## Notes of Maryna Viazovska’s Hadamard lectures 23-04-2019

Universal optimality of ${E_8}$ and Leech lattices

A set of points in Euclidean ${d}$-space, energy is a sum of contributions from each pair, which are functions of the distance. What are minimizers?

1. Potential and energy

Fix a potential function ${p:(0,+\infty)\rightarrow{\mathbb R}}$. The potential ${p}$-energy of a finite set ${C\subset{\mathbb R}^d}$ is

$\displaystyle \sum_{x,\,y\in C,\,x\not=y}p(|x-y|).$

1.1. Configurations

Motivated by cristallography, we extend the definition to a class of infinite sets.

A point configuration is a non-empty discrete subset of ${{\mathbb R}^d}$ (every closed ball ${B_r^d(x)}$ contains only finitely many points of ${C}$). Its lower ${p}$-energy is

$\displaystyle E_p(C):=\liminf_{r\rightarrow+\infty}\frac{1}{|C\cap B_r^d(0)|}\sum_{x,\,y\in C\cap B_r^d(0),\,x\not=y}p(|x-y|).$

If the limit exists, we call it the ${p}$-energy of ${C}$.

1.2. Density

We say that a configuration ${C}$ has density ${\rho}$ if

$\displaystyle \liminf_{r\rightarrow+\infty}\frac{1}{|C\cap B_r^d(0)|}=\rho.$

Example. Let ${\Lambda\subset{\mathbb R}^d}$ be a lattice. The density of ${\Lambda}$ is

$\displaystyle \frac{1}{vol({\mathbb R}^d/\Lambda)}.$

Furthermore, the ${p}$-energy of ${\Lambda}$ is

$\displaystyle \sum_{x\in\Lambda\setminus\{0\}} p(|x|).$

Example. Periodic configurations. Fix a lattice ${\Lambda}$. A configuration is ${\Lambda}$-periodic if ${C+\ell=C}$ for every ${\ell\in\Lambda}$. Such a configuration has density

$\displaystyle \frac{|C/\Lambda|}{vol({\mathbb R}^d/\Lambda)}$

and ${p}$-energy

$\displaystyle \frac{1}{|C/\Lambda|}\left(\sum_{x,\,y\in C/\Lambda,\,x\not=y}\sum_{\ell\in\Lambda} p(|x-y+\ell|)\right)+\sum_{\ell\in\Lambda\setminus\{0\}} p(|\ell|).$

Our goal is to minimize ${p}$-energy among configurations of fixed density ${\rho}$. We say that ${C}$ minimizes ${p}$-energy (or ${C}$ is a ground state for ${p}$) if its ${p}$-energy exists and every configuration in ${{\mathbb R}^d}$ with equal density has lower ${p}$-energy at least ${E_p(C)}$.

It sometimes happens that several different potentials admit the same ground states.

2. Universal optimality

How do the ground states depend on ${p}$?

2.1. Example: Gaussian core model in ${{\mathbb R}^3}$

Fix ${\alpha>0}$, let

$\displaystyle p_\alpha(r)=e^{-\alpha r^2}.$

Changing ${\alpha}$ turns out to be equivalent to changing density ${\rho}$.

Experiments show that for small densities, the fcc (face centered cubic) lattice wins. For large densities, the bcc (body centered cubic) lattice wins. In 1979, F. Stillinger observed that phase coexistence wins, i.e. fcc in a half space and bcc in the other, with suitable densities, does better than either fcc and bcc.

2.2. Completely monotonic functions

In the sequel, we shall stick to completely monotonic potentials. Say a function ${g}$ is completely monotonic if

1. infinitely differentiable,
2. derivatives satisfy ${(-1)^k g^{(k)}\geq 0}$.

Gaussian kernel, power laws are completely monotonic. Yukawa potential probably not.

