c_{3,(2np−1)/3}. We evaluate c_{1,2np+5} by using c_{3,(2np+5)/3} and then evaluate in turn c_{2,2np+4} = c_{1,2np+5}, c_{g}^{−}(1)c_{1,2np+3} (using the coefficient of r^{2}q^{2np+4}), c_{2,2np+3} = c_{1,2np+3}, c_{g}^{−}(1)c_{1,2np+1} which gives us the value of c_{1,2np+1}. We evaluate c_{2,2np} as in case 4.

Case 6. 2np ≡ 0 mod 3. By looking at c_{2,2np+2} and c_{2,2np+3} we can find expressions for the linear combinations c_{g}^{−}(1)c_{g}^{−}(2np + 1) + c_{g}^{−}(3)c_{g}^{−}(2np − 1) and c_{g}^{−}(2)c_{g}^{−}(2np + 1) + c_{g}^{−}(4)c_{g}^{−}(2np − 1). If p ≠ 59 or 71 then these two linear equations are independent so we can solve for c_{g}^{−}(2np + 1) and c_{g}^{−}(2np − 1), and then find c_{2,2np} as in case 4. If p = 59 or 71 then an even more tedious argument using c_{2,2np+2}, c_{3,2np+2}, and c_{3,2np+3} produces 3 independent linear relations between c_{g}^{−}(2pn − 1), c_{g}^{−}(2pn + 1), and c_{g}^{−}(4pn), so we can solve for them.

This proves lemma 4.7.

We can now complete the proof of theorem 4.1. If we look at the possible solutions for the coefficients c_{g}^{−}(n) given by proposition 4.6, we see that all the solutions except those congruent mod 3^{29} to the Hauptmoduls of the elements g ∈ M are ruled out by the inequalities of lemma 4.2. As 3^{29} > c_{1}^{+}(i) > c_{g}^{−}(i) ≥ 0 for i ≤ 5 we see that c_{g}^{+}(i) = c_{g}^{−}(i) for i ≤ 5. The coefficients c_{g}^{+}(n) also satisfy the identities of proposition 4.3 satisfied by the c_{g}^{−}(n)’s; for example, this follows easily from the identity

∑_{n∈Z} c_{g}^{+}(n)r^{n} − ∑_{n∈Z} c_{g}^{+}(n)q^{n} = r^{−1} ∏_{m>0,n∈Z}(1 − r^{m}q^{n})^{c_{g}^{+}(mn)}(1 − r^{pm}q^{pn})^{c_{g}^{+}(pmn)}

[B, page 434]. Therefore c_{g}^{−}(n) = c_{g}^{+}(n) for all n by lemma 4.7 because they both have the same values for n ≤ 5 and both satisfy the same recursion relations given in proposition 3.4. This proves theorem 4.1.

5. Calculation of some cohomology groups.

In this section we calculate the Tate cohomology groups of some spaces that appear in the construction of the monster vertex algebra. In the next section we will use these calculations to find the Tate cohomology Ĥ^{∗}(g, V [1/2]) for elements g of types 3B, 5B, 7B and 13B.

We will write g_{0}, g_{1}, …, g_{p−1} for the elements 1, g, …, g^{p−1} of the group ring Z_{p}[G] to avoid confusing the group ring multiplication with the multiplication in other rings containing it.

Suppose that A is a free Z_{p}-module. We write h(A) for the Z-graded Z_{p}-algebra generated by elements h_{n}(a) of degree n for a ∈ A, n ≥ 0 with the relations

1. h_{n}(λa) = λ^{n}h_{n}(a) if λ ∈ Z_{p}.

2. h_{0}(a) = 1, h_{n}(a) = 0 if n < 0.

3. h_{n}(a + b) = ∑_{i∈Z} h_{i}(a)h_{n−i}(b). If A has a basis a_{1}, …, a_{m} then h(A) is the polynomial algebra generated by elements h_{n}(a_{i}) for n > 0,

1 ≤ i ≤ m. An action of G on A induces an action of G on h(A) in the obvious way.

Lemma 5.1. The cohomology ring Ĥ^{∗}(g, h(Z_{p}[G])) is the polynomial algebra over F_{p} generated by the ordinary elements h_{n}(g_{0}) + h_{n}(g_{1}) + ⋯ + h_{n}(g_{p−1}) for n > 0.

