On theta series attached to maximal lattices and their adjoints

TL;DR

This paper studies theta series of adjoint lattices with specific properties, demonstrating their structure and extremality, and providing examples of dual extremal lattices for squarefree levels.

Contribution

It introduces the Weierstrass property for theta series of adjoint lattices and constructs examples of dual extremal lattices at squarefree levels.

Findings

Theta series of adjoint lattices have the Weierstrass property.

Existence of dual extremal lattices at squarefree levels.

Framework for defining extremality for arbitrary squarefree levels.

Abstract

The space spanned by theta series of adjoints of maximal even lattices of exact level $N$ and determinant $N^2$ has the Weierstrass property and hence allows to define extremality for arbitrary squarefree level $N$. We find examples of such dual extremal lattices.

Equations

$$\left(f{\mid }_k\gamma \right)(\tau )=det(\gamma {)}^{\frac{k}{2}}(c\tau +d{)}^{-k}f(\frac{a\tau +b}{c\tau +d})(\tau \in \mathbb{H}).$$

$$f⟼\left(f\mid V(p)\right)(\tau ):=f(p\cdot \tau ).$$

$$f(\tau )=\sum _n{a}_n{e}^{2\pi i\frac{n}{p}\tau }⟼f\mid U^0(p)(\tau )=\sum _n{a}_{np}{e}^{2\pi in\tau }.$$

$$\omega_{p}^{N}\equiv\left(\begin{array}[]{cc}0&-1\\ 1&0\end{array}\right)\bmod p$$

$${\omega }_p^N\equiv 1_2\mathrm{mod}\frac{N}{p}.$$

$$W_{p}^{N}:=\omega_{p}^{N}\cdot\left(\begin{array}[]{cc}p&0\\ 0&1\end{array}\right)$$

$$\theta (L)(\tau ):=\sum _{x\in L}{e}^{2\pi iQ(x)\cdot \tau }=\sum _{x\in L}{q}^{Q(x)}$$

$$\theta (L){\mid }_k{\omega }_p^N={\gamma }_p(d_p)s_p(V)D_p^{-\frac{1}{2}}\theta (L^{♯,p})$$

$${\gamma }_p(\delta )=1,{\gamma }_p(\delta \cdot p)=({\delta }_p,p{)}_p\cdot (-i{)}^{\frac{p(p-1)}{2}}$$

$$L_p\text{mbox}ismaximal$$

$$\theta (L){\mid }_k{\omega }_p^N\mid U^o(p)$$

$$\theta (L){\mid }_k{\omega }_p^N={\gamma }_p(d_p)s_p(V)D_p^{-\frac{1}{2}}\theta (L^{♯,p})$$

$$s_p(V)=-1,D_p=p^2\cdot d_p^{-1}.$$

$$L_p=L_p^{(0)}\perp L_p^{(1)}$$

$$\theta (L^{♯,p})\mid U^0(p)=\theta (L).$$

$$\theta (L){\mid }_k{\omega }_p^N\mid U^0(p)=-{\gamma }_p(d)D_p^{-\frac{1}{2}}\theta (L)$$

$${\theta }_P(L):=\sum _{x\in L}P(x)e^{2\pi iQ(x)\cdot \tau }.$$

$${\theta }_P(L){\mid }_{k+\nu }{\omega }_p^N\mid U^o(p)=-{\gamma }_p(d)p^{-1}{d}_p^{\frac{1}{2}}{\theta }_P(L)$$

$$s_p(V)=-1⟺p\mid N$$

$$s_{\infty}(V)=\left\{\begin{array}[]{ccc}-1&\mbox{ if }&m\equiv_{8}4\\ 1&\mbox{ if }&8\mid m\end{array}\right.$$

$$M_k(N{)}^{\ast }=\{f\in M_k(N)\mid \forall p\mid N:f\mid W_p^N+p^{1-\frac{k}{2}}f\mid U(p)=0\}.$$

$$\forall p\mid N:\text{mbox}{trace}_{\frac{N}{p}}^N(f\mid W_p^N)=0$$

$$f^{(N)}(\tau ):=\sum _{d\mid \frac{N}{M}}\mu (d)\frac{d{a}_f(d)}{{\sigma }_1(d)}f(d\cdot \tau )$$

$$S_k(M{)}^{new,N}:=\mathbb{C}\{f_i^{(N)}\},$$

$$S_k(N{)}^{\ast }={\oplus }_{M\mid N}{S}_k(M{)}^{new,N}.$$

$$\dim S_k(N{)}^{\ast }=\frac{(k-1)N}{12}-\frac{1}{2}-\frac{1}{4}\left(\frac{-1}{(k-1)N}\right)-\frac{1}{3}\left(\frac{-3}{(k-1)N}\right).$$

$$E_k^{(N)}:=\sum _{d\mid N}\mu (d)\frac{d{\sigma }_{k-1}(d)}{{\sigma }_1(d)}{E}_k(d\cdot \tau ).$$

$$M_k(N{)}^{\ast }=\Theta ({\mathfrak{S}}^{\ast }),$$

$$\Theta ({\mathfrak{S}}^{\ast })\subseteq M_k(N{)}^{\ast }$$

$$\Lambda:\left\{\begin{array}[]{ccc}S_{k}(p)&\longrightarrow&\Theta({\mathfrak{S}^{*}})\\ g&\longmapsto&\sum_{i}\frac{1}{m(L_{i})}<g,\theta(L_{i})>\theta(L_{i})\end{array}\right.$$

$$M(f):=\mathbb{C}\{f,f\mid V(p)\}.$$

Full text

On theta series attached to maximal lattices and their adjoints.

