A Composite Order Generalization of Modular Moonshine

A Composite Order Generalization
of Modular Moonshine

Satoru URANO

S. Urano

Division of Mathematics, University of Tsukuba,
1-1-1 Tennoudai, Tsukuba, Ibaraki 305-8571 Japan<EMAIL_ADDRESS>
Received March 31, 2021, in final form December 21, 2021; Published online
December 24, 2021

We introduce a generalization of Brauer character to allow arbitrary finite
length modules over discrete valuation rings. We show that the generalized
super Brauer character of Tate cohomology is a linear combination of trace
functions. Using this result, we find a counterexample to a conjecture of
Borcherds about vanishing of Tate cohomology for Fricke elements of the
Monster.

moonshine; modular function; Brauer character; vertex operator algebra

11F22; 11F85; 17B69; 20C11; 20C20

##  1 Introduction

Conway and Norton’s monstrous moonshine conjecture [6] stated the existence of
a graded representation $V=\bigoplus_{n\in\mathbb{Z}}V_{n}$ such that for any
element $g$ of the monster group $\mathbb{M}$, the McKay–Thompson series
$T_{g}(\tau)=\sum_{n\in\mathbb{Z}}\operatorname{Tr}(g|V_{n})q^{n-1}$, $q={\rm
e}^{2\pi{\rm i}\tau}$, is a Hauptmodul for some genus $0$ subgroup of ${\rm
SL}_{2}(\mathbb{R})$ of the real numbers $\mathbb{R}$. Frenkel, Lepowsky, and
Meurman [7] constructed a well-behaved graded representation as a vertex
algebra, and Borcherds [1] showed that this vertex algebra satisfied Conway
and Norton’s conjecture.

The original modular moonshine conjectures of Ryba asserted the existence of
vertex algebras ${}^{g}V$ over finite fields with finite group actions, such
that the graded Brauer characters are genus zero modular functions. He also
suggested a construction of the vertex algebra ${}^{g}V$ (see Section 3) for
an element $g\in\mathbb{M}$ of prime order $p$ and of conjugacy class $pA$
that is the largest conjugacy class of order $p$, in terms of a self-dual
integral form of the monster vertex algebra whose existence was proved later
by Carnahan [4]. Borcherds and Ryba reinterpreted the modular moonshine
conjectures in terms of Tate cohomology with coefficients in an integral form
$V$ of the monster vertex algebra. They proved that the 1-st Tate cohomology
$\hat{H}^{1}(g,V)$ for a cyclic group $\langle g\rangle$ generated by any
Fricke element $g$ (see Definition 2.9) of odd prime order vanishes. Borcherds
proved that if $h$ is a $p$-regular element of $C_{\mathbb{M}}(g)$, then
graded Brauer characters
$\widetilde{\operatorname{Tr}}\big{(}h|{\hat{H}}^{i}(g,V)\big{)}=\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}\big{(}h|{\hat{H}}^{i}(g,V_{n})\big{)}q^{n-1}$,
$i=0,1$, is a linear combination of Hauptmoduls for any non-Fricke element $g$
of odd prime order [2, 3]. At the end of [2], Borcherds proposed a conjecture
that asserts the existence of objects that unify modular moonshine with
generalized moonshine. In his conjecture, conditions 2, 4 and 7 together imply
$\hat{H}^{1}(g,V)$ vanishes for all Fricke elements $g$. We will call this
claim “Borcherds’s conjecture about vanishing of Tate cohomology”.

It is natural to consider Tate cohomology of $V$ for elements of composite
order, but there are no results in the literature. Tate cohomology does not
yield vector spaces over a field when the element has composite order, but
Brauer characters are only defined for such objects. Specifically, Tate
cohomology yields torsion modules over the ground ring. We can’t use the
Brauer character for such modules because it is defined only for
representations defined over a field and $p$-regular elements. So, we need to
define a generalized Brauer character.

In Section 2, we introduce some terminology in this paper. In Section 3, we
summarize the modular moonshine conjecture for elements of prime order. In
Section 4, we consider ramified extensions $R_{p}$ of the $p$-adic integers
$\mathbb{Z}_{p}$ and calculate Tate cohomology. In Section 5, to generalize
the modular moonshine conjectures, we define a generalized Brauer character
and show that the generalized super Brauer character of Tate cohomology is a
linear combination of traces. This is a generalization of [3, Proposition 2.2]
to the composite order case. Using this result, we give a counterexample to
Borcherds’s conjecture about vanishing of Tate cohomology. We propose a weaker
conjecture about vanishing of Tate cohomology.

##  2 Definitions and basic notions

The symbols $\mathbb{Z}$, $\mathbb{Q}$, $\mathbb{R}$, $\mathbb{C}$,
$\mathbb{F}_{p}$ will be used to denote the integers, rational numbers, real
numbers, complex numbers, the finite field of prime $p$ order, respectively.

In this section, we introduce some terminology in this paper and will give
brief definitions.

######  Definition 2.1 ([4]).

Let $R$ be a commutative ring. A vertex algebra over $R$ is an $R$-module $V$
equipped with a distinguished element $\mathbf{1}\in V$, a distinguished
linear transformation $T\colon V\rightarrow V$, and a left multiplication map
$Y(\cdot,z)\colon
V\rightarrow(\operatorname{End}V)\big{[}\big{[}z,z^{-1}\big{]}\big{]}$, where
$\operatorname{End}V$ is the endomorphisms on $V$, written
$Y(a,z)=\sum_{n\in\mathbb{Z}}a_{n}z^{-n-1}$ satisfying the following
conditions:

  * •

For any $a\in V$, $Y(a,z)\mathbf{1}\in a+zV[[z]]$ and
$Y\left(\mathbf{1},z\right)={\rm id}_{V}$, where ${\rm id}_{V}$ is the
identity map on $V$.

  * •

For any $a,b\in V$, $Y(a,z)b\in V((z))$, i.e., $a_{n}b=0$ for $n$ sufficiently
large.

  * •

(The Jacobi identity) For any $a,b\in V$,

$\displaystyle
x^{-1}\delta\left(\frac{y-z}{x}\right)Y(a,y)Y(b,z)-x^{-1}\delta\left(\frac{z-y}{-x}\right)Y(b,z)Y(a,y)$
$\displaystyle\qquad{}=z^{-1}\delta\left(\frac{y-x}{z}\right)Y(Y(a,x)b,z),$

where $\delta(z)=\sum_{n\in\mathbb{Z}}z^{n}$, called the formal delta
function.

  * •

For any $a\in V$, $[T,Y(a,z)]=\frac{{\rm d}}{{\rm d}z}Y(a,z)$.

$\mathbf{1}$ is called a vacuum vector and $Y(\cdot,z)$ is called a vertex
operator.

######  Definition 2.2.

The Leech lattice $\Lambda$ is the unique free abelian group of rank 24
equipped with a symmetric bilinear form
$\langle\cdot,\cdot\rangle\colon\Lambda\times\Lambda\rightarrow\mathbb{Z}$
that satisfies the following conditions:

  * •

(even) For all $a\in\Lambda,\langle a,a\rangle\in 2\mathbb{Z}$.

  * •

(positive definite) For all nonzero elements $a\in\Lambda$, $\langle
a,a\rangle>0$.

  * •

(no roots) For all $a\in\Lambda$, $\langle a,a\rangle\neq 2$.

  * •

(unimodular) If we choose a basis $(a_{1},\dots,a_{24})$ of $\Lambda$, then we
may form the Gram matrix $(\langle a_{i},a_{j}\rangle)_{1\leq i,j\leq 24}$ and
its determinant is 1.

######  Definition 2.3.

A Lie algebra over a commutative ring $R$ is an $R$-module $\mathfrak{g}$
equipped with an $R$-bilinear map
$[\cdot,\cdot]\colon\mathfrak{g}\times\mathfrak{g}\rightarrow\mathfrak{g}$
satisfying the following conditions:

  * •

For any $x\in\mathfrak{g}$, $[x,x]=0$.

  * •

For any $x,y,z\in\mathfrak{g}$, $[x,[y,z]]+[y,[z,x]]+[z,[x,y]]=0$.

We review the construction of an integral form $V_{\Lambda}$ of a vertex
algebra for the Leech lattice $\Lambda$. $\Lambda$ has a unique nonsplit
central extension $\hat{\Lambda}$ such that if we denote a lifting of
$a\in\Lambda$ by ${\rm e}^{a}\in\hat{\Lambda}$, then ${\rm e}^{a}{\rm
e}^{b}=\pm{\rm e}^{a+b}$ and ${\rm e}^{a}{\rm e}^{b}=(-1)^{\langle
a,b\rangle}{\rm e}^{b}{\rm e}^{a}$ so we have an exact sequence

$0\rightarrow\\{\pm 1\\}\rightarrow\hat{\Lambda}\rightarrow\Lambda\rightarrow
0.$

The group of automorphisms of $\hat{\Lambda}$ that preserve the inner product
on $\Lambda$ is a nonsplit extension $2^{24}.\operatorname{Aut}(\Lambda)$ of
the automorphism group $\operatorname{Aut}(\Lambda)$, and this group acts on
$V_{\Lambda}$ faithfully.

A lattice vertex algebra is described by using notions of affine Lie algebras
and twisted group rings. In this paper, we explain them briefly.

We consider a Lie algebra $\mathfrak{h}=\Lambda\otimes_{\mathbb{Z}}\mathbb{Q}$
with $[a,b]=0$ for any $a,b\in\mathfrak{h}$. Then we take the Lie algebra
$\hat{\mathfrak{h}}=\mathfrak{h}\otimes\mathbb{Q}\big{[}t,t^{-1}\big{]}\oplus\mathbb{Q}c$
satisfying relations $\big{[}c,\hat{\mathfrak{h}}\big{]}=0$ and $[x\otimes
t^{m},y\otimes t^{n}]=\langle x,y\rangle m\delta_{m+n,0}c$, where
$\delta_{i,j}$ is the Kronecker delta. Let $\hat{\mathfrak{h}}^{\pm}$ be a
subalgebra $\mathfrak{h}\otimes t^{\pm 1}\mathbb{Q}\big{[}t^{\pm 1}\big{]}$
and $\hat{\mathfrak{h}}^{0}$ be a subalgebra $\mathfrak{h}\otimes 1$. We view
$\mathbb{Q}$ as $\hat{\mathfrak{h}}$-module which $\hat{\mathfrak{h}}^{\geq
0}$ acts as 0 and $c$ acts as 1.

We view $\mathbb{Q}$ as $\\{\pm 1\\}$-module which $\pm 1$ acts as $\pm 1$.
The twisted group ring $\mathbb{Q}\\{\Lambda\\}$ of $\Lambda$ is defined to be
$\mathbb{Q}\big{[}\hat{\Lambda}\big{]}\otimes_{\mathbb{Q}[\\{\pm
1\\}]}\mathbb{Q}$ and is spanned by elements $\iota\big{(}{\rm
e}^{\lambda}\big{)}:={\rm e}^{\lambda}\otimes 1$ for ${\rm
e}^{\lambda}\in\hat{\Lambda}$. Then we have ${\rm e}^{a}\cdot\iota\big{(}{\rm
e}^{\lambda}\big{)}=\iota\big{(}{\rm e}^{a}{\rm e}^{\lambda}\big{)}$ and
$\\{\pm 1\\}\cdot\iota\big{(}{\rm e}^{\lambda}\big{)}=\pm\iota\big{(}{\rm
e}^{\lambda}\big{)}$, hence $\mathbb{Q}\\{\Lambda\\}$ is
$\hat{\Lambda}$-module.