Proof. This follows from the fact that h(Z_{p}[G]) is the tensor product of the rings A_{n}, where A_{n} is the polynomial algebra generated by the elements h_{n}(g_{i}) for 0 ≤ i < p. We see that A_{n} is isomorphic to the symmetric algebra S^{∗}(Z_{p}[G]) whose cohomology is given by lemma 2.8, and we then apply corollary 2.2 to finish the proof. (Notice that the space spanned by the elements h_{n}(g_{i}) for 0 ≤ i < p is not h_{n}(Z_{p}[G]); in fact, the latter is not even a Z_{p}-module.) This proves lemma 5.1.

Proposition 5.2. The cohomology ring Ĥ^{∗}(g, h(I)) is isomorphic to the free exterior algebra with one super generator in each positive degree not divisible by p. The automorphism −1 of I multiplies each generator by −1 and hence acts as −1 on Ĥ^{1}(g, h(I)) and as 1 on Ĥ^{0}(g, h(I)).

Proof. The algebra h(I) is the quotient of h(Z_{p}[G]) by the ideal generated by the elements h_{n}(N_{G}). We let R_{m} be the quotient of h(Z_{p}[G]) by the elements h_{n}(N_{G}) for n ≤ m, and we will prove by induction on m that Ĥ^{∗}(R_{m}) is the tensor product of the free polynomial algebra generated by the ordinary elements h_{n}(g_{0}) + h_{n}(g_{1}) + ⋯ + h_{n}(g_{p−1}) (of degree n) for n > m/p and the free exterior algebra generated by super elements of each degree n with 0 < n ≤ m, (p, n) = 1 (of degree n).

To do this we look at the exact sequence

0 → R_{m−1} → R_{m−1} → R_{m} → 0

where the map from R_{m−1} to R_{m−1} is given by multiplication by h_{m}(N_{G}), and work out the cohomology of R_{m} using the cohomology exact sequence of this short exact sequence. We have to consider the cases p|m and (p, m) = 1 separately.

If (p, m) = 1 then multiplication by h_{m}(N_{G}) induces the zero map on cohomology, so that Ĥ^{∗}(g, R_{m}) is the sum of Ĥ^{∗}(g, R_{m−1}) and Ĥ^{∗}(g, R_{m−1}) with the degree shifted by m and with the ordinary and super parts exchanged. In other words, Ĥ^{∗}(g, R_{m}) is the tensor product of Ĥ^{∗}(g, R_{m−1}) with an exterior algebra generated by one super element of degree m. This proves the induction step when (p, m) = 1.

If p|m then multiplication by h_{m}(N_{G}) induces an injective map on cohomology, so that Ĥ^{∗}(g, R_{m}) is the quotient of Ĥ^{∗}(g, R_{m−1}) by the image of h_{m}(N_{G}), in other words it is just Ĥ^{∗}(g, R_{m−1}) with a polynomial generator of degree m killed off.

Hence we see that to go from Ĥ^{∗}(h(R_{m−1})) to Ĥ^{∗}(h(R_{m})) we either kill off a polynomial generator or add an exterior algebra generator. This proves the statement about Ĥ^{∗}(h(R_{m})) for all m by induction, and hence proves proposition 5.2.

For completeness we remark that the cohomology ring Ĥ^{∗}(g, h(Z_{p})) is a polynomial algebra over F_{p} with one ordinary generator of each positive degree.

For any Z_{p}-module A we define ω to be the automorphism of h(A) taking h_{n}(a) to (−1)^{n}h_{n}(−a). We define h_{ω}(A) to be the largest quotient ring of h(A) on which the automorphism ω acts trivially. (Notice that this is smaller than the largest quotient module on which ω acts trivially.)

Lemma 5.3. The cohomology ring Ĥ^{∗}(g, h_{ω}(Z_{p}[G])) is the polynomial ring on ordinary generators of degrees (2n + 1)p for n ≥ 0.

Proof. The algebra h_{ω}(Z_{p}[G]) is the polynomial algebra generated by the elements h_{2n+1}(g_{i}) for 0 ≤ i < p, n ≥ 0, and lemma 5.3 follows from this.

Lemma 5.4. The cohomology ring Ĥ^{∗}(g, h_{ω}(I)) is the exterior algebra on generators of degrees (2n + 1) for n ≥ 0, (n, p) = 1. In particular it is generated (as a ring) by super elements of odd degrees, so that Ĥ^{0}(g, h_{ω}(I)) vanishes in odd degrees and Ĥ^{1}(g, h_{ω}(I)) vanishes in even degrees.

This follows from lemma 5.3 in the same way that lemma 5.2 follows from 5.1.