Definition 1 (Cohn, Kumar 2007) Say a configuration in ${{\mathbb R}^d}$ with density ${\rho>0}$ is universally optimal if it minimizes ${p}$-energy for every potential ${p}$ ofthe form ${p(r)=g(r^2)}$, where ${g}$ is completely monotonic.

Theorem 2 (Bernstein) Every completely monotonic function ${g:(0,+\infty)\rightarrow{\mathbb R}}$ can be written as an integral

$\displaystyle g(r)=\int_0^\infty e^{-\alpha r}\,d\mu(\alpha)$

for some positive measure ${\mu}$ on ${(0,+\infty)}$.

Corollary 3 ${C}$ is universally optimal ${\iff}$ ${C}$ minimizes all energies defined by Gaussian core models.

2.3. Results

Theorem 4 (Cohn, Kumar 2007) ${{\mathbb Z}\subset{\mathbb R}}$ is universally optimal.

They conjectured that the ${A_2}$ lattice in ${{\mathbb R}^2}$, the ${E_8}$ lattice in ${{\mathbb R}^8}$ and the Leeach lattice

The main result of the present lectures is

Theorem 5 (Cohn, Kumar, Miller, Radchenko, Viazovska 2019) ${E_8}$ and the Leech lattice are universally optimal.

${A_2}$ is still open.

The sphere packing theorem follows by a limiting argument.

3. Strategy of the proof

The key are linear programming bounds. These amount to relax the problem: replace it by a series of convex problems on functions on ${{\mathbb R}^d}$. It becomes infinite dimensional. Fourier interpolation enters the scene. This reduces the problem to the positivity of a function of 2 variables. The last step is handled numerically.

3.1. Linear programming bounds

A Schwarz function on ${{\mathbb R}^d}$ is

1. infinitely differentiable,
2. each derivative decays faster than any inverse polynomial.

Our choice of normalization of Fourier transform is

$\displaystyle \hat f(y)=\int_{{\mathbb R}^d}f(x)e^{-2\pi i c\cdot y}\,dx.$

The following linear programming formulation is due to Cohn and Kumar 2007.

Proposition 6 Let ${p}$ be a nonnegative potential. Let ${f}$ be a Schwarz function such that

1. ${f(x)\leq p(|x|)}$.
2. ${\hat f\geq \geq 0}$.

Then every configuration of ${{\mathbb R}^d}$ with density ${\rho}$ has lower ${p}$-energy at least ${\rho \hat f(0)-f(0)}$.

This reduces the problem to finding a clever radial Schwartz function ${f}$. Note that the assumptions on ${f}$ are convex.

3.2. Proof of Proposition 6

I explain the case of periodic configurations. Let ${C}$ be a ${\Lambda}$-periodic configuration. Then

$\displaystyle E_p(C)=\frac{1}{|C/\Lambda|}\left(\sum_{x,\,y\in C/\Lambda,\,x\not=y}\sum_{\ell\in\Lambda^*} p(|x-y+\ell|)\right)+\sum_{\ell\in\Lambda\setminus\{0\}} p(|\ell|)$

$\displaystyle \geq \frac{1}{|C/\Lambda|}\left(\sum_{x,\,y\in C/\Lambda,\,x\not=y}\sum_{\ell\in\Lambda^*} f(x-y+\ell)\right)+\sum_{\ell\in\Lambda\setminus\{0\}} f(\ell)$

$\displaystyle =\frac{1}{|C/\Lambda|}\left(\sum_{x,\,y\in C/\Lambda}\frac{1}{vol({\mathbb R}^d/\Lambda)}\sum_{\mu\in\Lambda^*} \hat f(\mu)e^{2\pi i(x-y)\cdot\mu}\right)-f(0)$

$\displaystyle =\frac{1}{|C/\Lambda|}\sum_{\mu\in\Lambda^*} \hat f(\mu)|\sum_{x\in C/\Lambda} e^{2\pi ix\cdot\mu}|^2-f(0)|$

$\displaystyle \geq \frac{1}{|C/\Lambda|}\frac{1}{vol({\mathbb R}^d/\Lambda)}\hat f(0)|C/\Lambda|^2-f(0)$

$\displaystyle =\rho \hat f(0)-f(0).$

The general case (nonperiodic) was proved by Cohn and Courcy-Ireland. Every configuration can be viewed as a limit of periodic ones.