$V_{\Lambda}\otimes\mathbb{Q}$ is isomorphic to
$\operatorname{Sym}_{\mathbb{Q}}\big{(}\hat{\mathfrak{h}}^{-}\big{)}\otimes\mathbb{Q}\\{\Lambda\\}$,
where $\operatorname{Sym}_{\mathbb{Q}}(\cdot)$ means the symmetric algebra
over $\mathbb{Q}$. $\hat{\Lambda}$ and $\hat{\mathfrak{h}}$ act on
$V_{\Lambda}\otimes\mathbb{Q}$ as following: For any $v\otimes\iota\big{(}{\rm
e}^{\lambda}\big{)}\in V_{\Lambda}$, $h\in\mathfrak{h}$, $n\neq 0$, and ${\rm
e}^{a}\in\hat{\Lambda}$, writing $h\otimes t^{n}$ by $h(n)$,

$\displaystyle h(n)\cdot\big{(}v\otimes\iota\big{(}{\rm
e}^{\lambda}\big{)}\big{)}=h(n)v\otimes\iota\big{(}{\rm
e}^{\lambda}\big{)},\qquad h(0)\cdot\big{(}v\otimes\iota\big{(}{\rm
e}^{\lambda}\big{)}\big{)}=\langle h,\lambda\rangle\big{(}v\otimes\iota({\rm
e}^{\lambda})\big{)},$ $\displaystyle c\cdot\big{(}v\otimes\iota\big{(}{\rm
e}^{\lambda}\big{)}\big{)}=v\otimes\iota\big{(}{\rm e}^{\lambda}\big{)},$
$\displaystyle{\rm e}^{a}\cdot\big{(}v\otimes\iota\big{(}{\rm
e}^{\lambda}\big{)}\big{)}=v\otimes\iota\big{(}{\rm e}^{a+\lambda}\big{)}.$

$V_{\Lambda}\otimes\mathbb{Q}$ is graded by $\Lambda$, called
$\Lambda$-degree, and we have
$V_{\Lambda}\otimes\mathbb{Q}=\bigoplus_{\lambda\in\Lambda}\operatorname{Sym}_{\mathbb{Q}}\big{(}\hat{\mathfrak{h}}^{-}\big{)}\otimes\mathbb{Q}\iota\big{(}{\rm
e}^{\lambda}\big{)}$.

Let $\mathbf{1}$ be a vacuum vector of $V_{\Lambda}$ and let $Y(\cdot,z)$ be a
vertex operator on $V_{\Lambda}$. We define the operator $D^{(n)}$ by
$Y(v,z)\mathbf{1}=\sum_{n\in\mathbb{Z}}D^{(n)}(v)z^{n}$ for any $v\in
V_{\Lambda}$. The operator $D^{(n)}$ has the properties $D^{(0)}(v)=v$,
$D^{(n)}D^{(m)}={n+m\choose n}D^{(n+m)}$, $D^{(n)}=0$ if $n<0$, and
$D^{(n)}(vw)=\sum_{m\in\mathbb{Z}}D^{(m)}(v)D^{(n-m)}(w)$ for any $v,w\in
V_{\Lambda}$. The derivation $D$ is a element of $\hat{\mathfrak{h}}$ and has
$\Lambda$-degree 0, i.e., $D^{(n)}(v)$ has the same $\Lambda$-degree as $v$.

Suppose that we choose a basis $(a_{1},\dots,a_{24})$ of $\Lambda$ and let $p$
be a prime number. The subring $V_{\Lambda,0}$ is generated as a commutative
ring by ${\rm e}^{-a_{j}}D^{(i)}\big{(}{\rm e}^{a_{j}}\big{)}$ for $i\geq 1$
and $1\leq j\leq 24$ [3]. Let $M$ be the free $\mathbb{Z}$-submodule of
$\hat{\mathfrak{h}}_{\mathbb{Q}}$ given by the basis $\big{\\{}{\rm
e}^{-a_{j}}D^{(i)}\big{(}{\rm e}^{a_{j}}\big{)}\big{\\}}_{i\geq 1,1\leq j\leq
24}$. Then $V_{\Lambda}$ is isomorphic to
$\operatorname{Sym}_{\mathbb{Z}}(M)\otimes\mathbb{Z}\\{\Lambda\\}$.

######  Definition 2.4.

Let $R$ be a subring of the complex numbers $\mathbb{C}$. An $R$-vertex
algebra $V$ is an $R$-form of the monster vertex algebra if
$V\otimes_{R}\mathbb{C}\cong V^{\natural}$, where $V^{\natural}$ is the
monster vertex algebra over $\mathbb{C}$.

######  Definition 2.5.

Let $G$ be a group and $N$ be an integer. An element $h$ of $G$ is $N$-regular
if $(|h|,N)=1$.

######  Definition 2.6.

Let $R$ be a discrete valuation ring and $\mathfrak{m}_{R}$ the maximal ideal
of $R$. Suppose that $k=R/\mathfrak{m}_{R}$ has prime characteristic $p$ and
that the fraction field $K$ of $R$ has characteristic $0$. The Brauer
character of a $k[h]$-module $M$ for a $p$-regular element $h$ is defined by

$\widetilde{\operatorname{Tr}}(h|M)=\operatorname{Tr}\big{(}h|\widetilde{M}\otimes_{R}K\big{)},$

where $\widetilde{M}$ is any $R$-free $R[h]$-module satisfying
$M\cong\widetilde{M}\otimes_{R[h]}k[h]$.

######  Definition 2.7 ([5, Chapter IV]).

Suppose that $G$ is a group acting on an abelian group $A$. Let ${\rm Nr}$ be
the norm map $A\rightarrow A$; $a\mapsto\sum_{g\in G}{g(a)}$. Then, the $i$-th
Tate cohomology ${\hat{H}}^{i}(G,A)$ is defined to be the following:

${\hat{H}}^{i}(G,A)=\begin{cases}H_{-i-1}(G,A),&i\leq-2,\\\
\operatorname{Ker}({\rm Nr})/\langle ga-a\,|\,a\in A,\,g\in G\rangle,&i=-1,\\\
A^{G}/\operatorname{Im}({\rm Nr}),&i=0,\\\ H^{i}(G,A),&i\geq 1,\end{cases}$

where $H_{i}(G,A)$ is the $i$-th homology, $H^{i}(G,A)$ is the $i$-th
cohomology and $A^{G}$ is the largest submodule of $A$ on which $G$ acts
trivially.

It has the following properties:

  1. (1)

If $0\rightarrow A\rightarrow B\rightarrow C\rightarrow 0$ is a short exact
sequence of $G$-modules, then we have a long exact sequence

$\cdots\rightarrow{\hat{H}}^{i-1}(G,C)\rightarrow{\hat{H}}^{i}(G,A)\rightarrow{\hat{H}}^{i}(G,B)\rightarrow{\hat{H}}^{i}(G,C)\rightarrow{\hat{H}}^{i+1}(G,A)\rightarrow\cdots.$

  2. (2)

$\bigoplus_{i\in\mathbb{Z}}{\hat{H}}^{i}(G,A)$ is $|G|$-torsion module. In
particular, if multiplication by $|G|$ is an isomorphism on $A$, then
$\hat{H}^{i}(G,A)$ vanishes for all $i\in\mathbb{Z}$.

Moreover, if $G$ is a cyclic group generated by $g$, then we have the
following properties, where we denoted the Tate cohomology
$\hat{H}^{i}(\langle g\rangle,A)$ for a cyclic group $\langle g\rangle$ by
$\hat{H}^{i}(g,A)$.

  1. (3)

${\hat{H}}^{i}(g,A)\cong{\hat{H}}^{i+2}(g,A)$ for any $i\in\mathbb{Z}$.

  2. (4)

${\hat{H}}^{\ast}(g,A)=\hat{H}^{0}(g,A)\oplus\hat{H}^{1}(g,A)$ has more or
less a super version of any $G$-invariant algebraic structure on $A$. In
particular, if $V$ is a vertex algebra, then $\hat{H}^{*}(g,V)$ is a vertex
superalgebra by the composite map

$\hat{H}^{*}(g,V)\otimes\hat{H}^{*}(g,V)\rightarrow\hat{H}^{*}(g,V\otimes
V)\rightarrow\hat{H}^{*}(g,V((z)))\rightarrow\hat{H}^{*}(g,V)((z)).$

  3. (5)

If $A$ is a finite module, then
$\big{|}{\hat{H}}^{0}(g,A)\big{|}=\big{|}{\hat{H}}^{1}(g,A)\big{|}$.

######  Definition 2.8.

Let $\mathbb{H}$ be the upper half plane of $\mathbb{C}$. A Hauptmodul is a
holomorphic function $f\colon\mathbb{H}\rightarrow\mathbb{C}$ satisfying the
following conditions:

  * •

$f$ is invariant under some discrete subgroup $\Gamma_{f}$ of ${\rm
SL}_{2}(\mathbb{R})$,

  * •

$f$ generates the meromorphic function field on $\mathbb{H}/\Gamma_{f}$.

######  Definition 2.9.

A monster element $g$ is Fricke if there is some $N\geq 1$ s.t.
$T_{g}\big{(}\frac{-1}{N\tau}\big{)}=T_{g}(\tau)$ for any $\tau\in\mathbb{H}$.

######  Definition 2.10.

Let $R$ be a discrete valuation ring of mixed characteristic $(0,p)$, i.e.,
$R$ has chracteristic 0 and the residue field $R/\mathfrak{m}_{R}$ of $R$ has
characteristic $p$. Let $S$ be the ring of integers of a finite extension of
the fraction field $\operatorname{Frac}(R)$ of $R$. Then
$\mathfrak{m}^{e}_{S}=\mathfrak{m}_{R}S$ for some positive integer $e$, called
the ramification index. When $e=1$, $S$ is called an unramified extension of
$R$. On the other hand, when $e\neq 1$, $S$ is called a ramified extension of
$R$.

######  Definition 2.11.

Let $R$ be a commutative ring with prime characteristic $p$. The Frobenius
endomorphism $\operatorname{Frob}_{p}\colon R\rightarrow R$ is defined by
$\operatorname{Frob}_{p}(r)=r^{p}$ for all $r\in R$.

If $R$ is a finite field, the Frobenius endomorphism is an automorphism.

######  Definition 2.12.

Let $A$ be a finite abelian group, $A^{*}$ be the homomorphisms
$\operatorname{Hom}(A,\mathbb{C}^{\times})$ from $A$ to $\mathbb{C}^{\times}$
and $f\colon A\rightarrow\mathbb{C}$ be a function. The discrete Fourier
transform $\hat{f}\colon A^{*}\rightarrow\mathbb{C}$ is defined by
$\hat{f}(\chi)=\sum_{a\in A}f(a)\chi(a)$ for any $\chi\in A^{*}$.

There is an inverse discrete Fourier transform, it is described by the
following

$f(a)=\frac{1}{|A|}\sum_{\chi\in A^{*}}\hat{f}(\chi)\overline{\chi}(a).$

##  3 Modular moonshine for elements of prime order

In this section, we summarize modular moonshine for elements of prime order
[2, 3].

Ryba proposed the modular moonshine conjecture. The existence of a self-dual
integral form of the monster vertex algebra was not proved when he suggested
it, but Carnahan proved it [4].

######  Theorem 3.1 (the original modular moonshine conjecture).

Suppose that an element $g\in\mathbb{M}$ has prime order $p$ and lies in
conjugacy class $pA$, and that $V$ is a self-dual integral form of the monster
vertex algebra. Let $V^{g}$ be the set of vectors fixed by $g$. We consider
the following vertex algebra with characteristic $p$:

${}^{g}V=\cfrac{V^{g}/pV^{g}}{(V^{g}/pV^{g})\cap(V^{g}/pV^{g})^{\perp}}.$

For ${}^{g}V=\bigoplus_{n\in\mathbb{Z}}{{}^{g}V_{n}}$, if $h$ is a $p$-regular
element of the centralizer $C_{\mathbb{M}}(g)$ of $g$, then

$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}(h|^{g}V_{n})q^{n-1}=T_{gh}(\tau).$

Here, we may take $R=\mathbb{Z}_{p}$ when computing Brauer character
$\widetilde{\operatorname{Tr}}$.