Remark. The ring h(Z_{p}) can be identified with the ring of symmetric functions over Z_{p} if we identify h_{n}(1) with the n’th complete symmetric function. In this case the automorphism ω taking h_{n}(1) to (−1)^{n}h_{n}(−1) is just the automorphism ω of [Mac, p. 14] taking the complete symmetric functions to the elementary symmetric functions.

Remark. More generally we can form a quotient ring associated to any automorphism ω of A of order n prime to p such that h_{ω}(A) is the quotient ring associated to the automorphism ω = −1 of A. To do this we let R_{p} be the unramified extension of Z_{p} generated by a primitive n’th root of unity ζ. We then form the largest quotient ring of h(A) ⊗ R_{p} whose elements are fixed by the automorphism taking h_{n}(a) to ζ^{n}h_{n}(ω(a)). For example this space appears when we try to construct the monster vertex algebra from a fixed point free automorphism ω of the Leech lattice Λ.

The extraspecial group 2^{1+24} occurring in the centralizer of an element of type 2B in M has a unique 2^{12}-dimensional irreducible module A over Z[1/2]. Any element of odd order in Aut(Λ) has an induced action on 2^{1+24} and on A.

Lemma 5.5. If g is an element of order 3, 5, 7, or 13 acting on the Leech lattice Λ with no nonzero fixed vectors then as a Z_{p}[G]-module, A ⊗ Z_{p} is the sum of Z_{p} and some copies of Z_{p}[G].

Proof. We let B be the 2^{24}-dimensional 2^{1+24}-module A ⊗ A. This is isomorphic to the group ring of 2^{24}. The element g acts fixed point freely on the group 2^{24} = Λ/2Λ because it acts fixed point freely on Λ and has odd order, so as a module over G, B ⊗ Z_{p} is the sum of Z_{p} and some copies of Z_{p}[G], and the Tate cohomology groups are therefore Ĥ^{0}(g, B ⊗ Z_{p}) = Z/pZ, Ĥ^{1}(g, B ⊗ Z_{p}) = 0. As Ĥ^{∗}(g, B ⊗ Z_{p}) = Ĥ^{∗}(g, A ⊗ Z_{p}) ⊗ Ĥ^{∗}(g, A ⊗ Z_{p}) we see that Ĥ^{∗}(g, A ⊗ Z_{p}) is either an ordinary or a super vector space over F_{p} of dimension 1.

Therefore as a G module, A ⊗ Z_{p} is the sum of some copies of Z_{p}[G] and either Z_{p} or I. But dim(A) = 2^{12} ≡ 1 mod p as (p − 1)|12 which rules out the second possibility. Therefore Ĥ^{0}(g, A ⊗ Z_{p})) = Z/pZ, Ĥ^{1}(g, A ⊗ Z_{p}) = 0. This proves lemma 5.5.

A similar argument can be used to calculate Ĥ^{∗}(g, A) for any element g of odd prime order p in Aut(Λ). If 2n is the dimension of the subspace of Λ fixed by g then n is an integer with 2^{n} ≡ ±1 mod p. Exactly one of the two cohomology groups Ĥ^{∗}(g, A) vanishes and the other has dimension 2^{n} over F_{p}. The group Ĥ^{1}(g, A) vanishes if 2^{n} ≡ 1 mod p, and the group Ĥ^{0}(g, A) vanishes if 2^{n} ≡ −1 mod p. There are some cases where it is Ĥ^{0}(g, A) and not Ĥ^{1}(g, A) that vanishes; for example, if g is the element of order 5 with trace −1 on Λ then Ĥ^{0}(g, A) = 0 and Ĥ^{1}(g, A) = (Z/5Z)^{4}.

6. Elements of type 3B, 5B, 7B, and 13B.

We use the result of the previous section to calculate the cohomology groups Ĥ^{i}(g, V [1/2]) when g ∈ M has type 3B, 5B, 7B, or 13B.

Theorem 6.1. If g ∈ M has type 3B, 5B, 7B, or 13B and σ is the element of order 2 generating the center of C_{M}(g)/O_{p}(C_{M}(g)) then O_{p}(C_{M}(g)) acts trivially on ^{g}V and σ fixes the ordinary part of ^{g}V and acts as −1 on the super part.

