Dependencies

We have

\(\Delta (z) \neq 0\) for all \(z \in \mathfrak {H}\).

\(\Delta (it) {\gt} 0\) for all \(t {\gt} 0\).

196 (resp. 197) is positive (resp. negative) on the (positive) imaginary axis.

191 holds.

192 holds.

There exists a constant \(C_0 {\gt} 0\) such that

for all \(z\) with \(\Im z {\gt} 1/2\).

We have

There exists a constant \(C_{-2} {\gt} 0\) such that

for all \(z\) with \(\Im z {\gt} 1/2\).

There exists a constant \(C_{-4} {\gt} 0\) such that

for all \(z\) with \(\Im z {\gt} 1/2\).

We have

All three functions \(t \mapsto H_2(it), H_3(it), H_4(it)\) are positive for \(t {\gt} 0\).

\(\Delta _8 = \Delta _{E_8}\).

All packing \(\mathcal{P} \subseteq \mathbb {R}^8\) has density satisfying \(\Delta _{\mathcal{P}} \leq \Delta _{E_8}\).

For \(x\in \mathbb {R}^8\) we define

The automorphy factor of weight \(k\) is defined as

For \(x\in \mathbb {R}^8\) define

A subgroup \(\Gamma \subset \Gamma _1\) is called a congruence subgroup if \(\Gamma (N)\subset \Gamma \) for some \(N\in \mathbb {N}\).

The Dedekind eta function is defined as

where \(q = e^{2\pi i z}\).

Let \(F\) be a quasimodular form. We define the (normalized) derivative of \(F\) as

The discriminant form \(\Delta (z)\) is given by

We set

For an even integer \(k\geq 4\) we define the weight \(k\) Eisenstein series as

Define two (quasi) modular forms as

The Fourier transform of an \(L^1\)-function \(f:\mathbb {R}^d\to \mathbb {C}\) is defined as

where \(\langle x, y \rangle = \frac12\| x\| ^2 + \frac12\| y\| ^2 - \frac12\| x - y\| ^2\) is the standard scalar product in \(\mathbb {R}^d\).

The modular group \(\Gamma _1:=\mathrm{SL}_2(\mathbb {Z})\) acts on \(\mathfrak {H}\) by linear fractional transformations

Define the matrices

It is easily verifiable that \(\alpha = T^2\) and \(\beta = -S\alpha ^{-1}S = -ST^{-2}S\).

Define

A (holomorphic) modular form of integer weight \(k\) and congruence subgroup \(\Gamma \) is a holomorphic function \(f:\mathfrak {H}\to \mathbb {C}\) such that:

1. (Slash invariant) \(f|_k\gamma =f\) for all \(\gamma \in \Gamma \)

2. (Holomorphic at \(i\infty \)) for each \(\alpha \in \Gamma _1\; f|_k\alpha \) has the Fourier expansion \(f|_k\alpha (z)=\sum _{n=0}^\infty c_f(\alpha ,\frac{n}{n_\alpha })\, e^{2\pi i \frac{n}{n_\alpha }z}\) for some \(n_\alpha \in \mathbb {N}\) and Fourier coefficients \(c_f(\alpha ,m)\in \mathbb {C}\).

The level \(N\) principal congruence subgroup of \(\Gamma _1\) is

Let \(M_k(\Gamma )\) be the space of modular forms of weight \(k\) and congruence subgroup \(\Gamma \).

The periodic sphere packing constant is defined to be

We define the following three functions

A \(C^\infty \) function \(f:\mathbb {R}^d\to \mathbb {C}\) is called a Schwartz function if it decays to zero as \(\| x\| \to \infty \) faster then any inverse power of \(\| x\| \), and the same holds for all partial derivatives of \(f\), ie, if for all \(k, n \in \mathbb {N}\), there exists a constant \(C \in \mathbb {R}\) such that for all \(x \in \mathbb {R}^d\), \(\left\lVert x \right\rVert ^k \cdot \left\lVert f^{(n)}(x) \right\rVert \leq C\), where \(f^{(n)}\) denotes the \(n\)-th derivative of \(f\) considered along with the appropriate operator norm. The set of all Schwartz functions from \(\mathbb {R}^d\) to \(\mathbb {C}\) is called the Schwartz space. It is an \(\mathbb {R}\)-vector space.