Let $G$ be a finite group including an element $g$ of prime order. Borcherds
and Ryba noticed that this definition of ${}^{g}V$ yields $\hat{H}^{0}(g,V)$.
They redefined ${}^{g}V$ to be $\hat{H}^{*}(g,V)$ and reinterpreted the
conjectures in terms of Tate cohomology. To prove the conjectures, they used
the fact that there are exactly 3 indecomposable finitely generated modules
over the group ring $\mathbb{Z}_{p}[G]$ which are free as
$\mathbb{Z}_{p}$-modules; there are $\mathbb{Z}_{p}$, $\mathbb{Z}_{p}[G]$ and
$I$, which is the kernel of the natural map
$\mathbb{Z}_{p}[G]\rightarrow\mathbb{Z}_{p}$. By the following lemma, the Tate
cohomology for any $\mathbb{Z}$-module is identified with the Tate cohomology
for $\mathbb{Z}_{p}$-module.

######  Lemma 3.2 ([3, Lemma 2.1]).

If $A$ is a free $\mathbb{Z}[1/n]$-module acted on by a $p$-group $G$ with
$(p,n)=1$, then the natural map from $\hat{H}^{i}(G,A)$ to
$\hat{H}^{i}(G,A\otimes\mathbb{Z}_{p})$ is an isomorphism for any
$i\in\mathbb{Z}$.

The next proposition is the main tool for calculating the Brauer character of
a modular moonshine vertex super algebra.

######  Proposition 3.3 ([3, Proposition 2.2]).

Let $G$ be a finite group. Suppose that an element $g$ of the center
$\operatorname{Cent}(G)$ of $G$ has prime order $p$ and that $h\in G$ is a
$p$-regular element. Let $A$ be a finitely generated $\mathbb{Z}$-free
$\mathbb{Z}[\langle g,h\rangle]$-module. Then
$\hat{H}^{*}(g,A)=\hat{H}^{0}(g,A)-\hat{H}^{1}(g,A)$ is a virtual
representation of $\langle h\rangle$, and

$\widetilde{\operatorname{Tr}}\big{(}h|\hat{H}^{*}(g,A)\big{)}=\operatorname{Tr}(gh|A).$

We have the following assertion by applying Proposition 3.3 to an element of
the monster group and the monster vertex algebra.

######  Corollary 3.4.

Suppose that $g$ is an element of $\mathbb{M}$ of odd prime order. Let $V$ be
an integral form of the monster vertex algebra. If $h\in C_{\mathbb{M}}(g)$ is
$p$-regular, then
$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}\big{(}h|\hat{H}^{*}(g,V_{n})\big{)}q^{n-1}$
is equal to the McKay–Thompson series $T_{gh}(\tau)$. In particular, it is a
Hauptmodul for some genus $0$ subgroup of ${\rm SL}_{2}(\mathbb{R})$.

Borcherds and Ryba proved that ${\hat{H}}^{1}(g,V)=0$ for any Fricke element
$g$ of odd prime order. There are two main methods for proving
${\hat{H}}^{1}(g,V)=0$.

First, when $g$ has small order, we can transfer a vanishing statement from
$V_{\Lambda}$, using a decomposition of $V$ into submodules isomorphic to
pieces of $V_{\Lambda}$ [3].

######  Lemma 3.5 ([3, Lemma 4.1]).

If $A$ is a permutation module of a finite group $G$ over a ring $R$ with no
$|G|$ torsion, then $\hat{H}^{-1}(G,A)=0$. $($A free $R$-module $A$ acted on
by a group $G$ is a permutation module if it has a basis which is closed under
the action of $G$.$)$

######  Lemma 3.6 ([3, Lemma 4.2]).

The set of permutation modules is closed under taking sums, tensor products,
and symmetric powers.

######  Lemma 3.7 ([3, Lemma 4.5 and Theorem 4.6]).

If $g$ is an element of odd prime order in a Mathieu group $M_{24}$ viewed as
a subgroup of $\operatorname{Aut}(\Lambda)$, then $\hat{H}^{1}(g,\Lambda)=0$
and $\hat{H}^{1}(g,V_{\Lambda})=0$.

######  Theorem 3.8 ([3, Theorem 4.7]).

Suppose that $g$ is an element of the monster of type $3A$, $3C$, $5A$, $7A$,
$11A$, or $23A$. Then $\hat{H}^{1}(g,V)=0$.

######  Remark 3.9.

Similarly, for an element $g$ of the monster type $2A$, they proved
$\hat{H}^{1}(g,V)=0$ by showing that $\hat{H}^{1}(g,\Lambda)$ and
$\hat{H}^{1}(g,V_{\Lambda})$ vanish. The method of proof is same as the case
of odd prime order, but slightly more complicated.

Borcherds and Ryba don’t use the assumption that $g$ has prime order in proofs
of Lemma 3.7 and Theorem 3.8. However, this assumption is applied to use Lemma
3.2. Hence, we will show that Lemma 3.2 holds without the assumption that $G$
is a $p$-group, and that $\hat{H}^{1}(g,V)=0$ for an element $g$ of the
monster of type $15A$ or $21A$ in Section 4.

Second, when $g$ has large order, it suffices to show that the coefficients

$c_{g}^{-}(n)=\dim\hat{H}^{0}(g,V_{n})+\dim\hat{H}^{1}(g,V_{n})$

are equal to the coefficients

$c_{g}^{+}(n)=\operatorname{Tr}(g|V_{n})=\dim\hat{H}^{0}(g,V_{n})-\dim\hat{H}^{1}(g,V_{n}).$

This was proved by combining Hodge theory, an improved “no-ghost” theorem, and
some explicit computation.

Borcherds proved that graded Brauer characters

$\widetilde{\operatorname{Tr}}(h|{\hat{H}}^{i}(g,V))=\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}\big{(}h|{\hat{H}}^{i}(g,V_{n})\big{)}q^{n-1},\qquad
i=0,1,$

is a linear combination of Hauptmoduls for any non-Fricke element $g$ of odd
prime order [2, 3].

######  Theorem 3.10 ([3, Theorem 5.3]).

If $g$ is an element of the monster of type $2B$, then $\hat{H}^{0}(g,V_{n})$
vanishes if $n$ is odd, and $\hat{H}^{1}(g,V_{n})$ vanishes if $n$ is even.

######  Theorem 3.11 ([2, Theorem 6.1]).

If $g$ is an element of the monster of type $3B$, $5B$, $7B$, and $13B$ and
$\sigma$ is the element of order $2$ generating the center of
$C_{\mathbb{M}}(g)/O_{p}(C_{\mathbb{M}}(g))$, where $O_{p}(C_{\mathbb{M}}(g))$
is the largest normal $p$-subgroup of $C_{\mathbb{M}}(g)$, then
$O_{p}(C_{\mathbb{M}}(g))$ acts trivially on
${}^{g}V=\hat{H}^{0}(g,V)\oplus\hat{H}^{1}(g,V)$ and $\sigma$ fixes
$\hat{H}^{0}(g,V)$ and acts as $-1$ on $\hat{H}^{1}(g,V)$.

We have the following complete description of
$\widetilde{\operatorname{Tr}}\big{(}h|{\hat{H}}^{i}(g,V)\big{)}$, $i=0,1$,

$\displaystyle\widetilde{\operatorname{Tr}}\big{(}h|{\hat{H}}^{0}(g,V)\big{)}=\begin{cases}T_{gh}(\tau),&g\in
pA,3C,\vspace{1mm}\\\ \dfrac{T_{gh}(\tau)+T_{gh}(\tau+1/2)}{2},&g\in
2B,\vspace{1mm}\\\ \dfrac{T_{gh}(\tau)+T_{gh\sigma}(\tau)}{2},&g\in
pB,2|(p-1),\end{cases}$
$\displaystyle\widetilde{\operatorname{Tr}}\big{(}h|{\hat{H}}^{1}(g,V)\big{)}=\begin{cases}0,&g\in
pA,3C,\vspace{1mm}\\\ \cfrac{-T_{gh}(\tau)+T_{gh}(\tau+1/2)}{2},&g\in
2B,\vspace{1mm}\\\ \cfrac{-T_{gh}(\tau)+T_{gh\sigma}(\tau)}{2},&g\in
pB,2|(p-1),\end{cases}$

where $\sigma\in C_{\mathbb{M}}(g)/O_{p}(C_{\mathbb{M}}(g))$ is an involution
which acts as $1$ on $\hat{H}^{0}(g,V)$ and as $-1$ on $\hat{H}^{1}(g,V)$.

##  4 Tate cohomology and ramified extensions

In this section, we will prove properties of Tate cohomology related to
ramified extensions and complete the proof of Theorem 4.7 of [3] for an
element of the monster of type $15A$ or $21A$.

Let $R$ be a discrete valuation ring. Let $g\in G$ be an element of order $N$,
$h\in G$ be an $N$-regular element and $A$ be an $R[g]$-module.

######  Proposition 4.1.

If a commutative ring homomorphism $R\rightarrow S$ is flat, then
$\hat{H}^{*}(g,A\otimes_{R}S)\cong\hat{H}^{*}(g,A)\otimes_{R}S$.

###### Proof.

Let ${\rm Nr}$ be a norm map as in Definition 2.7. Then,
$\hat{H}^{0}(g,A)\cong\operatorname{Ker}(g-1)/\operatorname{Im}({\rm Nr}).$ We
have an exact sequence of $R$-modules $0\rightarrow\operatorname{Im}({\rm
Nr})\rightarrow\operatorname{Ker}(g-1)\rightarrow\hat{H}^{0}(g,A)\rightarrow
0$. If $R\rightarrow S$ is flat, the sequence of $S$-modules
$0\rightarrow\operatorname{Im}({\rm
Nr})\otimes_{R}S\rightarrow\operatorname{Ker}(g-1)\otimes_{R}S\rightarrow\hat{H}^{0}(g,A)\otimes_{R}S\rightarrow
0$ is exact. We consider the following diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}({\rm
Nr}\otimes_{R}{\rm
id}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}((g-1)\otimes_{R}{\rm
id}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hat{H}^{0}(g,A\otimes_{R}S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{0}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}({\rm
Nr})\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}(g-1)\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hat{H}^{0}(g,A)\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Then, we have $\operatorname{Im}({\rm Nr}\otimes_{R}{\rm
id}_{S})\cong\operatorname{Im}({\rm Nr})\otimes_{R}S$ and
$\operatorname{Ker}((g-1)\otimes_{R}{\rm
id}_{S})\cong\operatorname{Ker}(g-1)\otimes_{R}S$. Indeed, we can take the
isomorphisms ${\rm Nr}\otimes_{R}{\rm id}_{S}(x\otimes s)\mapsto{\rm
Nr}(x)\otimes s$ and $x\otimes s\mapsto x\otimes s$, respectively. Also, the
above diagram is commutative. Hence, by the Five lemma, $\psi_{0}$ is an
isomorphism.

Like the degree 0 case, we have an exact sequence
$0\rightarrow\operatorname{Im}(g-1)\rightarrow\operatorname{Ker}({\rm
Nr})\rightarrow\hat{H}^{1}(g,A)\rightarrow 0$. As well above, we consider the
following commutative diagram:

$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}((g-1)\otimes_{R}{\rm
id}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}({\rm
Nr}\otimes_{R}{\rm
id}_{S})\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hat{H}^{1}(g,A\otimes_{R}S)\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\scriptstyle{\psi_{1}}$$\textstyle{0}$$\textstyle{0\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Im}(g-1)\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\operatorname{Ker}({\rm
Nr})\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{\hat{H}^{1}(g,A)\otimes_{R}S\ignorespaces\ignorespaces\ignorespaces\ignorespaces}$$\textstyle{0.}$

Then, we have $\operatorname{Im}((g-1)\otimes_{R}{\rm
id}_{S})\cong\operatorname{Im}(g-1)\otimes_{R}S$ and $\operatorname{Ker}({\rm
Nr}\otimes_{R}{\rm id}_{S})\cong\operatorname{Ker}({\rm Nr})\otimes_{R}S$.
Indeed, we can take the isomorphisms $(g-1)\otimes_{R}{\rm id}_{S}(x\otimes
s)\mapsto(gx-x)\otimes s$ and $x\otimes s\mapsto x\otimes s$, respectively. By
the Five lemma, $\psi_{1}$ is an isomorphism. ∎

Using the above proposition, we will prove that the 1-st Tate cohomology for
an element of the monster type $15A$ or $21A$ vanishes. To do this, we will
show the following lemma.