3.3. When can we hope for a sharp bound?

Did we lose much? Suppose that a lattice ${\Lambda}$ minimizes ${p}$-energy for some nonnegative potential ${p}$. Suppose that this can be proven by linear programming using a Schwartz function ${f}$. There is no loss in the above inequalities ${\iff}$

1. ${f(x)=p(|x|)}$ on ${\Lambda\setminus\{0\}}$.
2. ${\hat f=0}$ on ${\Lambda^*\setminus\{0\}}$.

We see that firqt derivatives must satisfy the above requirements as well. Can we recnstruct a Sxhwartz function from these conditions?

3.4. Fourier interpolation

Theorem 7 Let ${(d,n_0)=(8,1)}$ or ${(24,2)}$. Then any radial Schwartz function is uniquely determined by the values of

$\displaystyle f(\sqrt{2n}),\quad f'(\sqrt{2n}),\quad \hat f(\sqrt{2n}),\quad \hat f'(\sqrt{2n})$

for integer ${n\geq n_0}$.

More precisely, there exists an interpolation basis ${a_n}$, ${b_n}$, ${\tilde a_n}$, ${\tilde b_n}$, consisting of radial Schwartz functions, such that for every radial Schwartz function ${f}$ and every ${x\in {\mathbb R}^d}$,

$\displaystyle f(x)=\sum_{n=n_0}^{\infty} (f(\sqrt{2n})a_n(x)+f'(\sqrt{2n})b_n(x)+\hat f(\sqrt{2n})\tilde a_n(x)+\hat f'(\sqrt{2n})\tilde b_n(x)),$

where the series converges absolutely. Here are properties the interpolating basis.

1. $\displaystyle a_m(\sqrt{2n})=\delta_{mn},\quad a_m'(\sqrt{2n})=0,\quad \hat a_m(\sqrt{2n})=0,\quad \hat a_m'(\sqrt{2n})=0,$

2. $\displaystyle b_m(\sqrt{2n})=0,\quad b_m'(\sqrt{2n})=\delta_{mn},\quad \hat b_m(\sqrt{2n})=0,\quad \hat b_m'(\sqrt{2n})=0,$

Denote by ${\Lambda_d}$ the ${E_8}$ and Leech lattices respectively. Note that the shortest vector of ${\Lambda_d}$, ${d=8}$ or ${24}$, is ${\sqrt{2n_0(d)}}$.

3.5. Construction of the optimal auxiliary functions

The only possible auxiliary function that could prove a sharp bound for ${\Lambda_d}$, ${d=8,24}$ (among radial Schwartz functions) would be the following one,

$\displaystyle f(x)=\sum_{n=n_0}^{\infty}p(\sqrt{2n})a_n(x)+\sum_{n=n_0}^{\infty}p'(\sqrt{2n})b_n(x).$

Recall that, in addition to equality, one must prove inequalities

1. ${f(x)\leq p(|x|)}$ on ${\Lambda\setminus\{0\}}$.
2. ${\hat f\geq 0}$ on ${\Lambda^*\setminus\{0\}}$.

Therefore it suffices to check that

1. ${f(x)\leq p(|x|)}$ on ${{\mathbb R}^d\setminus\{0\}}$.
2. ${\hat f\geq 0}$ on ${{\mathbb R}^d}$.

Here, ${p}$ is a Gaussian.