For \(k \in \mathbb {R}\), define the weight \(k\) Serre derivative \(\partial _{k}\) of a modular form \(F\) as

Let \(F\) be a function on \(\mathfrak {H}\) and \(\gamma \in \mathrm{SL}_2(\mathbb {Z})\). Then the slash operator acts on \(F\) by

We define three different theta functions (so called “Thetanullwerte”) as

The two definitions above coincide, i.e. \(\Lambda _8 = \mathrm{span}_{\mathbb {Z}}(\mathcal{B}_8)\).

\(B_8\) is a \(\mathbb {R}\)-basis of \(\mathbb {R}^8\).

\(\Lambda _8\) is an additive subgroup of \(\mathbb {R}^8\).

(\(E_8\)-lattice, Definition 2)We define the \(E_8\) basis vectors to be the set of vectors

(\(E_8\)-lattice, Definition 1)We define the \(E_8\)-lattice (as a subset of \(\mathbb {R}^8\)) to be

All vectors in \(\Lambda _8\) have norm of the form \(\sqrt{2n}\), where \(n\) is a nonnegative inteeger.

The \(E_8\) sphere packing is the (periodic) sphere packing with separation \(\sqrt{2}\), whose set of centres is \(\Lambda _8\).

\(\operatorname {Vol}\! \left(\Lambda _8\right) = \mathrm{Covol}(\mathbb {R}^8 / \Lambda _8) = 1\).

We have \(\Delta _{\mathcal{P}(E_8)} = \frac{\pi ^4}{384}\).

\(c\Lambda _8\) is discrete, i.e. that the subspace topology induced by its inclusion into \(\mathbb {R}^8\) is the discrete topology.

\(c\Lambda _8\) is a \(\mathbb {Z}\)-lattice, i.e. it is discrete and spans \(\mathbb {R}^8\) over \(\mathbb {R}\).

We say that an additive subgroup \(\Lambda \leq \mathbb {R}^d\) is a lattice if it is discrete and its \(\mathbb {R}\)-span contains all the elements of \(\mathbb {R}^d\).

More explicitly, we have

The automorphy factor satisfies the chain rule

The Dedekind eta function transforms as

We have an equality of operators \(D = q \frac{\mathrm{d}}{\mathrm{d}q}\). In particular, the \(q\)-series of the derivative of a quasimodular form \(F(z) = \sum _{n \ge n_0} a_n q^n\) is \(F'(z) = \sum _{n \ge n_0} n a_n q^n\).

\(\Delta (z) \in M_{12}(\Gamma _1)\). Especially, we have

Also, it vanishes at the unique cusp, i.e. it is a cusp form of level \(\Gamma _1\) and weight \(12\).

We have

This function is not modular, however it satisfies

The Eisenstein series possesses the Fourier expansion

where \(\sigma _{k-1}(n)\, =\, \sum _{d|n} d^{k-1}\). In particular, we have

For all \(k\), \(E_k\in M_k(\Gamma _1)\). Especially, we have

\(F\) and \(G\) satisfy the following differential equations:

We have

For all \(t {\gt} 0\), we have \(F(it) {\gt} 0\) and \(G(it) {\gt} 0\).

The Fourier transform is a continuous, linear automorphism of the space of Schwartz functions.

We have \(\Gamma (1) = \langle S, T, -I \rangle \).

We have \(\Gamma (2) = \langle \alpha , \beta , -I \rangle \).

Inequality 184 and 186 are equivalent to

respectively.

Let \(X \subset \mathbb {R}^d\) be a set of sphere packing centres of separation \(1\) that is periodic with some lattice \(\Lambda \subset \mathbb {R}^d\). Then, there exists \(k \in \mathbb {N}\) sufficiently large such that

These three theta functions satisfy the Jacobi identity

For all \(R\), we have the following inequality relating the number of lattice points from \(\Lambda \) in a ball with the volume of the ball and the fundamental region \(\mathcal{D}\):

We have

Let \(f(z)\) be a holomorphic function with a Fourier expansion

with \(c_f(n_0) \ne 0\). Assume that \(c_f(n)\) has a polynomial growth, i.e. \(|c_f(n)| = O(n^k)\) for some \(k \in \mathbb {N}\). Then there exists a constant \(C_f {\gt} 0\) such that

for all \(z\) with \(\Im z {\gt} 1/2\).