######  Lemma 4.2.

If $A$ is a free $\mathbb{Z}[1/q]$-module acted on by a group $G$ with
$|G|=p^{k}q$ and $(p,q)=1$, then the natural map from $\hat{H}^{i}(G,A)$ to
$\hat{H}^{i}(G,A\otimes\mathbb{Z}_{p})$ is an isomorphism for any
$i\in\mathbb{Z}$.

###### Proof.

We show that multiplication by $|G|$ is an isomorphism on
$\mathbb{Z}_{p}/\mathbb{Z}[1/q]$. For any
$p^{l}\frac{a}{b}\in\operatorname{Ker}(|G|)$, where $l$ is a non-negative
integer and $p$, $a$, $b$ are pairwise coprime, we have $|G|\cdot
p^{l}\frac{a}{b}=p^{l+k}\frac{aq}{b}\in\mathbb{Z}[1/q]$. It follows that $b$
is a product of divisors of $q$ and then $p^{l}\frac{a}{b}\in\mathbb{Z}[1/q]$.
Hence, $|G|$ is injective. For any
$\sum_{i=0}^{\infty}a_{i}p^{i}\in\mathbb{Z}_{p}/\mathbb{Z}$, where
$a_{i}\in\\{0,\dots,p-1\\}$,we have
$\sum_{i=0}^{\infty}a_{i}p^{i}\equiv\sum_{i=k}^{\infty}a_{i}p^{i}$ $({\rm
mod}\ \mathbb{Z})$. Hence, we have
$|G|\cdot\frac{1}{q}\sum_{i=0}^{\infty}a_{k+i}p^{i}=\sum_{i=k}^{\infty}a_{i}p^{i}$
and then $|G|$ is surjective on $\mathbb{Z}_{p}/\mathbb{Z}$. By the
surjectivity of
$\mathbb{Z}_{p}/\mathbb{Z}\rightarrow\mathbb{Z}_{p}/\mathbb{Z}[1/q]$, $|G|$ is
surjective on $\mathbb{Z}_{p}/\mathbb{Z}[1/q]$. By property (2) of Tate
cohomology, $\hat{H}^{i}(G,A\otimes\mathbb{Z}_{p}/\mathbb{Z}[1/q])=0$ for all
$i\in\mathbb{Z}$. For the exact sequence

$0\rightarrow A\rightarrow A\otimes\mathbb{Z}_{p}\rightarrow
A\otimes(\mathbb{Z}_{p}/\mathbb{Z}[1/q])\rightarrow 0,$

taking the Tate cohomology, we have a long exact sequence because of property
(1) of Tate cohomology. It follows that
$\hat{H}^{i}(G,A)\cong\hat{H}^{i}(G,A\otimes\mathbb{Z}_{p})$ for any
$i\in\mathbb{Z}$. ∎

######  Lemma 4.3.

If $g$ is an element of odd order in $M_{24}$ viewed as a subgroup of
$\operatorname{Aut}(\Lambda)$, then $\hat{H}^{1}(g,\Lambda)=0$ and
$\hat{H}^{1}(g,V_{\Lambda})=0$.

###### Proof.

Let $|g|=p^{k}q$, where $p$ is a prime and $(p,q)=1$. By Proposition 4.1 and
Lemma 4.2, we have
$\hat{H}^{1}(g,\Lambda)\otimes\mathbb{Z}[1/q]\cong\hat{H}^{1}(g,\Lambda\otimes\mathbb{Z}[1/q])\cong\hat{H}^{1}\big{(}g,\Lambda\otimes_{\mathbb{Z}[1/q]}\mathbb{Z}_{p}\big{)}$.
By the same argument as the proof of Lemma 4.5 of [3],
$\hat{H}^{1}\big{(}g,\Lambda\otimes_{\mathbb{Z}[1/q]}\mathbb{Z}_{p}\big{)}=0$.
Hence, $\hat{H}^{1}(g,\Lambda)=0$.

Similarly, we have
$\hat{H}^{1}(g,V_{\Lambda})\otimes\mathbb{Z}[1/q]\cong\hat{H}^{1}\big{(}g,V_{\Lambda}\otimes_{\mathbb{Z}[1/q]}\mathbb{Z}_{p}\big{)}.$
We can decompose $V_{\Lambda}\otimes\mathbb{Z}_{p}$ into the following:

$V_{\Lambda}\otimes\mathbb{Z}_{p}=\bigoplus_{\operatorname{orbits}\langle
g\rangle\lambda}\bigg{(}\bigoplus_{g^{r}\lambda\in\langle
g\rangle\lambda}V_{\Lambda,g^{r}\lambda}\otimes\mathbb{Z}_{p}\bigg{)},$

where $\bigoplus_{{{\rm orbits}\langle g\rangle\lambda}}$ means the direct sum
over distinct $\operatorname{orbits}\langle g\rangle\lambda$,

$\bigoplus_{g^{r}\lambda\in\langle
g\rangle\lambda}V_{\Lambda,g^{r}\lambda}\otimes\mathbb{Z}_{p}\cong\mathbb{Z}_{p}[\langle
g\rangle/\operatorname{Stab}_{\langle
g\rangle}(\lambda)]\otimes_{\mathbb{Z}_{p}}(V_{\Lambda,0}\otimes\mathbb{Z}_{p})$

as $\mathbb{Z}_{p}[g]$-modules and $\operatorname{Stab}_{\langle
g\rangle}(\lambda)$ is the stabilizer subgroup of $\langle g\rangle$ with
respect to $\lambda$.

By the proof of Theorem 4.6 of [3], $V_{\Lambda,0}\otimes\mathbb{Z}_{p}$ is a
permutation module. By Lemma 3.6, $\bigoplus_{g^{r}\lambda\in\langle
g\rangle\lambda}V_{\Lambda,g^{r}\lambda}\otimes\mathbb{Z}_{p}$ is a
permutation module. By Lemmas 3.6 and 3.5, $V_{\Lambda}\otimes\mathbb{Z}_{p}$
is a permutation module and we have $\hat{H}^{1}(g,V_{\Lambda})=0$. ∎

######  Theorem 4.4.

Suppose that $g$ is an element of the monster of type $15A$ or $21A$. Then
$\hat{H}^{1}(g,V)=0$.

###### Proof.

In the same way as the proof of Theorem 4.7 of [3], this statement follows
from Lemma 4.3. ∎

Let $p$ be a prime factor of $N$ and $\zeta_{N}$ be a primitive $N$-th root of
unity. Now, we consider a ramified extension
$R_{p}=\mathbb{Z}_{p}[\zeta_{N|h|}]$ of $\mathbb{Z}_{p}$.

We define rank 1 $R$-free $R_{p}[\langle g,h\rangle]$-modules $R_{n,m}$ with
distinguished basis vectors $v_{n,m}\in R_{n,m}$, i.e.,

$R_{n,m}=\big{\\{}cv_{n,m}\,|\,c\in
R_{p},\,gv_{n,m}=\zeta_{N}^{n}v_{n,m},\,hv_{n,m}=\zeta_{|h|}^{m}v_{n,m}\big{\\}}.$

Any $R_{p}[\langle g,h\rangle]$-module that is $R_{p}$-free of finite rank is
an extension of rank 1 modules of the form $R_{n,m}$ for $0\leq n\leq N-1$,
$0\leq m\leq|h|-1$. Hence, it suffices to consider the Tate cohomology of
$R_{n,m}$.

######  Lemma 4.5.

For any $0\leq n\leq N-1$, $0\leq m\leq|h|-1$,

$\displaystyle\hat{H}^{0}(g,R_{n,m})=\begin{cases}R_{p}/NR_{p},&n=0,\\\
0,&n\neq 0,\end{cases}$
$\displaystyle\hat{H}^{1}(g,R_{n,m})=\begin{cases}0,&n=0,\\\
R_{p}/(1-\zeta_{N}^{n})R_{p},&n\neq 0,\end{cases}$

where $h$ acts as $\zeta_{|h|}^{m}$ on $R_{p}/NR_{p}$ and
$R_{p}/\big{(}1-\zeta_{N}^{n}\big{)}R_{p}$.

###### Proof.

$R_{0,m}^{g}=R_{0,m}$, $R_{n,m}^{g}=0$, $n\neq 0$. Hence,
$\hat{H}^{0}(g,R_{0,m})=R_{p}/NR_{p}$, $\hat{H}^{0}(g,R_{n,m})=0$, $n\neq 0$,
because the norm map $R_{n,m}\rightarrow R_{n,m}$ is
$v_{n,m}\mapsto\sum_{i=0}^{N-1}g^{i}v_{n,m}$.

By $\sum_{i=0}^{N-1}g^{i}v_{0,m}=Nv_{0,m}$, the kernel of the norm map on
$R_{0,m}$ is 0. Hence, $\hat{H}^{1}(g,R_{0,m})=0$. By
$\sum_{i=0}^{N-1}g^{i}v_{n,m}=\sum_{i=0}^{N-1}\zeta_{N}^{ni}v_{n,m}=0$, $n\neq
0$, the kernel of the norm map on $R_{n,m}$ is $R_{n,m}$. Therefore, by the
definition of Tate cohomology,
$\hat{H}^{1}(g,R_{n,m})=R_{p}/\big{(}1-\zeta_{N}^{n}\big{)}R_{p}$, $n\neq 0$.
∎

##  5 $\boldsymbol{p}$-Brauer character

In this section, we will define a new notion of generalized Brauer character
and prove properties of it. We will give a counterexample to a conjecture of
Borcherds about vanishing of Tate cohomology.

Let $p$ be a prime number. Let $R$ be a discrete valuation ring of mixed
characteristic $(0,p)$, $v\colon R\rightarrow\mathbb{Z}_{\geq
0}\cup\\{\infty\\}$ be a surjective valuation on $R$ and
$K=\operatorname{Frac}(R)$.

We can’t use the Brauer character for $R_{p}/(1-\zeta_{N}^{n})R_{p}$ because
this is not a vector space over a field. To solve this, we will define a
$p$-Brauer character.

######  Definition 5.1.

Let $G$ be a finite group and $M$ be a finite length $R[G]$-module. Let $h$ be
a $p$-regular element of $G$. Suppose that $M$ has a composition series
$0=M_{0}\subsetneq M_{1}\subsetneq\dots\subsetneq M_{n}=M$ as an
$R[h]$-module. We define the $p$-Brauer character of $M$ for $h$ by

$\widetilde{\operatorname{Tr}}_{p}(h|M)=\cfrac{1}{v(p)}\sum_{i=1}^{n}\operatorname{Tr}\big{(}h|\widetilde{M_{i}}\otimes
K\big{)},$

where $\widetilde{M_{i}}$ is an $R$-free $R[h]$-module satisfying
$M_{i}/M_{i-1}\cong\widetilde{{M}_{i}}\otimes_{R[h]}R/\mathfrak{m}_{R}[h]$.

######  Lemma 5.2.

This definition does not depend on our choices of $M_{i}$ and
$\widetilde{M}_{i}$.

