5.1 Even $p$ -rank

Let $L$ be an even lattice of prime level $p$ and genus $\mathit{II}_{n,2}(p^{\unicode[STIX]{x1D716}_{p}n_{p}})$ with $n>2$ and $n_{p}$ even carrying a holomorphic automorphic product $\unicode[STIX]{x1D6F9}(F)$ of singular weight.

Let $D$ be the discriminant form of $L$ . The oddity formula (see [Reference Conway and SloaneCS99, ch. 15, § 7.7])

implies

Hence, $n=\pm 2\;\text{mod}\;8$ and $k=1+n/2$ is an even integer. Note that $k\geqslant 4$ . Define

Let $E$ be the Eisenstein series of weight $k$ for $\overline{\unicode[STIX]{x1D70C}}_{D}$ corresponding to $0$ . Write

with

Define

Note that $c_{k,p,n_{p}}$ is positive. By explicit calculation we can derive the following formulas for the Fourier coefficients $a_{\unicode[STIX]{x1D6FE}}(m)$ (see also [Reference ScheithauerSch06, Theorem 7.1]).

Write

with

Pairing $F$ with the Eisenstein series $E$ (see Proposition 2.3) we obtain

In the following, we will often need that $L$ splits a hyperbolic plane $\mathit{II}_{1,1}$ . We give a criterion for this.

Let $d$ be a positive rational number such that $pd$ is integral. We define functions

where we assume $m$ to be integral. We have

This implies

if $q(\unicode[STIX]{x1D6FE})\neq d\;\text{mod}\;1$ .

The divisor of $\unicode[STIX]{x1D6F9}(F)$ is a linear combination of rational quadratic divisors $\unicode[STIX]{x1D6FE}^{\bot }$ , where $\unicode[STIX]{x1D6FE}$ is a primitive vector of positive norm in $L$ . The divisor $\unicode[STIX]{x1D6FE}^{\bot }$ has order $\sum _{m>0}c_{m\unicode[STIX]{x1D6FE}}(-m^{2}\unicode[STIX]{x1D6FE}^{2}/2)$ . Since $\unicode[STIX]{x1D6F9}(F)$ is holomorphic this is a non-negative integer.

We also define the multiplicative function

Now we expand the sum $-\sum _{\unicode[STIX]{x1D6FE}\in D}\,\sum _{m>0}c_{\unicode[STIX]{x1D6FE}}(-m)a_{\unicode[STIX]{x1D6FE}}(m)$ in terms of the non-negative divisor degrees $g_{\unicode[STIX]{x1D6FE}}$ .

Define $m_{\infty }=\max _{\unicode[STIX]{x1D6FE}\in D}(-\unicode[STIX]{x1D708}_{\infty }(F_{\unicode[STIX]{x1D6FE}}))$ . Note that $m_{\infty }>0$ .

We obtain the following inequalities.

We remark that the first inequality in the proposition is a consequence of the Riemann–Roch theorem and the second is a consequence of the Eisenstein condition.

Note that the proof is constructive. For example, taking $x=20$ gives the bounds $c=3.59750\ldots$ and $d=33.92899\ldots .$

5.2 Odd $p$ -rank

Now let $L$ be an even lattice of prime level $p$ and genus $\mathit{II}_{n,2}(p^{\unicode[STIX]{x1D716}_{p}n_{p}})$ with $n>2$ and $n_{p}$ odd. Suppose $\unicode[STIX]{x1D6F9}(F)$ is a holomorphic automorphic product of singular weight on $L$ .

Since $n_{p}$ is odd, it follows that $p$ is odd as well.

The oddity formula implies

so that

Define $k=1+n/2$ and

Then

Let $\unicode[STIX]{x1D712}(j)=(j/p)$ . Define the twisted divisor sum

and the generalised Bernoulli numbers $B_{m,\unicode[STIX]{x1D712}}$ by

(see [Reference IwasawaIwa72]). Let

The positivity of $L(k,\unicode[STIX]{x1D712})$ implies that $c_{k,p,n_{p}}$ is positive. We describe the Fourier coefficients $a_{\unicode[STIX]{x1D6FE}}(m)$ of the Eisenstein series $E$ .

We have the following result.

As above, we denote the Fourier coefficients of $F$ by $c_{\unicode[STIX]{x1D6FE}}$ and define the functions $g_{\unicode[STIX]{x1D6FE}}$ . We also define

The function $h_{\unicode[STIX]{x1D712}}$ is bounded below by $h_{\unicode[STIX]{x1D712}}(m)\geqslant (2-\unicode[STIX]{x1D701}(2))m^{k-1}\geqslant m^{k-1}/3$ .

Pairing $F$ with the Eisenstein series $E$ and applying the Riemann–Roch theorem to $F_{0}$ , we obtain the following result.

We can now derive a bound on $n$ .

Note that the constant $d$ is slightly larger here than in Theorem 5.7. Taking $x=20$ we obtain the bounds $c=3.59750\ldots$ and $d=40.52171\ldots .$

5.3 An example

Let $L$ be a lattice of genus $\mathit{II}_{n,2}(2_{\mathit{II}}^{+n_{2}})$ with $n>2$ and $n_{2}=2,~4$ or $6$ carrying a holomorphic automorphic product of singular weight. Then $n\leqslant 34,~42$ respectively $42$ and

by Theorem 5.7 and Proposition 5.6. The values of the bounds are given in the following table.

Since $m_{\infty }$ is half-integral we obtain the following theorem.

6.1 General results

We derive some general bounds and formulate the Eisenstein condition for reflective modular forms.

Let $L$ be an even lattice of prime level $p$ and genus $\mathit{II}_{n,2}(p^{\unicode[STIX]{x1D716}_{p}n_{p}})$ with $n>2$ . Let $F=\sum _{\unicode[STIX]{x1D6FE}\in D}F_{\unicode[STIX]{x1D6FE}}e^{\unicode[STIX]{x1D6FE}}$ be a non-zero reflective modular form on $L$ (see § 3). Then $F$ has weight $1-n/2$ ,

with $c_{0}(-1)=0$ or $1$ ,

with $c_{\unicode[STIX]{x1D6FE}}(-1/p)=0$ or $1$ if $q(\unicode[STIX]{x1D6FE})=1/p\;\text{mod}\;1$ and the other components $F_{\unicode[STIX]{x1D6FE}}$ of $F$ are holomorphic at $\infty$ . We define integers $c_{1}=c_{0}(-1)$ and $c_{p}=|\{\unicode[STIX]{x1D6FE}\in D\mid q(\unicode[STIX]{x1D6FE})=1/p\;\text{mod}\;1\text{ and }F_{\unicode[STIX]{x1D6FE}}\text{ singular}\}|$ .

Pairing $F$ with the Eisenstein series $E$ of weight $k=1+n/2$ we obtain (see Propositions 5.1 and 5.8) the following result.

We will also need the following result.

6.2 Symmetric forms

Here we classify reflective modular forms that are invariant under $O(D)$ .

Let $L$ be an even lattice of prime level $p$ and genus $\mathit{II}_{n,2}(p^{\unicode[STIX]{x1D716}_{p}n_{p}})$ with $n>2$ . Then the number of elements $\unicode[STIX]{x1D6FE}$ in $D$ of order $p$ and norm $q(\unicode[STIX]{x1D6FE})=1/p\;\text{mod}\;1$ is given by

if $n_{p}$ is even and by

if $n_{p}$ is odd (see [Reference ScheithauerSch06, Proposition 3.2]). Suppose $L$ carries a symmetric reflective modular form $F$ with $[F_{0}](0)=n-2$ . Then the Eisenstein condition takes the form

if $n_{p}$ is even and

if $n_{p}$ is odd. Here $d_{1}$ and $d_{p}$ can be $0$ or $1$ . In the case $n_{p}<n+2$ , the solutions of these equations have been determined in [Reference ScheithauerSch06].

6.3 Bounds in the non-symmetric case

In this section, we derive bounds on the signature for reflective modular forms which are not invariant under $O(D)$ .

First, we recall the Riemann–Roch theorem for $\unicode[STIX]{x1D6E4}_{1}(p)$ .

Let $p$ be prime. For $p\geqslant 3$ , the group $\unicode[STIX]{x1D6E4}_{1}(p)$ has $p-1$ classes of cusps which can be represented by $1/c$ with $c=1,\ldots ,(p-1)/2$ of width $p$ and $a/p$ with $a=1,\ldots ,(p-1)/2$ of width $1$ . The cusps of $\unicode[STIX]{x1D6E4}_{1}(2)$ can be represented by $1/2$ of width $1$ and $1/1$ of width $2$ . Let $f\neq 0$ be a meromorphic modular form on $\unicode[STIX]{x1D6E4}_{1}(p)$ of weight $m$ and finite-order character. For $p\geqslant 5$ , there are no torsion points and the Riemann–Roch theorem states

For $p=3$ , we have

with $e_{3}=(3+i\sqrt{3})/6$ and

with $e_{2}=(1+i)/2$ if $p=2$ .

Note that the bounds do not hold in the symmetric case.

Using Theorem 6.5, we can determine the non-symmetric forms on lattices of prime level by analysing the obstructions in a finite number of cases. For $p=3$ , which is the most complicated case, we describe this explicitly in the next section. The other cases are analogous.

6.4 Level $3$

In this section we determine the reflective forms on lattices of level $3$ and signature $(n,2)$ where $n=4,~6,~8$ or $10$ .

Let $L$ be a lattice of genus $\mathit{II}_{10,2}(3^{\unicode[STIX]{x1D716}_{3}n_{3}})$ and $F$ a reflective form on $L$ . Suppose $F_{0}$ has constant coefficient $[F_{0}](0)=8$ . Then $c_{1}=1$ (see Proposition 6.1) and the Eisenstein condition gives the following value for $c_{3}$ (see Proposition 6.2).

Since $c_{3}$ should be a non-negative integer, this already excludes the first three cases.

The space $S_{6}(\unicode[STIX]{x1D6E4}(3))$ has dimension $3$ and is spanned by the functions $\unicode[STIX]{x1D702}_{1^{8}}\unicode[STIX]{x1D703}_{A_{2}}^{2}$ , $\unicode[STIX]{x1D702}_{1^{8}}\unicode[STIX]{x1D703}_{A_{2}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D708}+A_{2}}$ and $\unicode[STIX]{x1D702}_{1^{6}3^{6}}$ . The liftings of these functions generate the obstruction space $S_{\overline{\unicode[STIX]{x1D70C}}_{D},6}$ .

Pairing $F$ with the lift $F_{\unicode[STIX]{x1D702}_{1^{6}3^{6}},0}$ of the $\unicode[STIX]{x1D702}$ -product $\unicode[STIX]{x1D702}_{1^{6}3^{6}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D70F})^{6}\unicode[STIX]{x1D702}(3\unicode[STIX]{x1D70F})^{6}$ we obtain

Next, we consider the case $n=8$ . Let $L$ be a lattice of genus $\mathit{II}_{8,2}(3^{\unicode[STIX]{x1D716}_{3}n_{3}})$ and $F$ a reflective modular form on $L$ with $[F_{0}](0)=6$ . Then we obtain the following for $c_{3}$ .

The discriminant form of type $3^{+1}$ contains no elements $\unicode[STIX]{x1D6FE}$ of norm $q(\unicode[STIX]{x1D6FE})=1/3\;\text{mod}\;1$ . Hence, this case can be excluded.

The space $S_{5}(\unicode[STIX]{x1D6E4}(3))$ has dimension $2$ and is spanned by the functions $\unicode[STIX]{x1D702}_{1^{8}}\unicode[STIX]{x1D703}_{A_{2}}$ and $\unicode[STIX]{x1D702}_{1^{8}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D708}+A_{2}}$ . The liftings of these functions generate the obstruction space $S_{\overline{\unicode[STIX]{x1D70C}}_{D},5}$ .

The lattice $A_{2}$ has genus $\mathit{II}_{2,0}(3^{-1})$ and is isomorphic to its rescaled dual $A_{2}^{\prime }(3)$ . The theta functions of $A_{2}$ can be written as

They transform under $S=\big(\!\begin{smallmatrix}0 & -1\\ 1 & 0\end{smallmatrix}\!\big)$ as

Let $\unicode[STIX]{x1D6FE}\in D$ be of norm $q(\unicode[STIX]{x1D6FE})=1/3\;\text{mod}\;1$ . Then the lift of $\unicode[STIX]{x1D702}_{1^{8}}\unicode[STIX]{x1D703}_{A_{2}}$ with respect to the dual Weil representation $\overline{\unicode[STIX]{x1D70C}}_{D}$ on $\unicode[STIX]{x1D6FE}$ is given by

with

and

where

and $g_{j}|_{5,T}=e(j/3)g_{j}$ . Note that $g_{0}=0$ . We obtain an analogous result for the lift of $\unicode[STIX]{x1D702}_{1^{8}}\unicode[STIX]{x1D703}_{\unicode[STIX]{x1D708}+A_{2}}$ with respect to $\overline{\unicode[STIX]{x1D70C}}_{D}$ on an element $\unicode[STIX]{x1D6FE}\in D$ of norm $q(\unicode[STIX]{x1D6FE})=2/3\;\text{mod}\;1$ .

Let

We assume now that $M$ is non-empty. Then $|M|=c_{3}=2\cdot 3^{(n_{3}-1)/2}$ and $M=-M$ because $F_{\unicode[STIX]{x1D6FE}}=F_{-\unicode[STIX]{x1D6FE}}$ . The crucial result to determine the structure of $M$ is the following proposition.

The proposition implies that $M^{\bot }$ is an isotropic subgroup of $D$ . Let $\unicode[STIX]{x1D6FE}\in M$ and $\unicode[STIX]{x1D707}\in M^{\bot }$ . Then $M\cap \unicode[STIX]{x1D6FE}^{\bot }=M\cap (\unicode[STIX]{x1D6FE}+\unicode[STIX]{x1D707})^{\bot }$ . Hence, the group $M^{\bot }$ acts on $M$ by translations.

A consequence of this result is the following proposition.

Let $H$ be an isotropic subgroup of $D$ of order $|H|=3^{(n_{3}-3)/2}$ . Then the lift of $9\unicode[STIX]{x1D702}_{1^{-9}3^{3}}$ , with respect to $\unicode[STIX]{x1D70C}_{D}$ on $H$ , is given by

with

and

where $\unicode[STIX]{x1D702}_{1^{3}3^{-9}}(\unicode[STIX]{x1D70F}/3)=g_{0}(\unicode[STIX]{x1D70F})+g_{1}(\unicode[STIX]{x1D70F})+g_{2}(\unicode[STIX]{x1D70F})$ and $g_{j}|_{-3,T}=e(j/3)g_{j}$ . Note that

The function $F_{9\unicode[STIX]{x1D702}_{1^{-9}3^{3}},H}$ has $0$ -component $F_{0}=6\unicode[STIX]{x1D702}_{1^{-9}3^{3}}$ and is reflective. The singular components are the $F_{\unicode[STIX]{x1D6FE}}$ with $\unicode[STIX]{x1D6FE}\in H^{\bot }$ and $q(\unicode[STIX]{x1D6FE})=1/3\hspace{0.6em}{\rm mod}\hspace{0.2em}1$ . The discriminant form $H^{\bot }/H$ is of type $3^{-3}$ . It is generated by elements $\{\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2},\unicode[STIX]{x1D6FE}_{3}\}$ with $q(\unicode[STIX]{x1D6FE}_{i})=1/3\;\text{mod}\;1$ and $(\unicode[STIX]{x1D6FE}_{i},\unicode[STIX]{x1D6FE}_{j})=0\;\text{mod}\;1$ for $i\neq j$ . We obtain the following result (see Theorem 2.1 and Proposition 2.2).

We can decompose $L=K\oplus \mathit{II}_{1,1}(3)$ , where $K$ has genus $\mathit{II}_{7,1}(3^{-\unicode[STIX]{x1D716}_{3}(n_{3}-2)})$ and assume that $H$ is a maximal isotropic subgroup of the discriminant form of $K$ . Then the embedding $K\subset K_{H}$ gives an embedding $L\subset L_{H}$ and identifies the corresponding domains ${\mathcal{H}}_{L}$ and ${\mathcal{H}}_{L_{H}}$ . Proposition 3.4 implies the following proposition.

We calculate the product expansions of the automorphic product $\unicode[STIX]{x1D6F9}$ corresponding to $F_{9\unicode[STIX]{x1D702}_{1^{-9}3^{3}},0}$ on $L$ of genus $\mathit{II}_{8,2}(3^{-3})$ .

First, we decompose $L=K\oplus \mathit{II}_{1,1}(3)$ . Then $K=E_{6}\oplus \mathit{II}_{1,1}$ . We choose a primitive norm $0$ vector $z$ in $\mathit{II}_{1,1}(3)$ .

This is the twisted denominator identity of the fake monster superalgebra [Reference ScheithauerSch00] corresponding to an element of class $3A$ in $O(E_{8})$ (see [Reference ScheithauerSch01, Proposition 6.1]).

Now, we decompose $L=K\oplus \mathit{II}_{1,1}$ with $K=E_{6}\oplus \mathit{II}_{1,1}(3)$ and choose a primitive norm $0$ vector $z$ in $\mathit{II}_{1,1}$ . Then we have the following result.

This is the twisted denominator identity of the fake monster algebra corresponding to an element of class $3C$ in $\mathit{Co}_{0}$ (see [Reference ScheithauerSch04, Proposition 10.7]).

Again, let $L$ be a lattice of genus $\mathit{II}_{8,2}(3^{\unicode[STIX]{x1D716}_{3}n_{3}})$ carrying a reflective modular form $F$ .

The lift of $\unicode[STIX]{x1D702}_{1^{3}3^{-9}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D70F})^{3}\unicode[STIX]{x1D702}(3\unicode[STIX]{x1D70F})^{-9}$ with respect to $\unicode[STIX]{x1D70C}_{D}$ on $0$ is given by

with

and

where $\unicode[STIX]{x1D702}_{1^{-9}3^{3}}(\unicode[STIX]{x1D70F}/3)=g_{0}(\unicode[STIX]{x1D70F})+g_{1}(\unicode[STIX]{x1D70F})+g_{2}(\unicode[STIX]{x1D70F})$ and $g_{j}|_{-3,T}=e(j/3)g_{j}$ . The modular form $F_{\unicode[STIX]{x1D702}_{1^{3}3^{-9}},0}$ is reflective and has $0$ -component

Suppose $L$ has genus $\mathit{II}_{8,2}(3^{-7})$ . Then the level $1$ expansion of the theta lift of $F_{\unicode[STIX]{x1D702}_{1^{3}3^{-9}},0}$ on $L$ is the twisted denominator identity of the fake monster superalgebra corresponding to an element in $O(E_{8})$ of class $3A$ and the level $3$ expansion gives the twisted denominator identity of the fake monster algebra corresponding to an element in $\mathit{Co}_{0}$ of class $3C$ .

The case $n_{3}=9$ has already been described above because we have the following result.

Now we consider the case $n=6$ . Let $L$ be a lattice of genus $\mathit{II}_{6,2}(3^{\unicode[STIX]{x1D716}_{3}n_{3}})$ and $F$ a reflective form on $L$ with $[F_{0}](0)=4$ . We find the following value for $c_{3}$ .

Hence, we can assume that $F_{0}$ is holomorphic at $\infty$ and $n_{3}=4$ or $6$ .

The space $S_{4}(\unicode[STIX]{x1D6E4}(3))$ has dimension $1$ and is spanned by the function $\unicode[STIX]{x1D702}_{1^{8}}$ . The liftings of this function generate the obstruction space $S_{\overline{\unicode[STIX]{x1D70C}}_{D},4}$ .

Let $\unicode[STIX]{x1D6FE}\in D$ be of norm $q(\unicode[STIX]{x1D6FE})=1/3\;\text{mod}\;1$ . Then the lift of $\unicode[STIX]{x1D702}_{1^{8}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D702}(\unicode[STIX]{x1D70F})^{8}$ with respect to the dual Weil representation $\overline{\unicode[STIX]{x1D70C}}_{D}$ on $\unicode[STIX]{x1D6FE}$ is given by

with

and

As above, we define

Then $|M|=4\cdot 3^{(n_{3}-4)/2}$ and $M=-M$ . Pairing $F$ with $F_{\unicode[STIX]{x1D702}_{1^{8}},\unicode[STIX]{x1D6FE}}$ we obtain the following result.

This excludes the case $n_{3}=4$ . We assume now that $n_{3}=6$ . Then the proposition shows that $M$ must be of the form

with $(\unicode[STIX]{x1D6FE}_{i},\unicode[STIX]{x1D6FE}_{j})=0\;\text{mod}\;1$ for $i\neq j$ . In particular, $M^{+}=\{\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{6}\}$ is a basis of $D$ . Let $\unicode[STIX]{x1D6FE}\in D$ be of norm $q(\unicode[STIX]{x1D6FE})=1/3\;\text{mod}\;1$ with $\unicode[STIX]{x1D6FE}\notin M$ . Then $\unicode[STIX]{x1D6FE}$ is a linear combination of four of the $\unicode[STIX]{x1D6FE}_{i}$ so that $|M\cap \unicode[STIX]{x1D6FE}^{\bot }|=4$ . Hence, the principal part of $F$ satisfies all obstructions coming from $S_{\overline{\unicode[STIX]{x1D70C}}_{D},4}$ . This implies that a reflective modular form with constant coefficient $4$ on $\mathit{II}_{6,2}(3^{+6})$ exists.

We give an explicit construction. Let $\unicode[STIX]{x1D6FE}\in D$ be of norm $q(\unicode[STIX]{x1D6FE})=1/3\;\text{mod}\;1$ . Then the lift of $\unicode[STIX]{x1D703}_{A_{2}}^{2}/\unicode[STIX]{x1D702}_{1^{8}}$ on $\unicode[STIX]{x1D6FE}$ with respect to $\unicode[STIX]{x1D70C}_{D}$ is given by

with

and

where $\unicode[STIX]{x1D703}_{A_{2}^{\prime }}^{2}/\unicode[STIX]{x1D702}_{1^{8}}=g_{0}+g_{1}+g_{2}$ and $g_{j}|_{-2,T}=e(j/3)g_{j}$ . Note that $g_{2}=\unicode[STIX]{x1D703}_{A_{2}}^{2}/\unicode[STIX]{x1D702}_{1^{8}}$ .

The function $\unicode[STIX]{x1D702}_{(1/3)^{-3}1^{2}3^{-3}}(\unicode[STIX]{x1D70F})=\unicode[STIX]{x1D702}_{1^{-3}3^{2}9^{-3}}(\unicode[STIX]{x1D70F}/3)$ is a modular form for $\unicode[STIX]{x1D6E4}(3)$ of weight $-2$ . If we decompose $\unicode[STIX]{x1D702}_{(1/3)^{-3}1^{2}3^{-3}}=h_{0}+h_{1}+h_{2}$ with $h_{j}|_{-2,T}=e(j/3)h_{j}$ , then $g_{2}=h_{2}$ , $g_{1}=4h_{1}$ and $g_{0}=4h_{0}$ . It follows

Now let $M^{+}=\{\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{6}\}\subset D$ such that $q(\unicode[STIX]{x1D6FE}_{i})=1/3\;\text{mod}\;1$ , $(\unicode[STIX]{x1D6FE}_{i},\unicode[STIX]{x1D6FE}_{j})=0\;\text{mod}\;1$ for $i\neq j$ and $M=M^{+}\cup (-M^{+})$ . Define

The components of $F_{3\unicode[STIX]{x1D703}_{A_{2}}^{2}/4\unicode[STIX]{x1D702}_{1^{8}},M^{+}}$ can be described as follows. Write $\unicode[STIX]{x1D707}\in D$ as $\unicode[STIX]{x1D707}=\sum _{i=1}^{6}c_{i}\unicode[STIX]{x1D6FE}_{i}$ and let $\operatorname{wt}(\unicode[STIX]{x1D707})$ denote the number of non-zero $c_{i}$ . Then

if $\unicode[STIX]{x1D707}\in M$ and

with $j_{\unicode[STIX]{x1D707}}/3=-q(\unicode[STIX]{x1D707})\;\text{mod}\;1$ otherwise. In particular,

and $F_{\unicode[STIX]{x1D707}}=0$ if $q(\unicode[STIX]{x1D707})=1/3\;\text{mod}\;1$ and $\unicode[STIX]{x1D707}\notin M$ . Hence, $F$ is reflective. Conversely, we have the following proposition.

Let $L$ be a lattice of genus $\mathit{II}_{6,2}(3^{+6})$ . We can decompose $L$ as $L=K\oplus \mathit{II}_{1,1}(3)$ with $K=A_{2}\oplus A_{2}\oplus \mathit{II}_{1,1}(3)$ . Then $K$ has genus $\mathit{II}_{5,1}(3^{-4})$ . We choose an orthogonal basis $\{\unicode[STIX]{x1D6FE}_{1},\unicode[STIX]{x1D6FE}_{2},\unicode[STIX]{x1D6FE}_{3},\unicode[STIX]{x1D707}_{4}\}$ of the discriminant form of $K$ satisfying $q(\unicode[STIX]{x1D6FE}_{1})=q(\unicode[STIX]{x1D6FE}_{2})=q(\unicode[STIX]{x1D6FE}_{3})=-q(\unicode[STIX]{x1D707}_{4})=1/3\;\text{mod}\;1$ and an orthogonal basis $\{\unicode[STIX]{x1D707}_{5},\unicode[STIX]{x1D6FE}_{6}\}$ of the discriminant form of $\mathit{II}_{1,1}(3)$ satisfying $-q(\unicode[STIX]{x1D707}_{5})=q(\unicode[STIX]{x1D6FE}_{6})=1/3\;\text{mod}\;1$ . We define $\unicode[STIX]{x1D6FE}_{4}=\unicode[STIX]{x1D707}_{4}+\unicode[STIX]{x1D707}_{5}$ , $\unicode[STIX]{x1D6FE}_{5}=\unicode[STIX]{x1D707}_{4}-\unicode[STIX]{x1D707}_{5}$ and $M^{+}=\{\unicode[STIX]{x1D6FE}_{1},\ldots ,\unicode[STIX]{x1D6FE}_{6}\}$ . Let $\unicode[STIX]{x1D6F9}$ be the theta lift of $F=F_{3\unicode[STIX]{x1D703}_{A_{2}}^{2}/4\unicode[STIX]{x1D702}_{1^{8}},M^{+}}$ on $L$ . We choose a primitive norm $0$ vector $z$ in $\mathit{II}_{1,1}(3)$ . Then $z$ has level $3$ and $\operatorname{wt}(z/3)=3$ .

This identity is a new infinite product identity. One can show that it can also be obtained by twisting the denominator identity of the fake monster algebra by an element of class $9C$ in $\mathit{Co}_{0}$ .

Finally, we consider the case $n=4$ . Let $L$ be a lattice of genus $\mathit{II}_{4,2}(3^{\unicode[STIX]{x1D716}_{3}n_{3}})$ and $F$ a reflective form on $L$ with $[F_{0}](0)=2$ . Then, the Eisenstein condition gives the following value for $c_{3}$ .

Since $S_{3}(\unicode[STIX]{x1D6E4}(3))$ is trivial, the obstruction space $S_{\overline{\unicode[STIX]{x1D70C}}_{D},3}$ vanishes. Hence, $L$ carries a reflective form with constant coefficient $2$ if and only if it has genus $\mathit{II}_{4,2}(3^{-5})$ .

Let $D$ be a discriminant form of type $3^{-5}$ and $\unicode[STIX]{x1D6FE}\in D$ of norm $q(\unicode[STIX]{x1D6FE})=1/3\;\text{mod}\;1$ . Then, the lift of $\unicode[STIX]{x1D702}_{1^{1}3^{-3}}$ on $\unicode[STIX]{x1D6FE}$ with respect to the Weil representation $\unicode[STIX]{x1D70C}_{D}$ is given by

with

and

where $\unicode[STIX]{x1D702}_{1^{-3}3^{1}}(\unicode[STIX]{x1D70F}/3)=g_{0}(\unicode[STIX]{x1D70F})+g_{1}(\unicode[STIX]{x1D70F})+g_{2}(\unicode[STIX]{x1D70F})$ and $g_{j}|_{-1,T}=e(j/3)g_{j}$ . Note that $F_{\unicode[STIX]{x1D702}_{1^{1}3^{-3}},\unicode[STIX]{x1D6FE}}$ is reflective and $F_{0}$ has constant coefficient $2$ .

Let $L$ be a lattice of genus $\mathit{II}_{4,2}(3^{-5})$ . We choose an element $\unicode[STIX]{x1D6FE}\in D$ of norm $q(\unicode[STIX]{x1D6FE})=1/3\;\text{mod}\;1$ . Let $\unicode[STIX]{x1D6F9}$ be the automorphic product corresponding to $F_{\unicode[STIX]{x1D702}_{1^{1}3^{-3}},\unicode[STIX]{x1D6FE}}$ on $L$ .

We decompose $L=K\oplus \mathit{II}_{1,1}(3)$ such that $\unicode[STIX]{x1D6FE}$ is in the discriminant form of $\mathit{II}_{1,1}(3)$ and choose a primitive norm $0$ vector $z$ in $\mathit{II}_{1,1}(3)$ . Then $(\unicode[STIX]{x1D6FE},z/3)\neq 0\;\text{mod}\;1$ . Note that $K=A_{2}\oplus \mathit{II}_{1,1}(3)$ .

This is the twisted denominator identity of the fake monster superalgebra corresponding to an element of class $9A$ .

We can also decompose $L=K\oplus \mathit{II}_{1,1}(3)$ such that $\unicode[STIX]{x1D6FE}$ is in the discriminant form of $K$ . Again we choose a primitive norm $0$ vector $z$ in $\mathit{II}_{1,1}(3)$ . Then we have the following result.

This is the twisted denominator identity of the fake monster algebra corresponding to an element in $\mathit{Co}_{0}$ of class $9B$ .

The automorphic product $\unicode[STIX]{x1D6F9}$ was first described in [Reference Dittmann, Hagemeier and SchwagenscheidtDHS15].