For all \(R\), we have the following inequality relating the number of points from \(X\) (periodic w.r.t. \(\Lambda \)) in a ball with the number of points from \(\Lambda \):

We have

There exists a constant \(C_I {\gt} 0\) such that

for all \(z\) with \(\Im z {\gt} 1/2\).

The Fourier expansions of these functions are

There exists a constant \(C_S {\gt} 0\) such that

for all \(z\) with \(\Im z {\gt} 1/2\).

\(\psi _S(z)\) can be written as

We have

Let \(f : \mathbb {R}^d \to \mathbb {C}\) be a Schwartz function and let \(X \subset \mathbb {R}^d\) be periodic with respect to some lattice \(\Lambda \subset \mathbb {R}^d\). Then,

For even \(k\), \(F|_{k}(-I) = F\).

The chain rule implies

For any \(R {\gt} 0\),

For all \(\gamma \in \Gamma _1\), \(H_{2}|_2 \gamma \), \(H_{3}|_2 \gamma \), and \(H_{4}|_2 \gamma \) are holomorphic at \(i\infty \).

\(H_{2}\), \(H_{3}\), and \(H_{4}\) belong to \(M_2(\Gamma (2))\).

\(H_{2}\), \(H_{3}\), and \(H_{4}\) are slash invariant under \(\Gamma (2)\), i.e. for all \(\gamma \in \Gamma (2)\) and \(i \in \{ 2, 3, 4\} \), we have \(H_i|\gamma = H_i|\gamma ^{-1} = H_i\).

These elements act on the theta functions in the following way

and

For any constant \(C {\gt} 0\), we have

For all \(d\), the periodic sphere packing constant in \(\mathbb {R}^d\) is equal to the sphere packing constant in \(\mathbb {R}^d\).

We say that a sphere packing \(\mathcal{P}(X)\) is (\(\Lambda \)-)periodic if there exists a lattice \(\Lambda \subset \mathbb {R}^d\) such that for all \(x \in X\) and \(y \in \Lambda \), \(x + y \in X\) (ie, \(X\) is \(\Lambda \)-periodic).

For \(r\geq 0\) we have

The integral converges absolutely for all \(r\in \mathbb {R}_{\geq 0}\).

For \(r{\gt}\sqrt{2}\) we can express \(a(r)\) in the following form

\(a(x)\) satisfies 110.

\(a(x)\) is a Schwartz function.

We have \(a(0) = -\frac{i}{8640}\).

For \(r\geq 0\) we have

The integral converges absolutely for all \(r\in \mathbb {R}_{\geq 0}\).

For \(r{\gt}\sqrt{2}\) function \(b(r)\) can be expressed as

\(b(x)\) satisfies 111.

\(b(x)\) is a Schwartz function.

We have \(b(0) = 0\).

\(H_2\) admits a Fourier series of the form

for some \(c_{H_2}(n) \in \mathbb {R}_{\ge 0}\), with \(c_{H_2}(1) = 16\) and \(c_{H_2}(n) = O(n^k)\) for some \(k \in \mathbb {N}\).

\(H_3\) admits a Fourier series of the form

for some \(c_{H_3}(n) \in \mathbb {R}_{\ge 0}\) with \(c_{H_3}(0) = 1\) and \(c_{H_3}(n) = O(n^k)\) for some \(k \in \mathbb {N}\). Especially, \(H_3\) is not cuspidal.

\(H_4\) admits a Fourier series of the form

for some \(c_{H_4}(n) \in \mathbb {R}\) with \(c_{H_4}(0) = 1\) and \(c_{H_4}(n) = O(n^k)\) for some \(k \in \mathbb {N}\). Especially, \(H_4\) is not cuspidal.