###### Proof.

For
$M_{i}/M_{i-1}\cong\widetilde{{M}_{i}}\otimes_{R[h]}R/\mathfrak{m}_{R}[h]\cong\widetilde{{M}_{i}^{\prime}}\otimes_{R[h]}R/\mathfrak{m}_{R}[h]$,
we have $\operatorname{Tr}\big{(}h|\widetilde{M}_{i}\otimes
K\big{)}=\widetilde{\operatorname{Tr}}(h|M_{i}/M_{i-1})=\operatorname{Tr}\big{(}h|\widetilde{M}_{i}^{\prime}\otimes
K\big{)}$. By the Jordan–Hölder theorem, the composition series of $M$ is
unique up to permutation and isomorphism. That is, for any two composition
series $0=M_{0}\subsetneq M_{1}\subsetneq\dots\subsetneq M_{n}=M$,
$0=M_{0}^{\prime}\subsetneq M_{1}^{\prime}\subsetneq\dots\subsetneq
M_{n^{\prime}}^{\prime}=M$, we have $n=n^{\prime}$ and some permutation
$\sigma$ of $\\{1,\dots,n\\}$ s.t. $M_{i}/M_{i-1}\cong
M_{\sigma(i)}^{\prime}/M_{\sigma(i)-1}^{\prime}$. Hence,
$\sum_{i=1}^{n}\widetilde{\operatorname{Tr}}(h|M_{i}/M_{i-1})=\sum_{i=1}^{n}\widetilde{\operatorname{Tr}}(h|M_{\sigma(i)}^{\prime}/M_{\sigma(i)-1}^{\prime})=\sum_{i=1}^{n}\widetilde{\operatorname{Tr}}(h|M_{i}^{\prime}/M_{i-1}^{\prime})$.
Therefore the $p$-Brauer character is well-defined. ∎

Note that it does depend on a choice of a prime number $p$.

We can consider the notion of an irreducible $p$-Brauer character, so that $M$
is a simple $R[h]$-module that $M\otimes_{R[h]}R/\mathfrak{m}_{R}[h]$ is a
simple $R/\mathfrak{m}_{R}[h]$-module. In particular, an irreducible
$p$-Brauer is an irreducible Brauer character and a character table of
$p$-Brauer characters is the same as of Brauer characters.

We will prove that the $p$-Brauer character is additive, where “additive”
means that for any short exact sequence $0\rightarrow M\rightarrow
M^{\prime}\rightarrow M^{\prime\prime}\rightarrow 0$,
$\widetilde{\operatorname{Tr}}_{p}$ satisfies
$\widetilde{\operatorname{Tr}}_{p}(h|M^{\prime})=\widetilde{\operatorname{Tr}}_{p}(h|M)+\widetilde{\operatorname{Tr}}_{p}(h|M^{\prime\prime})$.

######  Lemma 5.3.

The $p$-Brauer character is additive on short exact sequences.

###### Proof.

Let $0\rightarrow
M\xrightarrow{\varphi}M^{\prime}\xrightarrow{\psi}M^{\prime\prime}\rightarrow
0$ be an exact sequence of finite length $R[h]$-modules. Suppose that $M$ has
length $m$ and a composition series $0=M_{0}\subsetneq
M_{1}\subsetneq\dots\subsetneq M_{m}=M$. We use analogous notation for
$M^{\prime}$ and $M^{\prime\prime}$. Then the length $m^{\prime}$ of
$M^{\prime}$ is $m+m^{\prime\prime}$ and

$0=\varphi(M_{0})\subsetneq\varphi(M_{1})\subsetneq\dots\subsetneq\varphi(M_{m})=\psi^{-1}(M^{\prime\prime}_{0})\subsetneq\psi^{-1}(M^{\prime\prime}_{1})\subsetneq\dots\subsetneq\psi^{-1}(M^{\prime\prime}_{m^{\prime\prime}})=M^{\prime}$

is a composition series of $M^{\prime}$. By injectivity of $\varphi$,
$M_{i}\cong\varphi(M_{i})$ for any $i\in\\{0,\dots,m\\}$ and by surjectivity
of $\psi$, $\psi^{-1}(M^{\prime\prime}_{j})/\operatorname{Ker}(\psi)\cong
M^{\prime\prime}_{j}$ for any $j\in\\{0,\dots,m^{\prime\prime}\\}$. Hence, we
have $\varphi(M_{i})/\varphi(M_{i-1})\cong M_{i}/M_{i-1}$ and

$\psi^{-1}(M^{\prime\prime}_{j})/\psi^{-1}(M^{\prime\prime}_{j-1})\cong\big{(}\psi^{-1}(M^{\prime\prime}_{j})/\operatorname{Ker}(\psi)\big{)}/\big{(}\psi^{-1}(M^{\prime\prime}_{j-1})/\operatorname{Ker}(\psi)\big{)}\cong
M^{\prime\prime}_{j}/M^{\prime\prime}_{j-1}.$

Therefore,

$\displaystyle\widetilde{\operatorname{Tr}}_{p}(h|M^{\prime})$
$\displaystyle=\frac{1}{v(p)}\sum_{i=1}^{m}\widetilde{\operatorname{Tr}}\big{(}h|\varphi(M_{i})/\varphi(M_{i-1})\big{)}+\frac{1}{v(p)}\sum_{j=1}^{m^{\prime\prime}}\widetilde{\operatorname{Tr}}\big{(}h|\psi^{-1}(M^{\prime\prime}_{j})/\psi^{-1}(M^{\prime\prime}_{j-1})\big{)}$
$\displaystyle=\frac{1}{v(p)}\sum_{i=1}^{m}\widetilde{\operatorname{Tr}}(h|M_{i}/M_{i-1})+\frac{1}{v(p)}\sum_{j=1}^{m^{\prime\prime}}\widetilde{\operatorname{Tr}}(h|M^{\prime\prime}_{j}/M^{\prime\prime}_{j-1})$
$\displaystyle=\widetilde{\operatorname{Tr}}_{p}(h|M)+\widetilde{\operatorname{Tr}}_{p}(h|M^{\prime\prime}).$
∎

######  Lemma 5.4.

Let $S$ be the ring of integers of a finite extension of
$\operatorname{Frac}(R)$ and $A$ be a finite length $R[h]$-module. Then
$\widetilde{\operatorname{Tr}}_{p}(h|A\otimes_{R}S)=\widetilde{\operatorname{Tr}}_{p}(h|A)$.

###### Proof.

We have that the $p$-Brauer character of $A$ is described the sum of
$p$-Brauer characters of composition factors of $A$ by Definition 5.1. Hence,
it suffices to consider the case $A$ is a simple $R[h]$-module on which $h$
acts as $\zeta_{|h|}^{m}$. Now,

$\widetilde{\operatorname{Tr}}_{p}(h|A)=\frac{1}{v_{R}(p)}\operatorname{Tr}\big{(}h|\widetilde{A}\otimes\operatorname{Frac}(R)\big{)},$

where $v_{R}$ is a discrete valuation on $R$, $\widetilde{A}$ is an $R$-free
$R[h]$-module satisfying
$A\cong\widetilde{A}\otimes_{R[h]}R/\mathfrak{m}_{R}[h]$ and
$\mathfrak{m}_{R}$ is a maximal ideal of $R$. We have

$A\otimes_{R}S\cong\widetilde{A}\otimes_{R[h]}R/\mathfrak{m}_{R}[h]\otimes_{R}S\cong\widetilde{A}\otimes_{R[h]}S/\mathfrak{m}_{R}S[h].$

Let $v_{S}$ be a discrete valuation on $S$ and $\mathfrak{m}_{S}$ be a maximal
ideal of $S$. If $S$ is an unramified extension, then
$\mathfrak{m}_{S}=\mathfrak{m}_{R}S$ and $v_{S}(p)=v_{R}(p)$. Hence,

$\displaystyle\widetilde{\operatorname{Tr}}_{p}(h|A\otimes_{R}S)$
$\displaystyle=\frac{1}{v_{S}(p)}\operatorname{Tr}\big{(}h|\widetilde{A}\otimes_{R}S\otimes\operatorname{Frac}(S)\big{)}$
$\displaystyle=\frac{1}{v_{R}(p)}\operatorname{Tr}\big{(}h|\widetilde{A}\otimes\operatorname{Frac}(R)\big{)}=\widetilde{\operatorname{Tr}}_{p}(h|A).$

If $S$ is a ramified extension, then there is a positive integer $k>1$ s.t.
$\mathfrak{m}_{S}^{k}=\mathfrak{m}_{R}S$ and $v_{S}(p)=kv_{R}(p)$. Hence, we
have $A\otimes_{R}S=\widetilde{A}\otimes_{R[h]}S/\mathfrak{m}_{S}^{k}[h]$ and
a composition series $0\subsetneq\widetilde{A}\otimes
S/\mathfrak{m}_{S}\subsetneq\dots\subsetneq\widetilde{A}\otimes
S/\mathfrak{m}_{S}^{k}$. Its composition factors are all isomorphic to
$\widetilde{A}\otimes_{R[h]}S/\mathfrak{m}_{S}[h]$. Therefore,

$\displaystyle\widetilde{\operatorname{Tr}}_{p}(h|A\otimes_{R}S)$
$\displaystyle=\frac{1}{v_{S}(p)}\sum_{i=1}^{k}\operatorname{Tr}\big{(}h|\widetilde{A}\otimes_{R}S\otimes\operatorname{Frac}(S)\big{)}$
$\displaystyle=\frac{1}{kv_{R}(p)}k\operatorname{Tr}\big{(}h|\widetilde{A}\otimes\operatorname{Frac}(R)\big{)}=\widetilde{\operatorname{Tr}}_{p}(h|A).$
∎

By Proposition 4.1 and Lemma 5.4, any finitely generated $\mathbb{Z}$-free
$\mathbb{Z}[\langle g,h\rangle]$-module base changes to an $R_{p}[\langle
g,h\rangle]$-module that is $R_{p}$-free of finite rank and the $p$-Brauer
character of the Tate cohomology is unchanged. By Lemma 5.3, it suffices to
consider the $p$-Brauer character for the Tate cohomology of $R_{n,m}$.

We will calculate the Tate cohomology of $R_{n,m}$ by using the $p$-Brauer
character. To do this, we use the following lemma:

######  Lemma 5.5 ([5, Chapter III, Lemma 3]).

Let $q=p^{k}$ for a positive integer $k$. Then
$pR=(1-\zeta_{q})^{\varphi(q)}R$, where $\varphi$ is Euler’s totient function.

For a positive integer $s$ and $R=\mathbb{Z}_{p}[\zeta_{p^{s}}]$, a discrete
valuation $v(p)$ is given by $\varphi(p^{s})$ and maximal ideal of $R$ is
$(1-\zeta_{p^{s}})R$.

######  Lemma 5.6.

Suppose that the prime factorization of $N$ is $\prod p_{i}^{n_{i}}$. Then,

$\widetilde{\operatorname{Tr}}_{p_{i}}(h|\hat{H}^{*}(g,R_{n,m}))=\begin{cases}n_{i}\zeta_{|h|}^{m},&n=0,\vspace{1mm}\\\
\dfrac{-\zeta_{|h|}^{m}}{\varphi\big{(}p_{i}^{l}\big{)}},&|g^{n}|=p_{i}^{l},\
1\leq l\leq n_{i},\vspace{1mm}\\\ 0,&otherwise.\end{cases}.$

###### Proof.

Now, $h$ acts as $\zeta_{|h|}^{m}$ on $\hat{H}^{*}(g,R_{n,m})$ and
$v(p)=\varphi\big{(}p_{i}^{n_{i}}\big{)}$. By

$\widetilde{\operatorname{Tr}}_{p_{i}}\big{(}h|\hat{H}^{*}(g,R_{n,m})\big{)}=\widetilde{\operatorname{Tr}}_{p_{i}}\big{(}h|\hat{H}^{0}(g,R_{n,m})\big{)}-\widetilde{\operatorname{Tr}}_{p_{i}}\big{(}h|\hat{H}^{1}(g,R_{n,m})\big{)}$

and Lemma 4.5, it suffices to calculate the $p_{i}$-Brauer characters of
$R_{p_{i}}/NR_{p_{i}}$ and $R_{p_{i}}/(1-\zeta_{N}^{n})R_{p_{i}}$.