Siegfried Böcherer***Kunzenhof 4B, 79117 Freiburg, Germany, boecherer@t-online.de and Gabriele Nebe †††Lehrstuhl D für Mathematik, RWTH Aachen University, nebe@math.rwth-aachen.de

Abstract. The space spanned by theta series of adjoints of maximal even lattices of exact level $N$ and determinant $N^{2}$ has the Weierstrass property and hence allows to define extremality for arbitrary squarefree level $N$. We find examples of such dual extremal lattices.

keywords: theta series, modular forms, Weierstrass property, dual extremal lattices.

MSC: primary: 11F11, secondary: 11F33, 11H31

1 Introduction

This paper studies maximal even lattices from the geometric, arithmetic and analytic point of view. It is interesting to find even lattices $L$ such that the dual lattice $L^{\#}$ has the highest possible minimum. The most promising candidates for $L$ are clearly the maximal even lattices.

The maximal even lattices $L$ of level $N$ are characterized by the arithmetic property that the discriminant group $L^{\#}/L$ is an anisotropic quadratic abelian group of exponent $N$. If $m:=\dim(L)=2k$ is even, then this property can be translated in transformation rules of the theta series of $L$ under the Atkin-Lehner involutions for all prime divisors of $N$ (Theorem 3.1). If $\det(L)=N^{2}$ then the theta series of the adjoint lattice $\sqrt{N}L^{\#}$ lies in the space $M_{k}(N)^{*}$ introduced in [1]. This space has the Weierstrass property as defined in Definition 5.1 and hence allows to define extremality. The even lattice $L$ is called dual extremal if the theta series $\theta(\sqrt{N}L^{\#})$ of the adjoint lattice is the extremal modular form in $M_{k}(N)^{*}$. The dual extremal lattices of level $N$ are the maximal even lattices of level $N$ for which the minimum of the adjoint lattice is $\geq 2\dim(M_{k}(N)^{*})$. Remark 6.5 shows that in general this inequality may be strict. The dimension of $M_{k}(N)^{*}$ is calculated in [1]. It is interesting to note that for $k>2$ the space $M_{k}(N)^{*}$ is spanned by theta series of adjoint lattices of even maximal lattices of level $N$, so this space is as small as it can be to obtain bounds on the minimum with the theory of modular forms.

The last section of this paper lists some examples of dual extremal lattices. The level 2 case is remarkable. Its connection to the notion of s-extremal (odd) unimodular lattices in [10] allows to prove that for a dual extremal lattice $L$ of level $2$ and dimension $2k$ the minimum $\min(L^{\#})=\dim(M_{k}(2)^{*})$. Also for $k\equiv_{12}2$ the layers of $L^{\#}$ and of $L$ all form spherical 5-designs (Proposition 6.1) and hence both lattices are strongly perfect (see [27]) and therefore local maxima of the sphere packing density function.

2 Preliminaries

2.1 Modular forms

For basic facts about modular forms we refer to [19]. We denote by $M_{k}(N)$ and $S_{k}(N)$ the spaces of modular forms and cusp forms of weight $k$ for the congruence subgroup \Gamma_{0}(N)=\{\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right)\in\operatorname{SL}_{2}({\mathbb{Z}})\,\mid c\equiv_{N}0\}. Throughout the paper, we assume $N$ to be squarefree. For \gamma=\left(\begin{array}[]{cc}a&b\\ c&d\end{array}\right) and any function $f$ on the upper half plane ${\mathbb{H}}$ we define the slash operator $\mid_{k}$ by

[TABLE]

For primes $p$ we use the Hecke operators $T(p)$ (if $p\nmid N$), and $U(p)$ (for $p\mid N$) acting on $M_{k}(N)$ in the usual way. We also use the operator $V(p)$ defined by

[TABLE]

Occasionally we need a variant $U(p)^{0}$ of the operator $U(p)$, defined for functions $f$ on ${\mathbb{H}}$ periodic with respect to $p\cdot{\mathbb{Z}}$:

[TABLE]

Let $p$ be a prime with $p\mid N$. We denote by $\omega^{N}_{p}$ any element of $SL(2,{\mathbb{Z}})$ satisfying

[TABLE]

and

[TABLE]

For such a matrix $\omega_{p}^{N}$ we put

[TABLE]

and we recall that such a matrix defines an “Atkin-Lehner involution” on the space $M_{k}(N)$.

2.2 Lattices

We mainly consider even lattices $L$ in some positive definite quadratic space $(V,Q)$. Here $L$ is called even, if $Q(L)\subset{\mathbb{Z}}$. Then $L$ is automatically contained in its dual lattice $L^{\#}:=\{x\in V\mid(x,\ell)\in{\mathbb{Z}}\mbox{ for all }\ell\in L\}$ where $(x,y):=Q(x+y)-Q(x)-Q(y)$ is the associated bilinear form. The minimal number $N\in{\mathbb{N}}$ such that the adjoint lattice $\sqrt{N}L^{\#}:=(L^{\#},NQ)$ is again even is called the level of $L$. We also define the minimum $\min(L):=\min\{(\ell,\ell)\mid 0\neq\ell\in L\}$.

For a quadratic space $(V,Q)$ over ${\mathbb{Q}}$ we define the local Witt invariants $s_{p}(V)$ as in [23, p.80]. This normalization is very convenient for our purposes, in particular we will use the following lemma from [5].

Lemma 2.1**.**

*Let $L$ be an even lattice of level $N\cdot p$ with $p\nmid N$ in the quadratic space $(V,Q)$ Then the following statements are equivalent

i) $s_{p}(V)=1$

ii) $V$ carries (even) lattices of level $N$.

iii) If $L_{p}=L_{p}^{(0)}\perp L_{p}^{(1)}$ denotes the Jordan splitting of $L_{p}=L\otimes{\mathbb{Z}}_{p}$, then $L_{p}^{(1)}$ is an orthogonal sum of hyperbolic planes.*

3 Lattices maximal at $p$ and their theta series

We assume that $L$ is an even lattice in a positive definite quadratic space $(V,Q)$ of dimension $m=2k$ . We denote by $N$ the (exact) level of $L$. We put $D=det(L)$; then $(-1)^{k}D$ is a discriminant (i.e. it is congruent $1$ or [math] $\bmod 4$) and we denote by $(-1)^{k}d$ the corresponding fundamental discriminant (= a discriminant of a quadratic number field or equal to 1). Note that $d$ is odd because $N$ is squarefree.

We consider the theta series

[TABLE]

for $\tau\in{\mathbb{H}}$ and $q=e^{2\pi i\tau}$. Let $p$ be a prime with $p\mid N$.

We recall the transformation properties of $\theta(L)$ under $\omega_{p}^{N}$:

[TABLE]

Here $L^{\sharp,p}=L^{\sharp}\cap{\mathbb{Z}}[\frac{1}{p}]\cdot L$ is the lattice dualized only at $p$, $s_{p}(V)$ is the Witt invariant and $\gamma_{p}$ depends only on $d_{p}\cdot({\mathbb{Q}}_{p}^{\times})^{2}$, more precisely, $\gamma_{p}(1)=1$ and for odd primes $p$, $\delta\in{\mathbb{Z}}_{p}^{\times}$

[TABLE]

For details see [6, Lemma 8.2], [5], or in more classical language, [13], for the explicit determination of $\gamma_{p}$ see [9]. We do not need the more complicated $\gamma_{2}$ here.

Theorem 3.1**.**

Let $p$ be a prime divisor of $N$ with $p\mid\mid N$.

[TABLE]

We remark here that the statement of the theorem is local; actually the assumption that $N$ is squarefree is not necessary here.

Proof. ”$\Longleftarrow$”: The transformation properties of theta series imply

[TABLE]

Comparing constant terms on the right sides implies

[TABLE]

In any case, $(V,Q)$ does not carry a $p$-unimodular lattice and $D_{p}=p^{2}$ or $D_{p}=p$

“$\Longrightarrow$” Suppose that $L_{p}$ is maximal, in particular, $V_{p}$ does not carry a lattice, which is unimodular (at $p$), hence $s_{p}(V)=-1$. The local lattice $L_{p}$ has a decomposition

[TABLE]

such that $L_{p}^{(0)}$ is unimodular and the lattice $\sqrt{p}^{-1}L_{p}^{(1)}$ is anisotropic mod $p$ and of rank 1 or 2. This implies that any vector in $L_{p}^{\sharp}$ with length in ${\mathbb{Z}}_{p}$, is already in the sublattice $L_{p}$, which implies the global statement

[TABLE]

Taking into account that $s_{p}(V_{p})=-1$ and using the transformation formula from above, we therefore obtain

[TABLE]

Moreover, $D_{p}$ is either $p$ or $p^{2}$, i.e. $D_{p}=p^{2}\cdot d_{p}^{-1}$. The assertion follows. zzz $\square$

Remark 3.2**.**

We can more generally consider theta series with harmonic polynomials of degree $\nu$,

[TABLE]

Then we obtain again

[TABLE]

provided that $L_{p}$ is maximal and $p\mid\mid N$.

Remark 3.3**.**

Theorem 3.1 covers all maximal lattices except those where the fundamental discriminant $d$ is divisible by $2$ (where the level $N$ is divisible by $4$ and $8$ respectively).

We will mainly consider lattices which are maximal at all primes $p$. Concerning the existence we state

Proposition 3.4**.**

Suppose that $N$ is squarefree; then there is an even maximal lattice of even rank $m=2k$ with $\det(L)=N^{2}$ if and only if $m\equiv_{8}4$ and the number of prime divisors of $N$ is odd or $8\mid m$ and the number of prime divisors of $N$ is even.

Proof. Let $(V,Q)$ be a quadratic space over ${\mathbb{Q}}$ possibly carrying such a lattice. Then we have for finite primes

[TABLE]

and

[TABLE]

By the product formula for the Witt invariant, the number of prime divisors has to be odd ($m\equiv_{8}4$) or even (if $8\mid m$). In the other direction we prefer to give an explicit construction: For $N$ squarefree with an odd number of prime divisors, we choose a maximal order ${\mathcal{O}}(N)$ in the quaternion algebra over ${\mathbb{Q}}$ ramified exactly in the primes dividing $N$. We view it as usual as quadratic space (with the norm form). If $m\equiv_{8}4$ we may then take ${\mathcal{O}}(N)\oplus M$ as an example and for $8\mid m$ we take ${\mathcal{O}}(N_{1})\oplus{\mathcal{O}}(N_{2})\oplus M$. Here $M$ is an appropriate even unimodular lattice and $N=N_{1}\cdot N_{2}$ is a decomposition of $N$ into factors with an odd number of prime factors. The maximality of these lattices is then easily checked locally. zzz $\square$

4 The space $M_{k}(N)^{*}$

4.1 Definition and basic properties

The space of interest for us is (for any squarefree $N>1$ and even weight $k$)

[TABLE]

The subspace $S_{k}(N)^{*}$ of cuspforms in $M_{k}(N)^{*}$ was investigated in [1]. We recall some properties from there:

1. The definition may be rephrased in terms of the “trace”-operator (familiar from the theory of newforms [17]):

[TABLE]

We recall that trace${}_{\frac{N}{p}}^{N}:M_{k}(N)\longrightarrow M_{k}(\frac{N}{p})$ is defined by $f\longmapsto\sum_{\gamma}f\mid_{k}\gamma$, where $\gamma$ runs over $\Gamma_{0}(N)\backslash\Gamma_{0}(\frac{N}{p})$; using explicit representatives for the $\gamma$ we obtain the expression $\mbox{trace}^{N}_{\frac{N}{p}}(f)=f+p^{1-\frac{k}{2}}f\mid W^{N}_{p}\mid U(p)$

2. When we compare the definition of $S_{k}(N)^{*}$ with the characterization of newforms in terms of traces, we see that $S_{k}(N)^{*}$ satisfies half of the conditions describing newforms, see [17] for details. In particular, the space of newforms of level $N$ is contained in $S_{k}(N)^{*}$ and in fact it is easy to see from the theory of newforms that each eigenvalue system for the collection $\{T(p)\in End(S_{k}(N))\mid p\,\,\mbox{coprime to}\,\,N\}$ occurs with multiplicity one in $S_{k}(N)^{*}$. More precisely, $S_{k}(N)^{*}$ can be built out of the spaces of newforms of level $M\mid N$ as follows:

For a normalized Hecke eigenform $f=\sum_{n}a_{f}(n)q^{n}$ in $S_{k}(M)^{new}$ we put

[TABLE]

By the same reasoning as in [1], section 2.1, remark 2, we see that this defines an element of $S_{k}(N)^{*}$. We put

[TABLE]

where $f_{i}$ runs over the normalized Hecke eigenforms in $S_{k}(M)^{new}$. Then

[TABLE]

We computed the dimension of this space

[TABLE]

4. It is easy to see that $S_{k}(N)^{*}$ has codimension one in $M_{k}(N)^{*}$, so there is only one Eisenstein series in this space. Actually, we can (at least for $k\geq 4$) compute the Eisenstein series in $M_{k}(N)^{*}$ explicitly from the level one Eisenstein series $E_{k}$ by the same reasoning as above:

[TABLE]

4.2 The basis problem for $M_{k}(N)^{*}$

We want to span this space $M_{k}(N)^{*}$ by appropriate theta series. In [3] we already proved that $S_{k}(N)^{new}$ is always generated by linear combinations of theta series of quadratic forms from any fixed genus of quadratic forms with (exact) level $N$ and determinant $D$ such that $p^{2}\mid D$ and $p^{m}\nmid D$. The machinery developed in [3], section 8 can also be applied to oldforms in $M_{k}(N)$.

Theorem 4.1**.**

Suppose that the data $m=2k>4,N$ admit the existence of a genus ${\mathfrak{S}}$ of maximal lattices of determinant $N^{2}$ and rank $m$. Then

[TABLE]

where ${\mathfrak{S}}^{*}$ is the genus adjoint to ${\mathfrak{S}}$ and $\Theta({\mathfrak{S}}^{*})$ denotes the ${\mathbb{C}}$-vector space generated by the theta series $\theta(L)$, $L\in{\mathfrak{S}}^{*}$.

The statement above is false for $m=4$ unless $S_{k}(N)^{*}=S_{k}(N)^{new}$, as follows from the work of Eichler [8] and Hijikata-Saito [11] on the basis problem. Anyway, our proof would not work here (because of convergence reasons and because here (and only here) the genus of maximal lattices is equal to its adjoint genus).

Before we sketch the proof of this theorem, we recall from Theorem 3.1 that the inclusion

[TABLE]

holds. To simplify the exposition, we only consider the case $N=p$. We have to study the map

[TABLE]

Here $m(L)$ is the number of automorphisms of the lattice $L$ and the $L_{i}$ run over representatives of the classes in the genus (${\mathfrak{S}}^{*}$); the bracket $<,>$ denotes the Petersson product for modular forms. It is a general fact (“pullback formulas” for Eisenstein series) that this map can also be described completely in terms of Hecke operators, the explicit form of the contribution of the bad place $p$ depends on the genus at hand, see [3].

The case of newforms of level $p$ was discussed in [3].

We just have to add for a Hecke eigenform $f$ of level one an explicit description of the map $\Lambda$ for the two-dimensional space

[TABLE]

Indeed, it is of the form

[TABLE]

Here $c$ is an unimportant constant, $L_{2}(f,s)$ denotes the symmetric square $L$-function attached to $f$ and ${\mathcal{A}}_{p}$ is a certain $2\times 2$-matrix (involving the “Satake parameters” $\alpha_{p}$ and $\beta_{p}$ of $f$) which can be computed from [3]. The inclusion $\Theta({\mathfrak{S}}^{*})\subseteq M_{k}(p)^{*}$ already implies that the image of $M(f)$ under $\Lambda$ is at most one-dimensional. An inspection of ${\mathcal{A}}_{p}$ shows that it is always different from the zero matrix (i.e. of rank one), in other words, $M(f)$ will always be mapped onto the one-dimensional space ${\mathbb{C}}\cdot f^{(p)}\subseteq M_{k}(p)^{*}$.

Remark 4.2**.**

The case of an arbitrary squarefree number $N$ goes along the same line (Kronecker products of such $2\times 2$-matrices have then to be considered). A more detailed analysis of these matrices ${\mathcal{A}}_{p}$ for arbitrary genera ${\mathfrak{S}}$ will be given elsewhere [4].

By the same reasoning (or by applying the Fricke involution \left(\begin{array}[]{cc}0&-1\\ N&0\end{array}\right) to both sides of the theorem) we obtain

Corollary 4.3**.**

Under the same assumptions as in the theorem we have

[TABLE]

where

[TABLE]

Remark: Both the theorem and the corollary are remarkable because they describe precisely the “old” part of $\Theta({\mathfrak{S}}^{*})$ and $\Theta({\mathfrak{S}})$. From the point of view of [5] it may be of interest to study the trace of such an oldform: We consider the simplest case, $N=p$ and $f\in S_{k}(1)$ is a normalized Hecke eigenform. Then

[TABLE]

with $\lambda=p+1-\frac{p}{p+1}a_{f}(p)^{2}p^{-k+1}$. By Ramanujan-Petersson (see [7]) $|a_{f}(p)|\leq 2p^{(k-1)/2}$ and therefore $\lambda$ cannot be zero. On the other hand, $f^{(p)}$ is a linear combination of the $\theta(L)$ with $L\in{\mathfrak{S}}^{*}$. The trace of such theta series is not understood at all, see [5]. The situation is completely different for $f^{(p)}\mid W_{p}^{p}\in S_{k}(p)_{*}$: this function is in $\Theta({\mathfrak{S}})$ and the traces of the theta series are all zero. This fits well with the fact that $tr^{p}_{p}(f^{(p)}\mid W_{p}^{p})=0$.

4.3 $M(N)_{*}$ as a module over the ring of modular forms of level one

The orthogonal sum of a maximal lattice with an even unimodular lattice is again a maximal lattice. This elementary observation corresponds to fact that $M(N)_{*}=\bigoplus_{k}M_{k}(N)_{*}$ is a module over the ring of modular forms of level one. The corresponding module structure for $M(N)^{*}$ is defined by multiplying $f\in M_{k}(N)^{*}$ with g\mid_{\ell}\left(\begin{array}[]{cc}0&-1\\ N&0\end{array}\right) for $g\in M_{\ell}(1)$.

It is clear from the dimension formula for $M_{k}(N)^{*}$ that the number of generators grows with $N$.

In a few cases we can determine the module structure: From the point of view of lattices, we may consider the direct sum $\oplus_{k\equiv_{4}2}M_{k}(p)_{*}$ as a module over the graded ring $\oplus_{4\mid k}M_{k}(1)$. For $p=2$ and $p=3$ the module structure is already given in Chapter 10 of [21]. For these two primes $p$, the well known construction A establishes an isomorphism between $M_{k}(p)_{*}$ and the space spanned by the Hamming weight enumerators of maximal doubly-even self-orthogonal codes $C\leq{\mathbb{F}}_{2}^{2k}$ respectively maximal self-orthogonal codes $C\leq{\mathbb{F}}_{3}^{k}$. Note that these weight enumerators span the space of relative invariants of the associated Clifford Weil group. For details we refer to [21, Chapter 10]. To state the relevant result we need one construction.

Remark 4.4**.**

Let $R\neq\{0\}$ be a root lattice that is an orthogonal summand of the root sublattice of the 24-dimensional even unimodular lattice $L$. Then the lattice $M:=\{\ell\in L\mid(\ell,r)=0\mbox{ for all }r\in R\}$ is a lattice in dimension $24-\dim(R)$ with $M^{\#}/M\cong R^{\#}/R$. Though the isometry class of $M$ does depend on the choice of $L$, its theta series does not and we will denote it by $\theta(\operatorname{Comp}(R)):=\theta(M)$.

Proposition 4.5**.**

Let $R:=\bigoplus_{k\equiv_{4}0}M_{k}(1)={\mathbb{C}}[\theta({\mathbb{E}}_{8}),\theta(\Lambda_{24})]$ denote the ring spanned by theta series of even unimodular lattices.

• (i)

For any squarefree $N$ and any $i\in\{0,1,2,3\}$ the Fricke-involution is an $R$-module isomorphism between $\bigoplus_{k\equiv_{4}i}M_{k}(N)^{*}$ and $\bigoplus_{k\equiv_{4}i}M_{k}(N)_{*}$.

• (ii)

The module $\bigoplus_{k\equiv_{4}2}M_{k}(2)_{*}$ is the free $R$-module of rank $2$ with basis $(\theta({\mathbb{D}}_{4})$, $\theta(\operatorname{Comp}({\mathbb{D}}_{4})))$.

• (iii)

The module $\bigoplus_{k\equiv_{4}2}M_{k}(3)_{*}$ is the free $R$-module of rank $3$ with basis $(\theta({\mathbb{A}}_{2}\perp{\mathbb{A}}_{2})$, $\theta({\mathbb{E}}_{6}\perp{\mathbb{E}}_{6})$, $\theta(\operatorname{Comp}({\mathbb{A}}_{2}\perp A_{2})))$.

Proof. The first statement is clear, the second one is included in [21, Theorem 10.7.14] and the last one follows from [21, Corollary 10.7.7]. zzz $\square$

From the point of view of modular forms the full space $\oplus M_{k}(p)_{*}$ deserves attention as well as a module over the full graded ring of modular forms of level one.

We consider the cases $p=2,p=3$:

The case $p=2$: The dimension formulas show that we will certainly need $e_{2}$, $e_{4}$ as generators, where $e_{2}$ is the unique Eisenstein series in $M_{2}(2)_{*}$ and $e_{4}=E_{4}^{2}\mid W^{2}_{2}$ is the unique Eisenstein series in $M_{4}(2)_{*}$. The dimension formulas show that

[TABLE]

We can further show that the quotient $\frac{e_{4}}{e_{2}}$ is not a meromorphic modular form of weight $2$ for $SL(2,{\mathbb{Z}})$, therefore

Proposition 4.6**.**

The space $\oplus_{2\mid k}M_{k}(2)_{*}$ is a free module over the ring of modular forms of level one with basis $(e_{2},e_{4})$.

**The case $p=3$: ** Again the dimension formulas show that we need at least the generators $e_{2},e_{4},h_{6}$, where $e_{2}$ and $e_{4}$ again denote the Eisenstein series in the space $M_{2}(3)_{*}$ and $M_{4}(3)_{*}$ and $h_{6}$ is a nonzero element in the one-dimensional space $S_{6}(3)=S_{6}(3)_{*}$. An inspection of the Fourier expansions (in the cusps $\infty$ and [math] ) shows that a nontrivial relation

[TABLE]

with level one modular forms $E,F,H$ of weights $k-2,k-4$ and $k-6$ is not possible. On the other hand, the dimension formula gives the identity

[TABLE]

therefore we get

Proposition 4.7**.**

The space $\oplus_{2\mid k}M_{k}(3)_{*}$ is a free module over the ring of modular forms of level one with basis $(e_{2},e_{4},h_{6})$.

5 Extremality

5.1 Generalities on analytic extremality

Definition 5.1**.**

A subspace ${\mathcal{M}}\subseteq M_{k}(N)$ has the Weierstrass property $(\mathcal{W})$ if the projection ${\mathcal{M}}\longrightarrow{\mathbb{C}}^{r}$ to the first $r=dim{\mathcal{M}}$ coefficients of the Fourier expansion

[TABLE]

is injective. If this holds, the unique element

[TABLE]

with Fourier expansion

[TABLE]

is called the extremal modular form in ${\mathcal{M}}$.

If ${\mathcal{M}}$ contains (say, by definition) only modular forms with vanishing Fourier coefficient $a_{0}$, the definition of “Weierstrass property” has to be modified in the obvious way. Note that $(\mathcal{W})$ holds for ${\mathcal{M}}$ iff $(\mathcal{W})$ holds for the cuspidal subspace of $\mathcal{M}$, provided that the codimension of the cuspidal part in ${\mathcal{M}}$ is one.

The notion “Weierstrass property” is motivated by the connection of this property with $\infty$ being a Weierstrass points on the modular curve $X_{0}(N)$ if ${\mathcal{M}}=S_{2}(N)$, see e.g. [22].

Suppose now that we have a lattice $L$ such that $\theta(L)\in\mathcal{M}$ for a space $\mathcal{M}$ with property ($\mathcal{W}$). Then we may call the lattice $L$ analytically extremal with respect to $\mathcal{M}$ if

[TABLE]

In particular, such an analytically extremal lattice satisfies

[TABLE]

In this generality this definition was introduced in [24].

Of course these notions only make sense, if we know interesting classes of such distinguished subspaces ${\cal M}$.

Example 5.2**.**

*(1) Clearly, for any lattice $L$, the one-dimensional space ${\mathcal{M}}:={\mathbb{C}}\cdot\theta(L)$ has the property (${\mathcal{W}}$) and then $L$ is extremal with respect to this space.

(2) The full space $M_{k}(1)$ of modular forms of level 1 has the Weierstrass property and the well-known Leech lattice is then an ${\cal M}=M_{12}(1)$- extremal lattice.

(3) The spaces of modular forms for the Fricke groups considered by Quebbemann [25, 26] in his work on modular lattices.*

5.2 Analytic extremality with respect to $M_{k}(N)^{*}$

In general, neither the spaces $S_{k}(N)$ nor $S_{k}(N)^{new}$ (or versions of it appropriately enlarged by some Eisenstein series) have the Weierstrass property. In the case of squarefree level $N$ we showed in [1] that the intermediate space $S_{k}(N)^{*}$ (and therefore also $M_{k}(N)^{*}$) has the property $({\mathcal{W}})$, therefore there is an extremal modular form

[TABLE]

in this case.

Definition 5.3**.**

A maximal lattice $L$ of level $N$ and determinant $N^{2}$ in dimension $m=2k$ is called dual extremal, if $\theta(\sqrt{N}L^{\#})=F_{N,k}$.

Remark 5.4**.**

*(1) Our definition allows to define analytic extremality for all squarefree levels. This is in contrast to the situation studied by Quebbemann [25, 26].

(2) The additional information $\Theta({\mathfrak{S}}^{*})=M_{k}(N)^{*}$ is not necessary for the definition of dual extremal lattices, but it shows that the space ${\mathcal{M}}=M_{k}(N)^{*}$ the smallest one to be chosen for adjoints of maximal lattices.*

Example 5.5**.**

Let $D$ be a rational definite quaternion algebra ramified exactly at the prime $p$. Then any lattice $L$ of level $p$ in the quadratic space $(D,n)$, where $n$ is the norm form, is a maximal even lattice. These lattices $L$ are fractional left-ideals for some maximal order in $D$. The non principal $L$ satisfy $\min(L)\geq 4$. If the class number (the number of isomorphism classes of left-ideals for a fixed maximal order in $D$) is two, then $\dim M_{2}(p)^{*}=2$ since $S_{2}(p)^{*}=S_{2}(p)^{new}$ and any non-principal $L$ is dual extremal. Note that the definite quaternion algebras over ${\mathbb{Q}}$ with class number two are classified by the work of Kirschmer and Voight [14]: $N=p\in\{11,17,19\}$ and $N=2\cdot 3\cdot 5,N=2\cdot 3\cdot 7,N=2\cdot 3\cdot 13,N=2\cdot 5\cdot 7$. The condition $S_{2}(N)^{*}=S_{2}(N)^{new}$, which is quite special for the case $m=4$, is automatically satisfied if $N=p$, but never in the other cases of class number two as can be seen by evaluating the dimension formula for $S_{2}(N)^{*}$.

5.3 A remark about extremal modular forms of level $p$ and weight divisible by $p-1$

Proposition 5.6**.**

Let $p$ be a prime. Assume that the weight $k$ is divisible by $p-1$. Then any modular form $f\in M_{k}(1)$ with Fourier expansion

[TABLE]

satisfies

[TABLE]

Corollary 5.7**.**

Let $p\geq 5$ be a prime. Then any extremal modular form $g\in M_{k}(p)^{*}$ with $k$ divisible by $p-1$ satisfies

[TABLE]

Proof. (of Proposition 5.6) There exists a modular form $\mathcal{E}$ of weight $k$ with ${\mathcal{E}}\equiv 1\bmod p$. For $p\geq 5$ we may take an appropriate power of the Eisenstein series $E_{p-1}$ of weight $p-1$. For $p=2$ or $p=3$ we can take a suitable monom $E_{4}^{\alpha}\cdot E_{6}^{\beta}$. Therefore we can write $f$ as

[TABLE]

with

[TABLE]

such that the first $d-1$ coefficients $b_{i}$ are congruent zero mod $p$. For $1\leq i\leq d-1$ we choose $f_{j}\in S_{k}(1)$ with integral Fourier coefficients $c_{i,n}$ such that for §$1\leq i,j\leq d-1$

[TABLE]

Such cusp forms always exist, see e.g. [16, Theorem 4.4].

Then

[TABLE]

such that the first $d$ Fourier coefficients of $H$ are zero, hence $H$ is identically zero. The assertion follows. zzz $\square$

To prove the corollary we note that (by [1]) $g$ is equivalent mod $p$ to a modular form $G\in M_{k+(p-1)(k-1)}(1)$ provided that $p\geq 5$. We apply Proposition 5.6 to this $G$.

Remark: Using a suitable interpretation of the congruence of modular forms, it is not necessary in the statements above to assume that the Fourier coefficients of the modular forms are rational.

Remark: It would be desirable to include the cases $p=2$ and $p=3$ in the corollary.

6 Examples for dual extremal maximal lattices.

This section lists some examples of dual extremal maximal lattices of small level $N$ and small dimension $m$. For $N=2$ and $N=3$, one may deduce the classification of all dual extremal lattices from suitable known classifications of unimodular lattices. For the higher levels $N\geq 5$ we use Kneser’s neighboring method [15] to list the whole genus of maximal lattices together with the mass formula to double check the completeness of the result. The computer calculations where performed with MAGMA. Gram matrices for the new lattices are available in [20].

6.1 $N=2$.

Let $L$ be a maximal 2-elementary lattice of exact level 2 and even dimension $m:=\dim(L)=2k\equiv_{8}4$. Then $L$ is the even sublattice of an odd unimodular lattice $M$ and $L^{\#}=M\cup v+M$ where $2v\in M$ is a characteristic vector of $M$, i.e. $(2v,x)\equiv_{2}(x,x)$ for all $x\in M$. If $\mu=\min(M)$ and $4\sigma$ is the minimal norm of a characteristic vector in $M$, then $4\sigma\equiv_{8}m$ and

[TABLE]

Philippe Gaborit proved in [10] that for $m\neq 23$

[TABLE]

Lattices achieving this bound are called s-extremal. We use $\star$ to show that dual extremal lattices $L$ satisfy $\min(L^{\#})=\lfloor\frac{k+4}{6}\rfloor$.

Proposition 6.1**.**

Let $L$ be a dual-extremal maximal lattice of level $2$ and dimension $m=24\ell+4$. Then $L^{\#}$ has minimum $1+2\ell$ and all layers of $L$ and of $L^{\#}$ form spherical 5-designs. In particular $L$ and $L^{\#}$ are strongly perfect. If $M$ is one of the three odd unimodular lattices with even sublattice $L$, then $M$ is $s$-extremal of minimum $1+2\ell$.

Proof. Let $\mu:=\min(M)$ and $\sigma:=\min(L^{\#}-M)$. Since $L$ is dual-extremal $\mu$ and $\sigma$ are both $\geq 1+2\ell$. By the bound in [10] we obtain $\mu+\frac{\sigma}{2}\leq\frac{3}{2}+3\ell$ hence $\mu=\sigma=1+2\ell$. The design property follows from the fact that \dim(\mbox{M_{12\ell+2}(2)^{}})=\dim(\mbox{M_{12\ell+4}(2)^{}})=\dim(\mbox{M_{12\ell+6}(2)^{*}})=2\ell+1. zzz $\square$

Similarly we obtain

Proposition 6.2**.**

Let $L$ be a dual-extremal maximal lattice of level $2$ and dimension $m=24\ell-4$ and let $M$ be one of the three odd unimodular lattices with even sublattice $L$. Then $M$ is $s$-extremal of minimum $2\ell$. The minimum of $L^{\#}-M$ is $2\ell+1$ and the minimal vectors of $L^{\#}$ (which are also those of $L$ and those of $M$) form a spherical 3-design, which means that $L^{\#}$, $L$ and $M$ are all strongly eutactic. The lattice $M$ is $s$-extremal.

Proof. Let $\mu:=\min(M)$ and $\sigma:=\min(L^{\#}-M)$. Since $L$ is dual-extremal $\mu$ and $\sigma$ are both $\geq 2\ell$. Since $\sigma\equiv_{2}\frac{m}{4}$ it is odd $\sigma\geq 2\ell+1$. By the bound $\star$ above we obtain $\mu+\frac{\sigma}{2}\leq\frac{1}{2}+3\ell$ hence $\mu=2\ell,$ $\sigma=1+2\ell$ and $M$ is $s$-extremal. zzz $\square$

Proposition 6.3**.**

Let $L$ be a dual-extremal maximal lattice of level $2$ and dimension $m=24\ell+12$. Then $\min(L^{\#})=2\ell+1$.

Proof. Let $M$ be one of the three odd unimodular lattices with even sublattice $L$. Let $\mu:=\min(M)$ and $\sigma:=\min(L^{\#}-M)$. Since $L$ is dual extremal, $\min(\mu,\sigma)\geq 2\ell+1$. By Gaborit’s bound $\mu+\frac{\sigma}{2}\leq 3\ell+2+\frac{1}{2}$. If $\min(\mu,\sigma)\geq 2\ell+2$, then $\mu+\frac{\sigma}{2}\geq 3\ell+3$ contradicting the bound above. zzz $\square$

Corollary 6.4**.**

A dual extremal lattice $L$ of level $2$ and dimension $2k\equiv_{8}4$ satisfies $\min(\sqrt{2}L^{\#})=2\lfloor\frac{k+4}{6}\rfloor$.

6.1.1 $m=4$

Here the root lattice ${\mathbb{D}}_{4}$ is the unique maximal 2-elementary lattice and dual extremal.

6.1.2 $m=12$

The two root lattices ${\mathbb{D}}_{4}\perp{\mathbb{E}}_{8}$ and ${\mathbb{D}}_{12}$ are all maximal 2-elementary lattices and both are dual extremal.

6.1.3 $m=20$

Let $L$ be a maximal 2-elementary lattice of dimension 20. Then $L\perp{\mathbb{D}}_{4}$ is contained in some even unimodular lattice $U$ of dimension 24. Since $L$ is maximal it is the orthogonal supplement $\operatorname{Comp}({\mathbb{D}}_{4})$ of ${\mathbb{D}}_{4}$ in $U$ and $L^{\#}$ is the projection of $U$ to ${\mathbb{D}}_{4}^{\perp}$. Since $\min(\sqrt{2}L^{\#})\geq 4$, all roots of $U$ are either in ${\mathbb{D}}_{4}$ or perpendicular to this sublattice. Hence ${\mathbb{D}}_{4}$ is an orthogonal summand of the root system of $U$, which is therefore either ${\mathbb{D}}_{4}^{6}$ or ${\mathbb{D}}_{4}\perp{\mathbb{A}}_{5}^{4}$. Both lattices $U$ contain a unique $\operatorname{Aut}(U)$-orbit of such sublattices ${\mathbb{D}}_{4}$ yielding the two dual extremal 2-elementary lattices of dimension 20.

6.1.4 $m=28$

Let $L$ be a maximal 2-elementary lattice of dimension 28 and $M$ be an odd unimodular lattice containing $L$. If $L$ is dual extremal, then $\min(L^{\#})\geq 3$ and hence $M$ has minimum $3$. The 28-dimensional unimodular lattices of minimum 3 are all classified in [2]. There are 38 isometry classes of such lattices, two of which have a characteristic vector of norm 4. The other 36 lattices give rise to 31 even sublattices $L$ which are all dual extremal. By Proposition 6.1 the $6720$ minimal vectors of $L^{\#}$ as well as all layers of $L$ and $L^{\#}$ form spherical 5-designs and hence $L^{\#}$ is a strongly perfect lattice (see [27]). The next dimension where such a phenomenon occurs is $m=52$, where $\min(L^{\#})=5$. Then any unimodular sublattice $M$ (with even sublattice $L$) is an s-extremal lattice of minimum $5$ in the sense of [10]. Up to now, no such lattice is known.

6.2 $N=3$.

A dual extremal lattice $L$ of dimension $m=2k\equiv_{8}4$ satisfies $\min(\sqrt{3}L^{\#})\geq 2\frac{k+2}{4}$

6.2.1 $m=4$.

Here ${\mathbb{A}}_{2}\perp{\mathbb{A}}_{2}$ is the unique maximal 3-elementary lattice and this is dual extremal.

6.2.2 $m=12$.

The 3-elementary maximal lattices are ${\mathbb{A}}_{2}\perp{\mathbb{A}}_{2}\perp{\mathbb{E}}_{8}$ and ${\mathbb{E}}_{6}\perp{\mathbb{E}}_{6}$, the latter is dual extremal.

6.2.3 $m=20$.

Let $L$ be a dual extremal 3-elementary lattice of dimension 20. Then $L\perp{\mathbb{A}}_{2}\perp{\mathbb{A}}_{2}$ is contained in an even unimodular lattice of dimension 24. As for $N=2$ the dual extremality of $L$ implies that the root system of $U$ is ${\mathbb{A}}_{2}^{12}$ and there is a unique such lattice $L$.

6.2.4 $m=28$.

Let $L$ be a dual extremal 3-elementary lattice of dimension 28 and let $U$ be an even unimodular lattice of dimension 32 containing $L\perp{\mathbb{A}}_{2}\perp{\mathbb{A}}_{2}$. Then $\min(L^{\#})\geq 8/3>2$ implies that $L$ has no roots and that the root system of $U$ is ${\mathbb{A}}_{2}\perp{\mathbb{A}}_{2}$. By [12] the mass of such lattices $U$ is $>41610$ so there are more than $72\cdot 41610$ such lattices. Every lattice $L\perp{\mathbb{A}}_{2}\perp{\mathbb{A}}_{2}$ is contained in 8 unimodular lattices, so it follows from the discussion below that there are at least $9\cdot 41610$ dual extremal lattices. The lattice $L^{\#}$ is the projection of $U$ to $({\mathbb{A}}_{2}\perp{\mathbb{A}}_{2})^{\perp}$, so

[TABLE]

Here we may assume that $z$ is minimal in its class modulo ${\mathbb{A}}_{2}\perp{\mathbb{A}}_{2}$. Then $(z,z)\in\{0,\frac{2}{3},\frac{4}{3}\}$. If $x\neq 0$ then $(y,y)\geq 4$ and $(x,x)=(y,y)-(z,z)\geq 4-\frac{4}{3}=\frac{8}{3}$. This shows that for all these lattices $U$ the orthogonal $L$ of the root sublattice of $U$ is dual extremal.

We list these results and the ones found for level $N=5,7,11$ resp. $N=6,10$ in the following tables, with lines labeled by the level $N$ and columns labeled by the dimension $m$. Each entry is the triple $(h,h_{ext},min)$ giving the class number $h$ of the genus of maximal lattices, the number $h_{ext}$ of isometry classes of dual extremal maximal lattices as well as the minimum $\min(\sqrt{N}L^{\#})$. A “$\cdot$” instead of $h$ indicates that we did not compute the full genus. Note that for dimension $m=4$, the classification follows from Example 5.5.

[TABLE]

Remark 6.5**.**

It is interesting to note that for level $N=11$ and dimension $m=20$, the extremal theta series is $1+132q^{10}+660q^{12}+1320q^{13}+2640q^{14}+\ldots$ so any dual extremal lattice $L$ satisfies $\min(\sqrt{11}L^{\#})=20>2\dim(M_{10}(11)^{*})=18$. So Corollary 6.4 does not hold in general for arbitrary levels. Note that here the $132$ minimal vectors of $L$ form a spherical $2$-design. We constructed such a lattice $L$ as the orthogonal supplement $L=\operatorname{Comp}(D)\leq\Lambda_{24}$ in the Leech lattice, where $D$ is the dual extremal lattice of level $11$ and dimension $4$.

[TABLE]