Consider the function \(A:(0,\infty )\to \mathbb {C}\) defined as

Then

for all \(t {\gt} 0\).

Consider the function \(B:(0,\infty )\to \mathbb {C}\) defined as

Then

for all \(t {\gt} 0\).

The function \(t \mapsto Q(t)\) is monotone decreasing.

We have

or equivalently,

Given a set \(X \subset \mathbb {R}^d\) and a real number \(r {\gt} 0\) (known as the separation radius) such that \(\| x - y\| \geq r\) for all distinct \(x, y \in X\), we define the sphere packing \(\mathcal{P}(X)\) with centres at \(X\) to be the union of all open balls of radius \(r\) centred at points in \(X\):

We define the density of a packing \(\mathcal{P}\) as the limit superior

The finite density of a packing \(\mathcal{P}\) is defined as

Given a sphere packing \(\mathcal{P}(X)\) with separation radius \(r\), we defined the scaled packing with respect to a real number \(c {\gt} 0\) to be the packing \(\mathcal{P}(cX)\), where \(cX = \left\{ cx \in V \; \middle | \; x \in X \right\} \) has separation radius \(cr\).

Let \(\mathcal{P}(X)\) be a sphere packing and \(c\) a positive real number. Then, the density of the scaled packing \(\mathcal{P}(cX)\) is equal to the density of the original packing \(\mathcal{P}(X)\).

Let \(\mathcal{P}(X)\) be a sphere packing and \(c\) a positive real number. Then, for all \(R {\gt} 0\),

The sphere packing constant is defined as supremum of packing densities over all possible packings:

All periodic packing \(\mathcal{P} \subseteq \mathbb {R}^8\) has density satisfying \(\Delta _{\mathcal{P}} \leq \Delta _{E_8} = \frac{\pi ^4}{384}\), the density of the \(E_8\) sphere packing (see definition 3.9).

For a periodic sphere packing \(\mathcal{P} = \mathcal{P}(X)\) with centers \(X\) periodic to the lattice \(\Lambda \) and separation \(r\),

Let \(F\) be a holomorphic quasimodular cusp form with real Fourier coefficients. Assume that there exists \(k\) such that \((\partial _{k}F)(it) {\gt} 0\) for all \(t {\gt} 0\). If the first Fourier coefficient of \(F\) is positive, then \(F(it) {\gt} 0\) for all \(t {\gt} 0\).

(Cohn–Elkies [ 3 ] ) Suppose that \(f:\mathbb {R}^d\to \mathbb {R}\) is a Schwartz function that is not identically zero and satisfies the following conditions:

and

Then the density of \(d\)-dimensional sphere packings is bounded above by

Let \(\Gamma \) be a congruence subgroup. Then \(M_k(\Gamma )\) is finite-dimensional.

There exists a radial Schwartz function \(g:\mathbb {R}^8\to \mathbb {R}\) which satisfies:

The function

satisfies conditions 6–8.

Let \(k \in \mathbb {Z}\) with \(k \ge 0\) and even. Then \(\dim M_k(\Gamma _1) = \lfloor k / 12 \rfloor \) if \(k \equiv 2 \mod 12\) and \(\dim M_k(\Gamma _1) = \lfloor k / 12 \rfloor + 1\) if \(k \not\equiv 2 \mod 12\).

Let \(k \in \mathbb {Z}\) with \(k {\lt} 0\). Then \(M_k(\Gamma _1) = \{ 0\} \) and moreover \(\dim M_0(\Gamma (1)) = 1\).

Let \(\Lambda \) be a lattice in \(\mathbb {R}^d\), and let \(f:\mathbb {R}^d\to \mathbb {R}\) be a Schwartz function. Then, for all \(v \in \mathbb {R}^d\),

We have

Let \(F\) be a modular form of weight \(k\) and level \(\Gamma \). Then, \(\partial _{k}F\) is a modular form of weight \(k + 2\) of the same level.

The Serre derivative satisfies the following product rule: for any quasimodular forms \(F\) and \(G\),

The dual lattice of a lattice \(\Lambda \) is the set

The dual of a lattice is also a lattice.