When $n=0$, by Lemmas 4.5 and 5.5,

$R_{p_{i}}/NR_{p_{i}}\cong R_{p_{i}}/p_{i}^{n_{i}}R_{p_{i}}\cong
R_{p_{i}}/\big{(}1-\zeta_{p_{i}^{n_{i}}}\big{)}^{n_{i}\varphi(p_{i}^{n_{i}})}R_{p_{i}}.$

This module has a composition series

$0\subsetneq
R_{p_{i}}/(1-\zeta_{{p_{i}}^{n_{i}}})R_{p_{i}}\subsetneq\dots\subsetneq
R_{p_{i}}/\big{(}1-\zeta_{{p_{i}}^{n_{i}}}\big{)}^{n_{i}\varphi(p_{i}^{n_{i}})}R_{p_{i}}$

and its composition factors are all isomorphic to
$R_{p_{i}}/\big{(}1-\zeta_{{p_{i}}^{n_{i}}}\big{)}R_{p_{i}}$. Hence,

$\widetilde{\operatorname{Tr}}_{p_{i}}(h|R_{p_{i}}/NR_{p_{i}})=n_{i}\zeta_{|h|}^{m}.$

When $|g^{n}|=p_{i}^{l}$, $\zeta_{N}^{n}$ is a $p_{i}^{l}$-th primitive root
of unity. By Lemma 5.5, we have

$\big{(}1-\zeta_{p_{i}^{l}}\big{)}^{\varphi(p_{i}^{l})}=\big{(}1-\zeta_{p_{i}^{n_{i}}}\big{)}^{\varphi(p_{i}^{n_{i}})}.$

Hence, by Lemma 4.5, we have

$R_{p_{i}}/\big{(}1-\zeta_{{p_{i}}^{l}}\big{)}R_{p_{i}}\cong
R_{p_{i}}/\big{(}1-\zeta_{{p_{i}}^{n_{i}}}\big{)}^{\varphi(p_{i}^{n_{i}})/\varphi(p_{i}^{l})}R_{p_{i}}.$

This module has the length
$\varphi\big{(}p_{i}^{n_{i}}\big{)}/\varphi\big{(}p_{i}^{l}\big{)}$ and its
composition factors are all isomorphic to
$R_{p_{i}}/\big{(}1-\zeta_{{p_{i}}^{n_{i}}}\big{)}R_{p_{i}}$. Calculating the
$p_{i}$-Brauer character, we have

$\widetilde{\operatorname{Tr}}_{p_{i}}\big{(}h|R_{p_{i}}/\big{(}1-\zeta_{p_{i}^{l}}\big{)}R_{p_{i}}\big{)}=1/\varphi\big{(}p_{i}^{l}\big{)}.$

When $|g^{n}|=p_{i}^{k}q$, where $q\neq 1$, $(q,p_{i})=1$ and $0\leq k\leq
n_{i}$, $\zeta_{N}^{n}$ is a $p_{i}^{k}q$-th primitive root of unity. Let
$\pi$ be a surjective homomorphism
$R_{p_{i}}\rightarrow\mathbb{F}_{p_{i}}\big{[}\zeta_{N|h|/p_{i}^{n_{i}}}\big{]}$;
$\sum_{j=0}^{\infty}c_{j}p_{i}^{j}\mapsto c_{0}$,
$\zeta_{N|h|}\mapsto\zeta_{N|h|/p_{i}^{n_{i}}}$. Then
$\operatorname{Ker}(\pi)$ is the maximal ideal of $R_{p_{i}}$ and
$\operatorname{Frob}_{p_{i}}$ on
$\mathbb{F}_{p_{i}}[\zeta_{N|h|/p_{i}^{n_{i}}}]$ is an automorphism. We have
$\pi\big{(}\zeta_{p_{i}^{k}q}\big{)}=\operatorname{Frob}_{p_{i}}^{-k}\big{(}\zeta_{p_{i}^{k}q}^{p_{i}^{k}}\big{)}\neq
1$. Hence, $1-\zeta_{N}^{n}\notin\operatorname{Ker}(\pi)$. $1-\zeta_{N}^{n}$
is a unit of $R_{p_{i}}$ because every element of a local ring not included in
a maximal ideal is a unit. Therefore,
$R_{p}/\big{(}1-\zeta_{N}^{n}\big{)}R_{p}=0$. ∎

We generalize Proposition 3.3 [3, Proposition 2.2] using the $p$-Brauer
character.

######  Theorem 5.7.

Let $G$ be a finite group. Suppose that $g\in\operatorname{Cent}(G)$ has order
$N$ and that $h\in G$ is an $N$-regular element. Let $A$ be a finitely
generated $R$-free $R[\langle g,h\rangle]$-module. Then
$\hat{H}^{*}(g,A)=\hat{H}^{0}(g,A)-\hat{H}^{1}(g,A)$ is a virtual
representation of $\langle h\rangle$, and

$\displaystyle\widetilde{\operatorname{Tr}}_{p_{i}}(h|\hat{H}^{*}(g,A))=\sum_{k=1}^{N-1}a_{k,p_{i}}\operatorname{Tr}\big{(}g^{k}h|A\big{)},$
$\displaystyle
a_{k,p_{i}}=\begin{cases}\dfrac{1}{\sum_{(p_{i},d)=1,d|N}\varphi(N/d)}\left(n_{i}-l-\dfrac{n_{i}-l-1}{p_{i}}\right),&\big{(}k,p_{i}^{n_{i}}\big{)}=p_{i}^{l},\
0\leq l\leq n_{i}-1,\\\
0,&\big{(}k,p_{i}^{n_{i}}\big{)}=p_{i}^{n_{i}},\end{cases}$

where the prime factorization of $N$ is $\prod p_{i}^{n_{i}}$.

###### Proof.

Suppose

$\widetilde{\operatorname{Tr}}_{p_{i}}\big{(}h|\hat{H}^{*}(g,A)\big{)}=\sum_{k=1}^{N-1}a_{k,p_{i}}\operatorname{Tr}\big{(}g^{k}h|A\big{)}.$

It suffices to take $A=R_{n,m}$. By an inverse discrete Fourier transform and
Lemma 5.6, we have

$\displaystyle a_{k,p_{i}}$
$\displaystyle=\frac{1}{N}\sum_{b=0}^{N-1}\widetilde{\operatorname{Tr}}_{p_{i}}\big{(}h|\hat{H}^{*}(g,R_{b,m})\big{)}\overline{\operatorname{Tr}\big{(}g^{k}h|R_{b,m}\big{)}}$
$\displaystyle=\frac{1}{N}\Bigg{(}n_{i}+\sum_{|g^{b}|=p_{i}}\frac{-1}{\varphi(p_{i})}\zeta_{N}^{-kb}+\sum_{|g^{b}|=p_{i}^{2}}\frac{-1}{\varphi\big{(}p_{i}^{2}\big{)}}\zeta_{N}^{-kb}+\dots+\sum_{|g^{b}|=p_{i}^{n_{i}}}\frac{-1}{\varphi\big{(}p_{i}^{n_{i}}\big{)}}\zeta_{N}^{-kb}\Bigg{)}.$

Let $\big{(}k,p_{i}^{n_{i}}\big{)}=p_{i}^{l}$. If $s>l$, then
$\sum_{|g^{b}|=p_{i}^{s}}\zeta_{N}^{-kb}$ is a sum of primitive
$p_{i}^{s-l}$-th roots of unity and equals an integer multiple of the Möbius
function $\mu\big{(}p_{i}^{s-l}\big{)}$. If $s\leq l$, then we have

$\sum_{|g^{b}|=p_{i}^{s}}\zeta_{N}^{-kb}=\varphi(p_{i}^{s}).$

Hence, when $\big{(}k,p_{i}^{n_{i}}\big{)}=p_{i}^{n_{i}}$, we have

$a_{k,p_{i}}=\frac{1}{N}(n_{i}-1-\dots-1)=0.$

When $\big{(}k,p_{i}^{n_{i}}\big{)}=p_{i}^{l}$, because
$\big{(}N/p_{i}^{n_{i}}\big{)}=\sum_{d|(N/p_{i}^{n_{i}})}\varphi\big{(}N/p_{i}^{n_{i}}d\big{)}$,
we have

$\displaystyle a_{k,p_{i}}$
$\displaystyle=\frac{1}{N}\Bigg{(}n_{i}-1-\dots-1+\frac{p_{i}^{l}}{\varphi\big{(}p_{i}^{l+1}\big{)}}\Bigg{)}$
$\displaystyle=\frac{(n_{i}-l)\varphi\big{(}p_{i}^{l+1}\big{)}+p_{i}^{l}}{N\varphi\big{(}p_{i}^{l+1}\big{)}}=\frac{p_{i}^{l+1}(n_{i}-l)-p_{i}^{l}(n_{i}-l-1)}{p_{i}^{l+1}\varphi\big{(}p_{i}^{n_{i}}\big{)}\big{(}N/p_{i}^{n_{i}}\big{)}}$
$\displaystyle=\frac{1}{\sum_{(p_{i},d)=1,d|N}\varphi(N/d)}\bigg{(}n_{i}-l-\frac{n_{i}-l-1}{p_{i}}\bigg{)}.$
∎

######  Corollary 5.8.

Suppose that $g$ is an element of $\mathbb{M}$ of order $N=\prod
p_{i}^{n_{i}}$. Let $V$ be an integral form of the monster vertex algebra. If
$h\in C_{\mathbb{M}}(g)$ is $N$-regular, then
$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{p_{i}}\big{(}h|\hat{H}^{*}(g,V_{n}\otimes\mathbb{Z}_{p_{i}})\big{)}q^{n-1}$
is a linear combination of Hauptmoduls and is denoted by $\sum_{d|N,d\neq
N}a_{d,p_{i}}\varphi(N/d)T_{g^{d}h}(\tau).$

###### Proof.

By Theorem 5.7 and the fact that McKay–Thompson series depend only on
conjugacy classes. ∎

For example, we consider a case that $g\in\mathbb{M}$ has order $N=15=3\cdot
5$. Then by using the 3-Brauer character, we have

$\displaystyle
a_{1,3}=\frac{1}{\varphi(15)+\varphi(15/5)}\left(1-0-\frac{1-0-1}{3}\right)=\frac{1}{10},\qquad
a_{3,3}=0,$ $\displaystyle
a_{5,3}=\frac{1}{\varphi(15)+\varphi(15/5)}\left(1-0-\frac{1-0-1}{3}\right)=\frac{1}{10}.$

Hence, for any 15-regular element $h\in C_{\mathbb{M}}(g)$, we have

$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{3}\big{(}h|\hat{H}^{*}(g,V_{n}\otimes\mathbb{Z}_{3})\big{)}q^{n-1}=\frac{4}{5}T_{gh}(\tau)+\frac{1}{5}T_{g^{5}h}(\tau).$

Similarly, by using the 5-Brauer character, we have

$\displaystyle
a_{1,5}=\frac{1}{\varphi(15)+\varphi(15/3)}\left(1-0-\frac{1-0-1}{5}\right)=\frac{1}{12},$
$\displaystyle
a_{3,5}=\frac{1}{\varphi(15)+\varphi(15/3)}\left(1-0-\frac{1-0-1}{5}\right)=\frac{1}{12},\qquad
a_{5,5}=0.$

Hence, for any 15-regular element $h\in C_{\mathbb{M}}(g)$, we have

$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{5}\big{(}h|\hat{H}^{*}(g,V_{n}\otimes\mathbb{Z}_{5})\big{)}q^{n-1}=\frac{2}{3}T_{gh}(\tau)+\frac{1}{3}T_{g^{3}h}(\tau).$

In particular, for $g\in 15A$, we have the following relation from Theorem
4.4:

$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{3}\big{(}h|\hat{H}^{0}(g,V_{n}\otimes\mathbb{Z}_{3})\big{)}q^{n-1}=\frac{4}{5}T_{gh}(\tau)+\frac{1}{5}T_{g^{5}h}(\tau),$
$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{3}\big{(}h|\hat{H}^{1}(g,V_{n}\otimes\mathbb{Z}_{3})\big{)}q^{n-1}=0,$
$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{5}\big{(}h|\hat{H}^{0}(g,V_{n}\otimes\mathbb{Z}_{5})\big{)}q^{n-1}=\frac{2}{3}T_{gh}(\tau)+\frac{1}{3}T_{g^{3}h}(\tau),$
$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{5}\big{(}h|\hat{H}^{1}(g,V_{n}\otimes\mathbb{Z}_{5})\big{)}q^{n-1}=0.$

By a similar calculation, for $g\in 21A$ and any 21-regular element $h\in
C_{\mathbb{M}}(g)$, we have the following relation from Theorem 4.4:

$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{3}\big{(}h|\hat{H}^{0}(g,V_{n}\otimes\mathbb{Z}_{3})\big{)}q^{n-1}=\frac{6}{7}T_{gh}(\tau)+\frac{1}{7}T_{g^{7}h}(\tau),$
$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{3}\big{(}h|\hat{H}^{1}(g,V_{n}\otimes\mathbb{Z}_{3})\big{)}q^{n-1}=0,$
$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{7}\big{(}h|\hat{H}^{0}(g,V_{n}\otimes\mathbb{Z}_{7})\big{)}q^{n-1}=\frac{2}{3}T_{gh}(\tau)+\frac{1}{3}T_{g^{3}h}(\tau),$
$\displaystyle\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{7}\big{(}h|\hat{H}^{1}(g,V_{n}\otimes\mathbb{Z}_{7})\big{)}q^{n-1}=0.$

We propose the following conjecture about the graded $p$-Brauer character for
Tate cohomology.