3.6. Generating functions

Let

$\displaystyle F(\tau,x)=\sum_{n=n_0}^{\infty}a_n(x)e^{2\pi i n\tau}+2\pi i n\tau\sqrt{2n}b_n(x)e^{2\pi i n\tau}.$

and

$\displaystyle \tilde F(\tau,x)=\sum_{n=n_0}^{\infty}\tilde a_n(x)e^{2\pi i n\tau}+2\pi i n\tau\sqrt{2n}\tilde b_n(x)e^{2\pi i n\tau}.$

Substitute ${\tau=i\alpha}$. Then the inequalities

$\displaystyle F(i\alpha,x)\leq e^{-\pi \alpha|x|^2}$

for all ${\alpha>0}$ and ${x\in{\mathbb R}^d}$ and

$\displaystyle \hat F(i\alpha,y)\geq 0$

for all ${\alpha>0}$ and ${y\in{\mathbb R}^d}$, imply the universal optimality of ${\Lambda_d}$.

Apply the interpolation formula (IF) above to ${p_\tau(x):=e^{-\pi i \tau |x|^2}}$. It is equivalent to

$\displaystyle e^{\pi i \tau |x|^2}=F(\tau,x)+\tau^{-d/2}\tilde F(-\frac{1}{\tau},x).$

We secretly know that

$\displaystyle F(\tau+1,x)-2F(\tau,x)+F(\tau-1,x)=0$

and

$\displaystyle \tilde F(\tau+1,x)-2\tilde F(\tau,x)+\tilde F(\tau-1,x)=0.$

Note that the condition ${\hat F(i\alpha,y\geq 0}$ implies a control on the growth:

$\displaystyle F(i\alpha,x)\leq e^{-\pi \alpha|x|^2}.$

Indeed, using the functional equations, we get ${\hat F=\tilde F}$ and

$\displaystyle F(i\alpha,x)=e^{-\pi \tau |x|^2}-(i\alpha)^{-d/2}\tilde F(-\frac{1}{i\alpha},x)= e^{-\pi \tau |x|^2}-\alpha^{-d/2}\hat F(-\frac{1}{i\alpha},x)\leq e^{-\pi \alpha|x|^2}.$

4. Relating the Fourier interpolation formula with functional equations

4.1. Summary of previous episodes

Recall our goal:

Theorem 8 (Cohn, Kumar, Miller, Radchenko, Viazovska 2019) Let ${(d,n_0)=(8,1)}$ or ${(24,2)}$. There exists an interpolation basis ${a_n}$, ${b_n}$, ${\tilde a_n}$, ${\tilde b_n}$ for integer ${n\geq n_0}$, consisting of radial Schwartz functions, such that for every radial Schwartz function ${f}$ and every ${x\in {\mathbb R}^d}$,

$\displaystyle f(x)=\sum_{n=n_0}^{\infty} (f(\sqrt{2n})a_n(x)+f'(\sqrt{2n})b_n(x)+\hat f(\sqrt{2n})\tilde a_n(x)+\hat f'(\sqrt{2n})\tilde b_n(x)),$

where the series converges absolutely.

Denote by

$\displaystyle S({\mathbb N}):=\{(x_n)_{n=1}^{\infty}\;|\; x_n=o(n^{-k})\,\forall k>0\}.$

Theorem 9 (Cohn, Kumar, Miller, Radchenko, Viazovska 2019) Consider the maps

$\displaystyle \Psi:S_{rad}({\mathbb R}^d)\rightarrow S({\mathbb N})^4,\quad f\mapsto (f(\sqrt{2n}),f'(\sqrt{2n}),\hat f(\sqrt{2n}),\hat f'(\sqrt{2n})$

and

$\displaystyle \Phi:S({\mathbb N})^4\rightarrow S_{rad}({\mathbb R}^d),$

$\displaystyle (\alpha_n,\beta_n,\tilde\alpha_n,\tilde\beta_n)\mapsto f(x):=\sum_{n=n_0}^{\infty} (\alpha_n a_n(x)+\beta_n b_n(x)+\tilde \alpha_n\tilde a_n(x)+\tilde\beta_n \tilde b_n(x)).$

These maps are isomorphisms and inverses of each other.

Again, we form the generating series

$\displaystyle F(\tau,x)=\sum_{n=n_0}^{\infty}a_n(x)e^{2\pi i n\tau}+2\pi i n\tau\sqrt{2n}b_n(x)e^{2\pi i n\tau}.$

and

$\displaystyle \tilde F(\tau,x)=\sum_{n=n_0}^{\infty}\tilde a_n(x)e^{2\pi i n\tau}+2\pi i n\tau\sqrt{2n}\tilde b_n(x)e^{2\pi i n\tau}.$

The interpolation formula implies the following functional equations.

$\displaystyle F(\tau,x)+(i/\tau)^{d/2}\tilde F(-1/\tau,x)=e^{\pi i \tau |x|^2},$

$\displaystyle F(\tau+1,x)-2F(\tau,x)+F(\tau-1,x)=0,$

and

$\displaystyle \tilde F(\tau+1,x)-2\tilde F(\tau,x)+\tilde F(\tau-1,x)=0.$

4.2. Reversed procedure: functional equations imply interpolation formula

Conversely, we shall now check that the functional equations imply the interpolation formula.

Notations. For a radial Schwartz function ${f}$, expressed as ${f(x)=f_0(|x|)}$, let ${Df(x)=f'_0(|x|)}$.

Consider seminorms

$\displaystyle \|f\|^{rad}_{k,\ell}=\sup_{x\in {\mathbb R}^d}|x|^k|D^\ell f(x).$

Lemma 10 The complex Gaussians ${x\mapsto e^{\pi i \tau|x|^2}}$ span a dense subspace in ${S_{rad}({\mathbb R}^d)}$. For any ${y>0}$, the same is true if we consider only complex Gaussians with ${\Im m(\tau)=y}$.

Therefore the interpolation formula needs be proven only for complex Gaussians.

Here comes the converse statement.

Theorem 11 Suppose there exist functions ${F}$, ${\tilde F:H\times{\mathbb R}^d\rightarrow {\mathbb C}}$ such that

1. ${F}$ and ${\tilde F}$ are holomorphic in ${\tau\in H=}$ upper half plane.
2. ${F}$ and ${\tilde F}$ are radial in ${x}$.
3. For all ${k,\ell\in{\mathbb N}}$,

$\displaystyle \|F_\tau\|^{rad}_{k,\ell}\leq\alpha_{k,\ell}\Im m(\tau)^{-\beta_{k,\ell}}+\gamma_{k,\ell}|\tau|^{\delta_{k,\ell}}.$

$\displaystyle \|\tilde F_\tau\|^{rad}_{k,\ell}\leq\alpha_{k,\ell}\Im m(\tau)^{-\beta_{k,\ell}}+\gamma_{k,\ell}|\tau|^{\delta_{k,\ell}}.$

In the special case ${(k,\ell)=(0,0)}$, we make a stronger assumption

$\displaystyle \|F_\tau\|+\|\tilde F_\tau\|_\infty \le \alpha_{0,0}\Im m(\tau)^{-\beta_{0,0}}$

for ${-1\leq\Re e(\tau)\leq 1}$, with ${\beta_{0,0}>0}$.

4. ${F}$ and ${\tilde F}$ satisfy the three functional equations above.

Then ${F}$ and ${\tilde F}$ have expansions of the form above (with ${n_0=1}$), for some radial Schwartz functions ${a_n,b_n,\tilde a_n, \tilde b_n}$. Moreover, for every radial Schwartz function ${f}$, the interpolation formula

$\displaystyle f(x)=\sum_{n=n_0}^{\infty} (f(\sqrt{2n})a_n(x)+f'(\sqrt{2n})b_n(x)+\hat f(\sqrt{2n})\tilde a_n(x)+\hat f'(\sqrt{2n})\tilde b_n(x)),$

holds. Finally, for every ${k,\ell\in{\mathbb N}}$, the radial seminorms ${\|a_n\|^{rad}_{k,\ell}}$, …, grow at most polynomially in ${n}$. The n=1 functions ${a_1}$,… vanish if and only if ${F}$ and ${\tilde F}$ are ${o(e^{-2\pi \Im(\tau)})}$ in the strip ${-1\leq\Re e(\tau)\leq 1}$ with ${x}$ fixed.

4.3. Proof of Theorem 11

Step 1. As functions of ${\tau}$, ${F(\tau+1,x)-F(\tau,x)}$ is holomorphic, invariant under translation by 1 and goes to ${0}$ as ${\Im m(\tau)}$ tends to ${0}$. Therefore it can be viewed as a function (of ${e^{2\pi i\tau}}$) on the disk, hence it admits an expansion in powers of ${e^{2\pi i\tau}}$, with coefficients which we denote by ${2\pi i\sqrt{2n}b_n(x)}$.

Same argument with ${F(\tau,x)-\tau(F(\tau+1,x)-F(\tau,x))}$ provides the ${A_n(x)}$ coefficients. The same for ${\tilde F}$.

Step 2. The functions ${a_n}$,… are radial Schwartz functions.

Express ${a_n(x)}$ as a Fourier coefficient,

$\displaystyle a_n(x)=\int_{iy}^{1+iy}(F(\tau,x)-\tau(F(\tau+1,x)-F(\tau,x)))e^{-2\pi in\tau}\,d\tau$

and idem for ${b_n}$. Part (3)of the assumptions, we see that the radial seminorms of ${a_n}$… are finite. To show that they grow polynomially, we take ${y=1/n}$ in the integral. This moderate growth implies that the interpolation series converges absolutely for every radial Schwartz function ${f}$. This defines a continuous linear functional on ${S_{rad}({\mathbb R}^d)}$.

Step 3. Proof of the interpolation formula.

Fix ${x}$ and consider the linear functional

$\displaystyle \lambda(f)=\sum_{n=n_0}^{\infty} (f(\sqrt{2n})a_n(x)+f'(\sqrt{2n})b_n(x)+\hat f(\sqrt{2n})\tilde a_n(x)+\hat f'(\sqrt{2n})\tilde b_n(x))-f(x).$

By density, we need merely prove that ${\lambda}$ vanishes on complex Gaussians. This is equivalent to the first functional equation for ${F}$ and ${\tilde F}$. This concludes the proof of Theorem 11.

Remark. Suppose that ${H}$ and ${\tilde H}$ are solutions of the homogeneous equations (i.e. removing right hand sides)

$\displaystyle H(\tau)+(i/\tau)^{d/2}\tilde H(-1/\tau)=0,$

$\displaystyle H(\tau +1)-2H(\tau)+H(\tau-1)=0,$

$\displaystyle \tilde H(\tau +1)-2\tilde H(\tau)+\tilde H(\tau-1)=0.$

Then

$\displaystyle H(\tau)=\sum_{n=1}^\infty (c_n+2\pi i \tau\sqrt{2n}d_n)e^{2\pi i n\tau},$

$\displaystyle \tilde H(\tau)=\sum_{n=1}^\infty (\tilde c_n+2\pi i \tau\sqrt{2n}\tilde d_n)e^{2\pi i n\tau},$

The proof of Theorem 11 shows that for any radial Schwartz function ${f}$,

$\displaystyle \sum_{n=n_0}^\infty (f(\sqrt{2n})c_n+f'(\sqrt{2n})d_n+\hat f(\sqrt{2n})\tilde c_n+\hat f'(\sqrt{2n})\tilde d_n)=0.$

This shows that the orthogonal of the image of ${\Psi}$ is the finite dimensional space of solutions of the set of homogeneous functional equations.

5. Construction of solutions of functional equations: preliminary steps

How to construct ${F}$ and ${\tilde F}$?

5.1. ${PSL_2({\mathbb Z})}$ invariance

Remember that the group ${PSL_2({\mathbb Z})}$ acts on the upper half plane ${H}$.

Definition 12 Let ${f:H\rightarrow{\mathbb C}}$ be a function, let ${k\in 2{\mathbb Z}}$ be an even integer. Fix an element ${\gamma\in PSL_2({\mathbb Z})}$. The slash operator is defined by

$\displaystyle (f|_k\gamma)(\tau):=(c\tau+d)^{-k}f(\gamma(\tau)),$

where ${c\tau+d}$ is the denominator of ${\gamma(\tau)}$.

It has the following property: ${f|_k \gamma_1\gamma_2=(f|_k\gamma_1)|_k\gamma_2}$.

Notation. Let ${R={\mathbb C}[PSL_2({\mathbb Z})]}$ denote the group algebra (with the right action of the group). The slash operator extends to ${R}$ by linearity.

Definition 13 A function on ${H}$ has moderate growth if there exist ${\alpha,\beta,\gamma>0}$ such that

$\displaystyle |F(\tau)|\leq\alpha(\Im m(\tau)^{-\beta}+|\tau|^\gamma).$

The space of holomorphic functions with moderate growth is denoted by ${P}$.

${P}$ is invariant under the slash operator.

Notation. The involution ${S}$ and the translation by ${1}$ ${T}$ are the usual generators for ${PSL_2({\mathbb Z})}$, with relators ${S^2}$ and ${(ST)^3}$.

The functional equation for ${F}$ and ${\tilde F}$ can be written

$\displaystyle F|_{d/2}(T-2+T^{-1})=0,$

$\displaystyle \tilde F|_{d/2}(T-2+T^{-1})=0,$

$\displaystyle F+\tilde F|_{d/2}S=e^{\pi i\tau|x|^2}.$

We use the last equation to express ${\tilde F}$ in terms of ${F}$ and reduce to a set of 2 equations with only one unknown function,

$\displaystyle F|_{d/2}(T-2+T^{-1})=0,$

$\displaystyle F|_{d/2}S(T-2+T^{-1})=e^{\pi i\tau|x|^2}|_{d/2}S(T-2+T^{-1}).$

We want to solve this equation in ${P}$, and eventually find a unique solution.

5.2. The ideal associated to functional equations

In the group algebra ${R}$, consider the right ideal

$\displaystyle I:=(T-2+T^{-1})R+S(T-2+T^{-1})R.$

It has the following special properties.

1. ${I}$ has complex codimension ${6}$ in ${R}$.
2. ${I}$ is the free ${R}$-module generated by ${(T-2+T^{-1})}$ and ${S(T-2+T^{-1})}$.
3. The right action of ${PSL_2({\mathbb Z})}$ on the ${6}$-dimensional complex vectorspace ${I\setminus R}$ has “polynomial growth”.

The last statement means that in a basis of ${I\setminus R}$, the absolute values of the ${36}$ matrix coefficients of an element ${\gamma\in PSL_2({\mathbb Z})}$ are bounded above by powers of the matrix coefficients of ${\gamma}$. It follows from the following description of the representation ${\sigma}$. Let ${\rho_3}$ denote the restriction to ${SL_2({\mathbb Z})}$ of the ${3}$-dimensional representation ${S^2}$ of ${SL_2({\mathbb R})}$. Let ${\rho_2}$ denote the ${2}$-dimensional representation of ${SL_2({\mathbb Z})}$ such that the kernel of ${\rho_2}$ is the congruence subgroup ${\Gamma_2}$ (its image has ${6}$ elements, it is isomorphic to the dihedral group ${D_3}$). