%This is a plain tex paper. \def\Aut{{\rm Aut}} \def\Im{{\rm Im}} \def\Ker{{\rm Ker}} \def\F{{\bf F}} \def\H{{\hat H}} \def\m{{\bf m}} \def\R{{\bf R}} \def\Tr{{\rm Tr}} \def\Z{{\bf Z}} \def\Q{{\bf Q}} \vbadness=10000 \hbadness=10000 \tolerance=10000 \proclaim Modular Moonshine III. \hfill 11 October 1994, corrected 18 Nov 1997 Duke Math. J. 93 (1998), no. 1, 129--154. Richard E. Borcherds,\footnote{$^*$}{ Supported by NSF grant DMS-9401186 and by a Royal Society research professorship.} D.P.M.M.S., 16 Mill Lane, Cambridge, CB2 2SB, England. \bigskip Mathematics department, University of California at Berkeley, CA 94720-3840, U. S. A. \bigskip reb@dpmms.cam.ac.uk www.dpmms.cam.ac.uk/\~{}reb \bigskip 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. \proclaim 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 $p\ge 13$. 5.~Calculation of some cohomology groups. 6.~Elements of type $3B$, $5B$, $7B$, and $13B$. 7.~Open problems and conjectures. \proclaim 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 $^gV$ 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 $\Tr(h|^gV)=\sum_n\Tr(h|^gV_n)q^n$ is equal to the Hauptmodul $\Tr(gh|V)=\sum_n\Tr(g|V_n)q^n$ (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 $^gV$ was defined for any element $g\in M$ of odd prime order to be the sum of the Tate cohomology groups $\H^0(g,V[1/2])\oplus \H^1(g,V[1/2])$ for a suitable $\Z[1/2]$ form $V[1/2]$ of $V$, and it was shown that $\Tr(h|^gV)=\Tr(h|\H^0(g,V[1/2]))-\Tr(h|\H^1(g,V[1/2]))$ was equal to the Hauptmodul of $gh\in M$. Hence to prove the modular moonshine conjecture for an element $g$ of type $pA$ it is enough to prove that $\H^1(g,V[1/2])=0$. In [B-R] this was shown by explicit calculation for the elements of type $pA$ for $p\le 11$, using the fact that these elements commute with an element of type $2B$. For $p\ge 13$ 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 $p\ge 13$ not of type $13B$ then $\H^1(g,V[1/2])=0$ (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 $3C$, $\H^1(g,V[1/2])$ is again zero, and for type $2B$, $\H^1(g,V[1/2])$ is zero in even degree and $\H^0(g,V[1/2])$ is zero in odd degree. This leaves the cases where $g$ is of type $3B$, $5B$, $7B$, or $13B$, when the Tate cohomology groups $\H^0(g,V[1/2])$ and $\H^1(g,V[1/2])$ 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 $\sigma$ of order 2 in the center of $C_M(g)/O_p(C_M(g))$ acts as $1$ on $\H^0(g,V[1/2])$ and as $-1$ on $\H^1(g,V[1/2])$. This determines the modular characters of both cohomology groups because $\Tr(h|\H^0(g,V[1/2]))+\Tr(h|\H^1(g,V[1/2]))$ is then given by $\Tr(h\sigma|\H^0(g,V[1/2]))-\Tr(h\sigma|\H^1(g,V[1/2]))$ which is the Hauptmodul of the element $gh\sigma\in M$. Hence both $\Tr(h|\H^0(g,V[1/2]))$ and $\Tr(h|\H^1(g,V[1/2]))$ can be written explicitly as linear combinations of two Hauptmoduls for any $p$-regular element $h\in C_M(g)$. 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 $\H^i(g,V)$ 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 $g\in M$ 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 $\H^*(g,V[1/2])$ 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 $p\ge 13$ 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 $\sum_n\Tr(h|\H^0(g,V[1/2]_n))q^n$ and $\sum_n\Tr(h|\H^1(g,V[1/2]_n))q^n$. The reason we get more information by using $\Z_p$-forms rather than $\Q_p$-forms is that the group ring $\Z_p[\Z/p\Z]$ has 3 indecomposable modules which are free over $\Z_p$, while the group ring $\Q_p[\Z/p\Z]$ 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 ($\ge 13$) 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 $\H^1(g,V[1/2])=0$ 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 $(p,mn)\ne 0$ in proposition 3.4. We summarize the results of this paper and of [B-R] about the vertex superalgebras $^gV$ 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 $^gV$ is just a vertex algebra and not a superalgebra. If $p\ge 13$ 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 $2\le p\le 13$ then the super part of $^gV$ does not vanish. If $p=2$ then $^gV$ 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 $3\le p\le 13$ then $^gV$ 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 $^gV$. 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 $^gV$ in terms of the coefficients of certain Hauptmoduls. \proclaim Notation. \item{$[A]$} is the element of $K$ represented by the module $A$. \item{$A,B,C$} $G$-modules. \item{$A_n$} The alternating group on $n$ symbols. \item{$\Aut$} The automorphism group of something. \item{$c^+_g(n)$} The $n$'th coefficient of the Hauptmodul of $g\in M$, equal to $\Tr(1|^gV_n)=\dim(\H^0(g,V[1/2]_n))-\dim(\H^1(g,V[1/2]_n))$. \item{$c^-_g(n)$} $\dim(\H^0(g,V[1/2]_n))+\dim(\H^1(g,V[1/2]_n))$. \item{$c_{m,n}$} Defined in proposition 4.3. \item{${\bf C}$} The complex numbers. \item{$C_M(g)$} The centralizer of $g$ in the group $M$. \item{$Co_1$} Conway's largest sporadic simple group. \item{$E$} The positive subalgebra of the monster Lie algebra. \item{$F$} The negative subalgebra of the monster Lie algebra. \item{$\F_p$} The finite field with $p$ elements. \item{$f$} A homomorphism from $K$ to $\Q$ defined in lemma 2.3. \item {$g$} An element of $G$, usually of order $p$. \item {$g_i$} The element $g^i$ of $G$, used when it is necessary to distinguish the multiplication in the group ring from some other multiplication. \item{$\langle g\rangle$} The group generated by $g$. \item{$G$} A group, often cyclic of prime order $p$ and generated by $g$. \item{$h(A),h_n$} See section 5. \item{$H$} The Cartan subalgebra of the monster Lie algebra. \item{$\H^i(G,A)$} A Tate cohomology group of the finite group $G$ with coefficients in the $G$-module $A$. \item{$\H^i(g,A)$} means $\H^i(\langle g\rangle,A)$, where $\langle g\rangle$ is the cyclic group generated by $g$. \item{$\H^*(g,A)$} The sum of the Tate cohomology groups $\H^0(g,A)$ and $\H^1(g,A)$, considered as a super module. \item{$HJ$} The Hall-Janko sporadic simple group (sometimes denoted $J_2$). \item{$I$} 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 $\Z_p[G]/N_G\Z_p$. \item{$\Im$} The image of a map. \item{$K$} A ring which is a free $\Q$-module with a basis of 3 elements $[\Z_p]=1$, $[\Z_p[G]]$, and $[I]$. \item{$\Ker$} The kernel of a map. \item{$\Lambda,\hat\Lambda$} The Leech lattice and a double cover of the Leech lattice. \item{$\Lambda^n(A)$} The $n$'th exterior power of $A$. \item{$\Lambda^*(A)$} The exterior algebra $\oplus_n\Lambda^n(A)$ of $A$. \item{$L$} An even lattice. \item{$\m$} The monster Lie algebra. \item{$M$} The monster simple group. \item{$N_G$} The element $\sum_{g\in G}g$ of $\Z_p[G]$. \item{$O_p(G)$} The largest normal $p$-subgroup of the finite group $G$. \item{$p$} A prime, usually the order of $g$. \item{$\Q,\Q_p$} The rational numbers and the field of $p$-adic numbers. \item{$\rho$} The Weyl vector of the monster Lie algebra. \item{$\R $} The real numbers. \item{$R_p$} A finite extension of the $p$-adic integers. \item{$R(i)$} Defined just before lemma 3.2. \item{$\sigma$} An automorphism of type $2B$ in the monster or the automorphism $-1$ of the Leech lattice. \item{$S_n$} A symmetric group. \item{$S^n(A)$} The $n$'th symmetric power of $A$. \item{$S^*(A)$} The symmetric algebra $\oplus_nS^n(A)$ of $A$. \item{$Suz$} Suzuki's sporadic simple group. \item{$\Tr$} $\Tr(g|A)$ 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. \item{$V[1/n]$} A $\Z[1/n]$-form of the monster vertex algebra. \item{$V_\Lambda$} The integral form of the vertex algebra of $\hat\Lambda$. \item{$V_n$} The degree $n$ piece of $V$. \item{$V^n$} An eigenspace of some group acting on $V$. \item{$^gV$} A modular vertex algebra or superalgebra given by $\H^*(g,V[1/n])$ for some $n$ coprime to $|g|$. \item{$\Z$} The integers. \item{$\Z_p$} The ring of $p$-adic integers. \item{$\omega$} A cube root of 1 or a conformal vector. \item{$\Omega$} The Laplace operator on $\Lambda^*(E)$. \proclaim 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 $N_G=1+g+g^2+\cdots+g^{p-1}$). Their Tate cohomology groups are given by $\H^0(g,\Z_p)=\Z/p\Z$, $\H^1(g,\Z_p)=0$, $\H^0(g,\Z_p[G])=0$, $\H^1(g,\Z_p[G])=0$, $\H^0(g,I)=0$, and $\H^1(g,I)=\Z/p\Z$. \proclaim Lemma 2.1. The tensor products of these modules are given as follows: $$\eqalign{ \Z_p\otimes X&=X \hbox{ (for any $X$)}\cr \Z_p[G]\otimes X&=\Z_p[G]^{\dim(X)} \hbox{ (the sum of $\dim(X)$ copies of $\Z_p[G]$)}\cr I\otimes I&= \Z_p[G]^{p-2}\oplus \Z_p.\cr }$$ Proof. The case $\Z_p\otimes X$ is trivial. If $X$ has a basis $x_1,\ldots,x_n$ then for any fixed $i$ the elements $g^j\otimes g^j(x_i)$ ($0\le j
p$}\cr \Lambda^n(I)&=\Z_p\oplus \Z_p[G]^{({p-1\choose n}-1)/p} \hbox{ if $n$ is even and $0\le n
0$, 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: \proclaim Assumption. If $m
0,n\in \Z}r^mq^n [V_{mn}])$$ lies in $R(1)$. Proof. The Laplace operator $\Omega=dd^*+d^*d$ acts as multiplication by $(m-1)n$ on the degree $(m,n)$ piece of $\Lambda^*(E[1/2])$, so the result follows from lemma 2.9 if we let $A_i$ be the degree $(m,n)$ piece of $\Lambda^i(E[1/2])$. This proves lemma 3.2. \proclaim Lemma 3.3. If $A$ is a $\Z_p[G]$-module with $\Tr(g|A)=c^+$, $f([A])=c^-$, (and $m
0,n\in \Z}(1-r^mq^n)^{(c^-_g(mn)+c^+_g(mn))/2} (1+r^mq^n)^{(c^+_g(mn)-c^-_g(mn))/2}$$ 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 $r^mq^n$ 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 $r^mq^n$ in proposition 3.4 vanish when $p|mn$. Some numerical calculations suggest that they usually do not. \proclaim 4.~The modular moonshine conjectures for $p\ge 13$. 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 $p=2$.) \proclaim Theorem 4.1. If $g\in M$ is an element of prime order $p\ge 17$ or an element of type $13A$ then $\H^1(g,V[1/2])=0$. This implies that $^gV=\H^0(g,V[1/2])=\H^*(g,V[1/2])$ 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 $\H^1(g,V[1/2]_n)$. We start by summarizing what we know about these numbers. \proclaim Lemma 4.2. \item {1.} The numbers $c^-_g(n)$ and $c^+_g(n)$ satisfy the relations given in proposition 3.4. \item {2.} The numbers $c^-_g(n)$ are integers, with $c^-_g(n)\equiv c^+_g(n)\bmod 2$. \item {3.} $c^-_g(n)\ge |c^+_g(n)|$ and $(p-2)c^+_g(n)+pc^-_g(n)\le 2c^+_1(n)$, (where $c^+_1(n)$ is the coefficient of $q^n$ in the elliptic modular function). \item {4.} The numbers $c^+_g(n)$ are the coefficients of the Hauptmodul of the element $g\in M$. Proof. These properties follow easily from proposition 3.4 together with the fact that $\Z_p\otimes V_n$ 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. \proclaim Proposition 4.3. There are integers $c_{m,n}$ defined for $m,n>0$ (and $m
0,n>0}c_{m,n}r^mq^n\right) \equiv
\sum_{m>0,n>0}\sum_{d|(m,n)} {c^{(-)^d}_g(mn/d^2)\over d}r^mq^n\bmod r^p$$
\item{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.
\proclaim Lemma 4.4. If $p>13$ then the values
of $c^-_g(n)$ for $n\le 21$, $c^-_g(2n)$ for $2n\le 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\le 21$ and $c(2n),n\le
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^4q^9$ in
proposition 4.3 and using the fact that we know $c_{m,n}$ for $m\le
3$, $n\le 8$. Similarly we can evaluate $c(45)$ by looking at the
coefficient of $r^5q^9$. Notice that the term
$(\sum_{m,n}c_{m,n}r^mq^n)^3/3$ has 3's only in the denominators of
coefficients of $r^mq^n$ when $3|m$ and $3|n$, so we do not get
problems from this for the coefficients of $r^4q^9$ and $r^5q^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^3q^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].
\proclaim 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\ge 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\le 16$, $c^-_g(2n)$ for $2n\le 32$, and
$c^-_g(36)$ and $c^-_g(45)$.
But now if we look at the coefficient of $r^3q^{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\bmod 3^n$
depends only on $c^-_g(m)\bmod 3^n$ if $n\ge 1$, so we can determine
$c^-_g(m)/3\bmod 3^n$ and hence $c^-_g(m)\bmod 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.
\proclaim 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.)
\vfill\line{}
$$\matrix{
p=71&\Gamma_0(71)+&
\quad& q^{-1} &+q &+q^2 &+q^3 &+q^4 &+2q^5 &+\cdots\cr
p=59&\Gamma_0(59)+&
\quad& q^{-1} &+q &+q^2 &+2q^3 &+2q^4 &+3q^5 &+\cdots\cr
p=47&\Gamma_0(47)+&
\quad& q^{-1} &+q &+2q^2 &+3q^3 &+3q^4 &+5q^5 &+\cdots\cr
p=47&\Gamma_0(94)+&
\quad& q^{-1} &+q & &+q^3 &+q^4 &+q^5 &+\cdots\cr
p=41&\Gamma_0(41)+&
\quad& q^{-1} &+2q &+2q^2 &+3q^3 &+4q^4 &+7q^5 &+\cdots\cr
p=41&\Gamma_0(82|2)+&
\quad& q^{-1} & & &+q^3 & &+q^5 &+\cdots\cr
p=31&\Gamma_0(31)+&
\quad& q^{-1} &+3q &+3q^2 &+6q^3 &+9q^4 &+13q^5 &+\cdots\cr
p=31&\Gamma_0(62)+&
\quad& q^{-1} &+q &+q^2 &+2q^3 &+q^4 &+3q^5 &+\cdots\cr
p=29&\Gamma_0(29)+&
\quad& q^{-1} &+3q &+4q^2 &+7q^3 &+10q^4 &+17q^5 &+\cdots\cr
p=29&\Gamma_0(58|2)+&
\quad& q^{-1} &+q & &+q^3 & &+q^5 &+\cdots\cr
p=23&\Gamma_0(23)+&
\quad& q^{-1} &+4q &+7q^2 &+13q^3 &+19q^4 &+33q^5 &+\cdots\cr
p=23&\Gamma_0(46)+23&
\quad& q^{-1} & &-q^2 &+q^3 &-q^4 &+q^5 &+\cdots\cr
p=23&\Gamma_0(46)+&
\quad& q^{-1} &+2q &+q^2 &+3q^3 &+3q^4 &+5q^5 &+\cdots\cr
p=19&\Gamma_0(19)+&
\quad& q^{-1} &+6q &+10q^2 &+21q^3 &+36q^4 &+61q^5 &+\cdots\cr
p=19&\Gamma_0(38)+&
\quad& q^{-1} &+2q &+2q^2 &+5q^3 &+4q^4 &+9q^5 &+\cdots\cr
p=19&\Gamma_0(38|2)+&
\quad& q^{-1} &+2q & &+q^3 & &+3q^5 &+\cdots\cr
p=17&\Gamma_0(17)+&
\quad& q^{-1} &+7q &+14q^2 &+29q^3 &+50q^4&+92q^5&+\cdots\cr
p=17&\Gamma_0(34)+&
\quad& q^{-1} &+3q &+2q^2 &+5q^3 &+6q^4 &+12q^5 &+\cdots\cr
p=17&\Gamma_0(34|2)+&
\quad& q^{-1} &+q & &+3q^3 & &+4q^5 &+\cdots\cr
p=13&\Gamma_0(13)+&
\quad& q^{-1} &+12q &+28q^2 &+66q^3 &+132q^4&+258q^5&+\cdots\cr
p=13&\Gamma_0(26)+&
\quad& q^{-1} &+4q &+4q^2 &+10q^3 &+12q^4 &+26q^5 &+\cdots\cr
p=13&\Gamma_0(26|2)+&
\quad& q^{-1} &+2q & &+4q^3 & &+6q^5 &+\cdots\cr
}$$
It is also possible (but unlikely) that there are other solutions for
$p=13$ for which $c^-_g(1)$ does not satisfy the inequalities
$0\le c^-_g(1)< 3^{10}$.
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
$c^-_g(5)\bmod 3$. 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 $c^-_g(1)\ge 3^{10}$. (If
$c^-_g(1)$ is not divisible by 3 then even when $p=13$ the numbers
$c^-_g(i)\bmod 3^n$ determine the numbers $c^-_g(n)\bmod 3^{n+1}$.
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 $3c^-_g(5)\bmod 3^n$ 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.)
\proclaim 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 $r^mq^n$ of
both sides of proposition 4.3 are equal.
Case 1. $c^-_g(2n)$ for $2n\not\equiv 0\bmod p$. By looking at the
coefficient of $r^2q^n$ 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 $1\le i m/p$
and the free exterior algebra generated by super elements of each
degree $n$ with $0