######  Conjecture 5.9.

For any element $g\in\mathbb{M}$ of order $N$ with prime factor $p$ and
$N$-regular element $h\in C_{\mathbb{M}}(g)$,
$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{p}\big{(}h|\hat{H}^{i}(g,V_{n}\otimes\mathbb{Z}_{p})\big{)}q^{n-1}$,
$i=0,1$, is a linear combination of Hauptmoduls.

If we have that the series

$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{p}\big{(}h|\hat{H}^{0}(g,V_{n}\otimes\mathbb{Z}_{p})\big{)}q^{n-1}+\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{p}\big{(}h|\hat{H}^{1}(g,V_{n}\otimes\mathbb{Z}_{p})\big{)}q^{n-1}$

is a linear combination of Hauptmoduls, then we describe clearly

$\sum_{n\in\mathbb{Z}}\widetilde{\operatorname{Tr}}_{p}\big{(}h|\hat{H}^{i}(g,V_{n}\otimes\mathbb{Z}_{p})\big{)}q^{n-1},\qquad
i=0,1,$

by a linear combination of Hauptmoduls.

If conditions 2, 4 and 7 in Borcherds’s conjecture at the end of [2] are true,
then $\hat{H}^{1}(g,V)$ vanishes for all Fricke elements $g$. From Corollary
5.8, we can give a counterexample to this conjecture of Borcherds about
vanishing of Tate cohomology $\hat{H}^{1}(g,V)$.

######  Theorem 5.10.

Let $V$ be an integral form of the monster vertex algebra. Then there exists a
Fricke element $g$ s.t. $\hat{H}^{1}(g,V)\neq 0$.

###### Proof.

Let $g$ be an element of the monster in class $8A$. Then $g$ is a Fricke
element. By Lemma 4.2 and Corollary 5.8, we have

$\displaystyle\sum_{n\in\mathbb{Z}}\operatorname{length}_{2}\big{(}\hat{H}^{*}(g,V_{n})\big{)}q^{n-1}$
$\displaystyle=a_{1,2}\varphi(8)T_{g}(\tau)+a_{2,2}\varphi(4)T_{g^{2}}(\tau)+a_{4,2}\varphi(2)T_{g^{4}}(\tau)$
$\displaystyle=2T_{8A}(\tau)+\frac{3}{4}T_{4C}(\tau)+\frac{1}{4}T_{2B}(\tau),$

where $\operatorname{length}_{2}(\cdot)$ means length of a composition series
of the $\mathbb{Z}_{2}$-module. Comparing coefficients of $q^{2}$, we have

$\operatorname{length}_{2}\big{(}\hat{H}^{0}(g,V_{3})\big{)}-\operatorname{length}_{2}\big{(}\hat{H}^{1}(g,V_{3})\big{)}=-256.$

This means $\hat{H}^{1}(g,V_{3})\neq 0$. Hence, $\hat{H}^{1}(g,V)\neq 0$. ∎

By Corollary 5.8, if $g$ is a Fricke element such that $g^{j}$ is a Fricke
element for any $j>0$, then we have that all coefficients of
$\sum_{n\in\mathbb{Z}}\operatorname{length}_{p_{i}}\big{(}\hat{H}^{*}(g,V_{n}\otimes\mathbb{Z}_{p_{i}})\big{)}q^{n-1}$
are positive, where $\operatorname{length}_{p_{i}}(\cdot)$ means length of a
composition series of the $\mathbb{Z}_{p_{i}}$-module. Hence, we propose a
weaker version of Borcherds’s conjecture about vanishing of Tate cohomology
$\hat{H}^{1}(g,V)$.

######  Conjecture 5.11.

Let $V$ be an self-dual integral form of the monster vertex algebra. If $g$ is
a Fricke element such that $g^{j}$ is a Fricke element for any $j>0$, then
$\hat{H}^{1}(g,V)=0$.

Suppose that $g$ has order $N$ and that the prime factorization of $N$ is
$\prod p_{i}^{n_{i}}$. If we have that

$\operatorname{length}_{p_{i}}\big{(}\hat{H}^{0}(g,V_{n}\otimes\mathbb{Z}_{p_{i}})\big{)}-\operatorname{length}_{p_{i}}\big{(}\hat{H}^{1}(g,V_{n}\otimes\mathbb{Z}_{p_{i}})\big{)}$

is equal to

$\operatorname{length}_{p_{i}}\big{(}\hat{H}^{0}(g,V_{n}\otimes\mathbb{Z}_{p_{i}})\big{)}+\operatorname{length}_{p_{i}}\big{(}\hat{H}^{1}(g,V_{n}\otimes\mathbb{Z}_{p_{i}})\big{)}$

for any $p_{i}$ and $n\in\mathbb{Z}$, then it will follow that
$\hat{H}^{1}(g,V)=0$.

### Acknowledgements

I would like to thank Scott Carnahan for many helpful comments and advice. I
would also like to thank the anonymous referees for many helpful comments.

## References

  * [1] Borcherds R.E., Monstrous moonshine and monstrous Lie superalgebras, Invent. Math. 109 (1992), 405–444.
  * [2] Borcherds R.E., Modular moonshine. III, Duke Math. J. 93 (1998), 129–154, arXiv:math.AG/9801101.
  * [3] Borcherds R.E., Ryba A.J.E., Modular Moonshine. II, Duke Math. J. 83 (1996), 435–459.
  * [4] Carnahan S., A self-dual integral form of the Moonshine module, SIGMA 15 (2019), 030, 36 pages, arXiv:1710.00737.
  * [5] Cassels J.W.S., Fröhlich A. (Editors), Algebraic number theory, Academic Press, London, Thompson Book Co., Inc., Washington, D.C., 1967.
  * [6] Conway J.H., Norton S.P., Monstrous moonshine, Bull. London Math. Soc. 11 (1979), 308–339.
  * [7] Frenkel I., Lepowsky J., Meurman A., Vertex operator algebras and the Monster, Pure and Applied Mathematics, Vol. 134, Academic Press, Inc., Boston, MA, 1988.


The Reasonable Effectiveness of Mathematics in the Natural Sciences
Alex Harveya)
Visiting Scholar

New York University

New York, NY 10003

###### Abstract

Mathematics and its relation to the physical universe have been the topic of
speculation since the days of Pythagoras. Several different views of the
nature of mathematics have been considered: Realism \- mathematics exists and
is discovered; Logicism \- all mathematics may be deduced through pure logic;
Formalism: mathematics is just the manipulation of formulas and rules invented
for the purpose; Intuitionism: mathematics comprises mental constructs
governed by self evident rules. The debate among the several schools has major
importance in understanding what Eugene Wigner called, “The Unreasonable
Effectiveness of Mathematics in the Natural Sciences.” In return, this
‘Unreasonable Effectiveness’ suggests a possible resolution of the debate in
favor of Realism. The crucial element is the extraordinary predictive capacity
of mathematical structures descriptive of physical theories.111It is a
pleasure to dedicate this paper to Josh Goldberg, long a valued member of the
community of Generl Relativists and a good friend.

PACS: 0170w *43.10.10Mq

##  1 Introduction

In an essay which appeared in his collection, Symmetries and Reflections,
Wigner [1] explored the connection between mathematics and science. He
wondered that,

> “$\dots$the enormous usefulness of mathematics in the natural sciences is
> something bordering on the mysterious and that there is no rational
> explanation for this.”

And, later,

> “$\ldots$ the mathematical formulation of the physicist’s often crude
> experience leads in an uncanny number of cases to an amazingly accurate
> description of a large class of phenomena.”

A similar view was expressed by Bochner222It is noteworthy that Wigner was a
Nobel Laureate in physics and Bochner, a distinguished mathematician. The
talent for the two skills does not seem to be the same. Only a few gifted
individuals such as Archimedes, Newton, or Gauss had supreme talents in both.
This difference in talents can be observed in college upperclassmen. Expert
knowledge and superior skill in the use of mathematics is the hallmark of the
gifted physicist. [2]

> “What makes mathematics so effective when it enters science is a mystery of
> mysteries and the present book wants to achieve no more than explicate how
> deep this mystery is.”

And again, in Chapter 22 of the London Philosophical Study Guide [3].

> “Mathematical reality is in itself mysterious: how can it be highly abstract
> and yet applicable to the physical world? How can mathematical theorems be
> necessary truths about an unchanging realm of abstract entities and at the
> same time so useful in dealing with the contingent, variable and inexact
> happenings evident to the senses?”

None mention the central mystery, which is the ability of an appropriate
formulation to predict phenomena yet undiscovered. Indeed, as often noted, the
principal problem in progress in physics is to find this formulation. It is
our purpose to explore this “mysterious” relation between physical theories
and their mathematical formulation and suggest an answer. Firstly an
understanding of mathematics is essential.

##  2 Mathematics

Mathematics arose from that most fundamental of mathematical activities -
counting objects. In the most ancient of times, a person might enumerate
objects of the same kind - say goats or sheaves of wheat and if he wanted to
record the results, would have to invent a system of symbols to identify and
differentiate two quantites. We can be sure that he would use the same symbol
system to record different quantities of apples. He would quickly learn to add
two collections of objects. This could well have happened anywhere groups of
people progressed beyond being hunter-gatherers.

This simple skill evolved to the sophisticated arithmetic of the Babylonians
[4]. They exhibited great ingenuity in their ability to use tables of squares
to solve various arithmetic problems and could solve quadratic and cubic
equations. Similarly, a direct descent for spatial relations can be discerned.
The Pharaonic Egyptians devised an elementary geometry for purposes of
resurveying fields after the recession of the annual Nile floodwaters. This
skill was elaborated to the extent that construction of temples and pyramids
became possible. To accomplish all this, a unit of length was defined. In this
way geometry was conflated with arithmetic. Despite their achievements,
neither the Babylonians nor the Egyptians considered their mathematics to be
more than a computational scheme. Their mathematics was devoted primarily to
mensuration, computing taxes, keeping accounts and making astronomical
calculations. They had no concepts of theorems or proofs. This was left to the
Greeks.

The genius of the Greeks, beginning with the Pythagorean school was to discern
an abstract coherent structure beneath the arithmetic of the Babylonians and
Egyptians. They introduced the concept of theorem and proof. In this way, from
the simple act of counting and the drawing of geometric figures, an elaborate
system of mathematics was constructed in Classical Greece.

The Pythagoreans [5] left no written record of their accomplishments. These
are known largely through the writings of later Greek philosophers [6]. They
were besotted with number per se effectively founding number theory. But, it
was their belief in the esoteric character of mathematics and its relation to
the physical universe which is of interest here. They believed that all that
could be known of the physical universe would be known through number. Of the
Pythagoreans, Aristotle commented [7],

> “Contemporaneously with these philosophers and before them, the so-called
> Pythagoreans, who were the first to take up mathematics, not only advanced
> this study, but also having been brought up in it they thought its
> principles were the principles of all things. Since of these principles
> numbers are by nature the first, and in numbers they seemed to see many
> resemblances to the things that be modeled on numbers, and numbers seemed to
> be the first things in the whole of nature, they supposed the elements of
> numbers to be the elements of all things, and they exist and come into being
> - more than in fire and earth and water $\ldots$ all other things seemed in
> their whole nature to hold heaven to be a musical scale and a number.”

Many centuries later, Galileo [8] put it more precisely (and famously).

> “Philosophy is written in this grand book–I mean the universe–which stands
> continually open to our gaze, but it cannot be understood unless one first
> learns to comprehend the language and interpret the characters in which it
> is written. It is written in the language of mathematics, and its characters
> are triangles, circles, and other geometrical figures, without which it is
> humanly impossible to understand a single word of it; without these, one is
> wandering around in a dark labyrinth.”

Whether his view is Pythagorean, that is, does he imply an identity between
the mode of description and the object described or that the mode and object
are distinct, is not clear.

A similar sentiment, with clearer distinction, was expressed some three
centuries later by Weyl [9] in connection with the “language” appropriate for
application to gravitation theory.

> “As everyone has to work hard learning language and writing before he can
> use them freely for expressing his thoughts, so here too the only way to
> shift the load of formulas from one’s own shoulders is to be well acquainted
> with tensor analysis so that one can turn unimpeded by formalities to the
> true problems: Insight into the nature of space, time and matter insofar as
> they contribute to the construction of the objective reality.”

##  3 Nature of Mathematics

For the scientist, mathematics is a tool; for the mathematician, it is an end
in itself; for the philosopher, the philosophy of mathematics is, as Körner
[10] puts it,

> “… not mathematics. It is a reflection on mathematics.”

The nature of mathematics has been debated by philosophers333For a concise
introduction and useful bibliography, see the Wikipedia article, “The
Philosophy of Mathematics”. For a detailed introduction and discussion,
consult Körner’s [11]“The Philosophy of Mathematics”. beginning with the
Pythagoreans. They considered numbers to be inextricably connected with the
physical world. Plato refined Pythagorean ideas to a scheme of ideal abstract
structures mirrored in the immediate world as discovered through human
activity. Subsequent modifications led to Realism in which the severe concept
of ideals is modified. Simply put, mathematical structures have an independent
existence and are uncovered by observation and ratiocination. This view has
been held by many mathematicians such as Paul Erdös who often referred to an
imaginary “Book” in which God had written down all the “beautiful”
mathematical proofs. This was expressed more picturesquely by Everett [12]

> “In the pure mathematics we contemplate absolute truths which existed in the
> divine mind before the morning stars sang together, and which will continue
> to exist there when the last of their host shall have fallen from heaven.”

A more temperate exposition of the same view was expressed by the
distinguished British mathematician, Hardy [13],

> “I have myself always thought of a mathematician in the first instance as an
> observer, who gazes at a distant range of mountains and writes down his
> observations.”

In the late 19th Century, a radically new approach by Frege [14], subsequently
termed Logicism claimed that all mathematics was just a logical structure.

> “Arithmetic thus becomes simply a development of logic, and every
> proposition of arithmetic a law of logic albeit a derivative one. To apply
> arithmetic in the physical sciences is to bring logic to bear on observed
> facts; calculation becomes deduction. The laws of number will not, $\ldots$,
> need to stand up to practical tests if they are to be applicable to the
> external world; for in the external world, in the whole of space and all
> that therein is, there are no concepts, no properties of concepts, no
> numbers. The laws of number, therefore, are not really applicable to
> external things; they are not laws of nature.”

A modified view was earlier given by Kronecker [15].

> “God created the numbers, all the rest is the work of man.”

This avoids the extreme difficulty of establishing the system of numbers by
purely logical arguments encountered later by Whitehead and Russell in their
Principia Mathematica [16]. They do not establish that $1+1=2$ until page 362
of Volume 1.

The concept of Formalism is due to Hilbert who described it simply as444Though
widely quoted, the attribution of this comment to Hilbert is apocryphal. [17].

> “Mathematics is a game played according to certain simple rules with
> meaningless marks on paper.”

Wigner, early in his essay, in the section titled What is Mathematics?,
provides the same view of mathematics.

> “$\ldots$ mathematics is the science of skillful operations with concepts
> and rules invented for just this purpose.”

Intuitionism, created by the Dutch mathematician Brouwer [18], proposes that
mathematical objects are mental constructions governed by self-evident laws.

In a class by itself is the school of thought that considers the brain to have
been hard-wired with the number concept at birth555For an example of this
approach see Dahaene [19].

The various schools of thought have enabled intense debates among philosophers
of science and mathematics. However, a conclusive argument strongly in favor
of Realism can be constructed out the interaction of mathematics and physics,
This will be described below.

##  4 The Practice of Physics

The physicist attempts to understand physical universe. The endeavor is
enormously facilitated through employment of Galileo’s “language of
mathematics”. The use of mathematics entails representing each physical entity
of interest with a mathematical object. This is done by mapping these entities
uniquely onto a mathematical concept. A velocity is a polar vector; a moment
is an axial vector; time is a one-dimensional manifold; flat space-time is a
four-dimensional manifold with a Minkowski metric. The process is a species of
conceptual isomorphism. It can’t be a homeomorphism if ambiguity is to be
avoided. Once this mapping has been completed, the mathematics takes over
completely; it has a life of its own. The rules of the particular mathematical
structure govern. There is no longer any physics per se. At the end of the
calculation, the results are mapped back onto the physical universe.

At intermediate steps it should be possible to map the mathematics back onto
the physics. That is, it should be possible to interpret every mathematical
symbol with a physical construct. Dirac [20] expressed it this way,

> “The new scheme [the representation of quantum states and variables] becomes
> a precise physical theory when all the axioms and rules of manipulation
> governing the mathematical quantities are specified and when in addition
> certain laws are laid down connecting physical facts with the mathematical
> formalism, so that from any given physical conditions equations between the
> mathematical quantities may be inferred and vice versa. In an application of
> the theory one would be given certain physical information, which one would
> proceed to express by equations between the mathematical quantities. One
> would then deduce new equations with the help of the axioms and rules of
> manipulation and would conclude by interpreting these new equations as
> physical conditions.”

The “Unreasonable Effectiveness” is two-fold. Firstly, there is the incredible
richness of phenomena which can be accurately described by a mathematical
structure. Maxwell’s equations furnish the means to accurately describe
electromagnetic phenomena from power transformers to radio transmission.
Einstein’s equations of general relativity provide the basis for calculating
gravitational phenomena from remarkably accurate solar system ephemerides to
the gravitational radiation from pulsar PSR1913+16. The wealth of accurate
description of atomic phenomena by the Schrödinger equation needs no
elaboration. This brief list could be extended dramatically.

Secondly, and of decisive importance, the mathematics may describe phenomena
completely unforeseen when the problem was originally formulated. think of the
Dirac equation for the electron. What Dirac [21] said about it later is
precisely to the point.

> “It was found that this equation gave the particle a spin of half a quantum.
> And also gave it a magnetic moment. It gave just the properties that one
> needed for an electron. That was really an unexpected bonus for me,
> completely unexpected.”

[Emphasis added] What he did not know at the time he published the equation
[1927] was that, quite remarkably, it also described the positron which had
yet to be discovered [1932].

The Schwarzschild singularity in the solution of the Einstein equations for an
isolated point mass is another example. So was the yet to be observed radio
waves in Maxwell’s equations. Physics is full of these surprises. A less well
known far more simple example is that of the exact solution of the simple
pendulum. This is an elliptic integral [22], the real value of which
corresponds to the given initial conditions. The imaginary period corresponds
to a reversed gravitational field and an initial displacement of the bob which
is supplementary to that in the original problem. The latter is an example of
the principle that every prediction of a mathematical formulation of a
physical has a “real” significance.

##  5 Reflections

The practice of physics is based on the concept that the physical universe,
locally and globally, evolves in accordance with fixed rules. This is attested
to by the consistent ability of independent observers to replicate each others
results. It is this regularity which admits the use of mathematics to describe
these results.

All of mathematics starts from a set of integers used for counting. The
historical record shows that the ability to count evolved in different areas
independently. The Babylonian and Mayan civilizations, among others, developed
this skill. Contrarily, some cultures developed counting no further than ”one,
two, many.” This suggests an objective independence of the integers and that
they were discovered rather than created or, as suggested, that the number
concept is hard-wired in the brain [19]. Further, evidence provided by
astrophysical observations implies that the laws of physics are invariant with
respect to location and independent of era. This implies that the mathematical
models created by extraterrestrials (if such exist) are identical, modulo
symbology, to those here on earth. In turn, these imply the universal common
existence of mathematics.

It was noted earlier that the relation between mathematics and physical
phenomena is a species of isomorphism. The phenomena is mapped uniquely onto a
mathematical structure and then mapped in the reverse direction. The extent to
which the initial choice of mathematical structure is an accurate mapping is
determined by how well the evolving physical phenomena remains isomorphic to
the mathematics The relationship between mathematics and phenomena is
reciprocal; each is a representation of the other. It is in this sense that
mathematics has an objective reality. The problem of the physicist is to find
that representation.

The identification of mathematics as objectively existent entails an
interesting problem. Hawking [23] has noted that mathematical structures are
subject to the incompleteness theorems of Gödel:

> “… we and our models are both part of the universe we are describing. Thus a
> physical theory is self referencing, … . One might therefore expect it to be
> either inconsistent or incomplete. … The theories we have so far are both
> inconsistent and incomplete.”

This is certainly correct but only of theoretical interest. There has yet to
constructed mathematical models which are precisely descriptive of the
phenomena we do observe. Despite an incredible accuracy of ten parts in a
billion, the present formulation of quantum electrodynamics is notoriously
lacking in rigor [24]; the “standard model”, however accurate its predictions,
is substantially more a protocol than a theory; a quantized version of the
theory of general relativity has yet to be found. In brief, there is nothing
yet in the mathematical models of observed physical phenomena to which Gödel’s
theorems might be applied. Moreover, our theories are certainly incomplete and
will always remain so. With respect to the cosmos this is obvious. The
strictures imposed by the principle of special relativity limits what
information we can collect to the interior of the past light-cone. That is
finite in extent, some 13+ billion years at the present time. The best that a
physicist may hope for is a model which is both a) accurate in description b)
possessed of predictive capability and c) mathematically rigorous. While the
success of (a) and (b) have been substantial, this is not the case for (c).
The approach to a final, mathematically rigorous “theory of everything” is at
best, asymptotic. Thus, for the finite future, Gödel’s theorem is not a
problem.

Physicists do not (with rare exception such as Newton’s fluxions, the
aforementioned Dirac delta function, or Heaviside’s operational calculus)
create any of the mathematics they utilize. The elaboration of existing and
discovery of new mathematics is mostly the province of mathematicians. We thus
have a very large apparatus of which only part is presently used by
physicists. This suggested the question [25], “Does any piece of mathematics
exist for which there is no application whatsoever in physics?” The question
has been addressed several times, most recently in 2001 [26] [27]. The answers
were uniformly no. This, though suggestive, is not conclusive.

If either Formalism or Constructivism were accurate characterizations of
mathematics then every mathematician could construct his own universe. There
could be as many different universes as there were mathematicians engaged in
the enterprise. The consequences of this are not easily described. The
decisive evidence that Realism is the proper characterization is the
predictive capability of the mathematical structures in which physical
theories are couched. This strongly infers that mathematics and the reality it
describes are both part of the objective universe and await discovery.

##  6 Acknowledgments

Thanks: to (the late) Professor Arthur Komar for stimulating conversations; to
Professor Hilail Gildin for useful information on the philosophy of
mathematics and comments on the manuscript; and especially to Professor
Engelbert Schucking who not only provided the translation of the quotation of
Weyl, page 4, above but also read the manuscript critically.
a)Professor Emeritus, Queens College, City University of New York,
<EMAIL_ADDRESS>

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