Proof. We prove this by calculating the action of σ on Ĥ^{∗}(g, V [1/2]) explicitly. We recall from [B-R] that the Z[1/2]-form V [1/2] of Frenkel, Lepowsky and Meurman’s monster vertex algebra [FLM] is the sum of two pieces V^{0} and V^{1} which are the +1 and −1 eigenspaces of an element σ of type 2B. (Unfortunately the details for the construction of this Z[1/2]-form have not been published anywhere.) We now show that Ĥ^{0}(g, V^{1}) = Ĥ^{1}(g, V^{0}) = 0.

The space V^{0} is the subspace of V_{Λ}[1/2] fixed by the automorphism induced by −1 acting on the Leech lattice Λ, where V_{Λ}[1/2] = V_{Λ} ⊗ Z[1/2], and V_{Λ} is the integral form of the vertex algebra of the Leech lattice, described in [B-R]. As in [B-R], the space V_{Λ} is the tensor product of a twisted group ring Z[Λ̂] and a space isomorphic to h(Λ). The cohomology Ĥ^{∗}(g, Z[Λ̂]) is just Z/pZ as in [B-R] because g acts fixed point freely on Λ. By proposition 5.2, the element −1 of Aut(Λ̂) acts as +1 on Ĥ^{0}(g, h(Λ)) and as −1 on Ĥ^{1}(g, h(Λ)). Therefore Ĥ^{1}(g, V^{0}) vanishes because it is isomorphic to the subspace of Ĥ^{1}(g, h(Λ)) fixed by −1 ∈ Aut(Λ).

The space V^{1} is the subspace of elements of integral weight of the tensor product A ⊗ h_{ω}(Λ ⊗ Z_{p}), where A is the 2^{12} dimensional module for 2^{1+24} and h_{ω}(Λ ⊗ Z_{p}) has its grading halved and then shifted by 3/2. By lemma 5.5, Ĥ^{∗}(g, A) = F_{p}, so Ĥ^{∗}(g, A ⊗ h_{ω}(Λ ⊗ Z_{p})) = Ĥ^{∗}(g, h_{ω}(Λ ⊗ Z_{p})). By lemma 5.4 the cohomology ring Ĥ^{∗}(g, h_{ω}(Λ ⊗ Z_{p})) is generated by super elements of odd degree, so Ĥ^{1}(g, h_{ω}(Λ ⊗ Z_{p})) vanishes in even degrees and Ĥ^{1}(g, h_{ω}(Λ ⊗ Z_{p})) vanishes in odd degrees. Therefore Ĥ^{0}(g, A ⊗ h_{ω}(Λ ⊗ Z_{p})) vanishes in integral degrees, so Ĥ^{0}(g, V^{1}) = 0.

This shows that the element σ acts as 1 on Ĥ^{0}(g, V [1/2]) = Ĥ^{0}(g, V_{Λ}) = Ĥ^{0}(g, h(Λ ⊗ Z_{p})) and as −1 on Ĥ^{1}(g, V [1/2]) = Ĥ^{1}(g, A ⊗ h_{ω}(Λ ⊗ Z_{p})).

Finally we show that O_{p}(C_{M}(g)) acts trivially on ^{g}V. The element σ acts trivially on the center ⟨g⟩ of the extraspecial group p^{1+24/(p−1)} and acts as −1 on the quotient p^{24/(p−1)}. The element g acts trivially on gV so we get an action of p^{24/(p−1)} on ^{g}V. As σ acts as 1 on the even part of ^{g}V and as −1 on the odd part, it commutes with all automorphisms of ^{g}V, so the group p^{24/(p−1)} must act trivially on ^{g}V. This proves theorem 6.1.

The group C_{M}(g) has structure 3^{1+12}.2.Suz, 5^{1+6}.2.HJ, 7^{1+4}.2.A_{7}, or 13^{1+2}.2.A_{4}, so we get modular vertex superalgebras acted on faithfully by the groups 2.Suz, 2.HJ, 2.A_{7}, and 2.A_{4} whose modular characters are given by Hauptmoduls. We can calculate the characters of both the ordinary and super parts of these superalgebras by using the fact that the characters of their sum and difference are both given in terms of known Hauptmoduls.

It seems likely that if g has type 2B then the vertex superalgebra acted on by 2^{24}.Co_{1} constructed in [B-R] is acted on trivially by the 2^{24} subgroup, but this has not yet been proved. The proof given above for the corresponding fact for elements of odd prime order fails for order 2 because an automorphism mapping each element of 2^{24} to its inverse is still the identity automorphism.

As an example, we work out the dimensions of the ordinary and super parts of ^{g}V in the case when g has type 3B. The function ∑_{n} Tr(1|^{g}V) is equal to the Hauptmodul of the element g of type 3B in the monster, which is

q^{−1} + 54q − 76q^{2} − 243q^{3} + 1188q^{4} − 1384q^{5} − ⋯,

and the function ∑_{n} Tr(σ|^{g}V) is equal to the Hauptmodul of the element σg of type 6B in the monster, which is

q^{−1} + 78q + 364q^{2} + 1365q^{3} + 4380q^{4} + 12520q^{5} + ⋯

so we find that the dimension of the ordinary part of ^{g}V_{n} is the coefficient of q^{n} of

q^{−1} + 66q + 144q^{2} + 561q^{3} + 2784q^{4} + 5568q^{5} + ⋯

and the dimension of the super part of ^{g}V_{n} is the coefficient of q^{n} of

12q + 220q^{2} + 804q^{3} + 1596q^{4} + 6952q^{5} · · · .

These series can be written as sums of dimensions of ordinary representations of the groups 3.Suz and 6.Suz as

q^{−1}+66q+(1+143)q^{2}+(429+66+66)q^{3}+(66+66+78+429+2145)q^{4}+(4×1+4×143+2×780+3432)q^{5}+⋯

and

12q + 220q^{2} + (780 + 12 + 12)q^{3} + (780 + 780 + 12 + 12 + 12)q^{4} + (4 × 220 + 2 × 572 + 4928)q^{5} + · · · .

(Some of the decompositions are ambiguous; for example 1 + 143 could be 66 + 78. We have chosen 1 + 143 instead of 66 + 78 for a reason that will appear in a moment.) At first sight this seems to rule out a vertex algebra structure on any lift to characteristic 0, as the homogeneous components V_{n}, V_{n+1}, … have to increase. However we can get around this by rescaling the grading by a factor of 1/3. The data above seem consistent with the existence of a \frac{1}{3}Z-graded vertex superalgebra W = ∑_{3n∈Z} W_{n} such that the mod 3 reduction of W_{n} is V_{3n}. The vertex superalgebra W is probably acted on by 6.Suz, with the element of order 3 in the center acting as ω^{3n} on W_{n}.

There is a second modular vertex super algebra acted on by each of the groups 2.Suz, 2.HJ, 2.A_{7}, and 2.A_{4}, which can be constructed as Ĥ^{∗}(g, V_{Λ}) where g is a fixed point free automorphism of order 3, 5, 7, or 13 of the Leech lattice Λ. The ordinary part of this superalgebra is the same as the ordinary part of the superalgebra constructed from the monster vertex algebra, but its super part is different; for example, its degree 0 piece does not vanish.

7. Open problems and conjectures.

For more open problems in this area see section 7 of [B-R]. At the end of section 6 we saw that the vertex superalgebra of an element of type 3B looks as if it

might be the reduction mod 3 of a superalgebra in characteristic 0. More generally something similar seems to happen for some other elements of the monster, as described in the following conjecture.

Conjecture. For each element g ∈ M of order n there is a \frac{1}{n}Z-graded super module ^{g}V̂ = ∑_{i∈ \frac{1}{n} Z} V_{i} over the ring Z[e^{2πi/n}] with the following properties.

1. ^{g}V̂ is often a vertex superalgebra with a conformal vector and a self dual invariant symmetric bilinear form. (For example, it seems to be a vertex algebra when g is of type 1A, 2B, 3B, 5B, 7B, 13B, or 3C, but possibly not if g has type 2A or 3A.)

2. ^{1}V̂ is a self dual integral form of the monster vertex algebra (probably the one conjecturally described in lemma 7.1 of [B-R]).

3. If g has prime order then the reduction mod p of ^{g}V̂ is the modular vertex superalgebra ^{g}V associated to g (at least when its grading is multiplied by n.)

4. If all coefficients of the Hauptmodul of g are nonnegative then the super part of ^{g}V̂ vanishes. In this case ^{g}V̂ should be the same as the “twisted sector” of g that appears in Norton’s generalized moonshine conjectures [N]. (But notice that if the Hauptmodul of g has negative coefficients then ^{g}V̂ is certainly not the twisted sector of g; for example, ^{g}V̂ is a super vector space and has a nontrivial piece of degree 0.)

5. ^{g}V̂ is acted on by a central extension (Z/nZ).C_{M}(g) of C_{M}(g) by Z/nZ. This central extension contains an element of order 1 or 2 acting as −1 on the super part of ^{g}V̂ and as 1 on the ordinary part.

6. If g and h are commuting elements of M of coprime orders and ĝ is the lift of g to the central extension of C_{M}(h) with the same order as g, then

Tr(ĝ|^{h}V̂) = Tr(gh|^{1}V̂).

(This identity is often false if the orders of g and h are not coprime.)

7. If g and h commute then Ĥ^{∗}(ĝ,^{h} V̂) is (essentially) the reduction mod |g| of ^{gh}V. (Actually this is not quite correct as stated, because the reduction mod |g| will be defined over some nilpotent extension of Z/|g|Z, and we should then take the tensor product over this ring with Z/|g|Z.)

These conjectures are similar to Norton’s generalized moonshine conjectures [N] because both associate some graded space with each element g of the monster. However they treat the problem of negative coefficients in the Hauptmodul T_{g}(τ) in a different way: Norton’s conjectures use the fact that T_{g}(−1/τ) has nonnegative coefficients, which are supposed to be the dimensions of some vector spaces, while the conjectures above allow super vector spaces which may have negative super dimension. Perhaps both Norton’s conjectures and the conjectures above are special cases of some conjecture which associates a Z/nZ×Z-graded super space acted on by a central extension Z/nZ.C_{M}(g) to each element g ∈ M of order n.

We finish by listing a few other questions.

1. Is the group C_{M}(g)/O_{p}(C_{M}(g)) the full automorphism group of the modular vertex algebra ^{g}V? If g has type 3C it seems possible that it is not, and the full automorphism group may be E_{8}(F_{3}). Notice that if g has type 2B then it is not yet known that O_{2}(C_{M}(g)) acts trivially on ^{g}V.

2. Replace the revolting argument of section 4 that involves computer calculations with a more conceptual proof. More generally, suppose we have coefficients c(n) of some function such that the coefficient of r^{m}q^{n} in

r^{−1} ∏_{m>0,n∈Z}(1 − r^{m}q^{n})^{c(mn)}

vanishes whenever (mn, N) = 1 for some fixed integer N. What does this imply about the coefficients c(n)? (There are many solutions to this given by taking the c(n)’s to be the coefficients of some Hauptmodul of level N.)

3. What happens when we apply the argument of section 3 to some automorphism of prime order of the fake monster Lie algebra? Do the infinite products we get represent interesting functions?

4. In [B section 10] it was suggested that there should be Lie superalgebras associated with elements g of the monster, but it was not clear what the dimensions of homogeneous components of these superalgebras should be in the cases when the coefficients of the Hauptmodul of g were neither all positive nor alternating in sign. The results of section 6 suggest what to do in this case: the sum of the dimensions of the ordinary and super parts should also be given by the coefficients of another Hauptmodul.

References.

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[B] R. E. Borcherds, Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109, 405-444

(1992). [B-R] R. E. Borcherds, A. J. E. Ryba, Modular Moonshine II, Duke Math Journal Vol. 83 No. 2, 435-459,

1996.

[CG] C. J. Cummins, T. Gannon, Modular equations and the genus zero property of moonshine functions, Invent. Math. 129, 413-443 1997.

[DM] C. Dong and G. Mason, On the construction of the moonshine module as a Z_{p}-orbifold, Mathematical aspects of conformal and topological field theories and quantum groups (South Hadley, MA, 1992), 37–52, Contemp. Math., 175, Amer. Math. Soc., Providence, RI, 1994.

[FLM] I. B. Frenkel, J. Lepowsky, A. Meurman, Vertex operator algebras and the monster, Academic press 1988.

[Mac] I. G. Macdonald, “Symmetric functions and Hall polynomials”, Oxford University press, 1979.

[Ma] Y. Martin, On modular invariance of completely replicable functions. Moonshine, the Monster, and related topics (South Hadley, MA, 1994), 263–286, Contemp. Math., 193, Amer. Math. Soc., Providence, RI, 1996.

[M] P. Montague, Third and Higher Order NFPA Twisted Constructions of Conformal Field Theories from Lattices, Nuclear Phys. B 441 (1995), no. 1-2, 337–382.

[N] S. P. Norton, Generalized Moonshine, Proc. Symp. Pure Math. 47 (1987) p. 208-209.

[R] A. J. E. Ryba, Modular Moonshine?, In “Moonshine, the Monster, and related topics”, edited by Chongying Dong and Geoffrey Mason. Contemporary Mathematics, 193. American Mathematical Society, Providence, RI, 1996. 307-336.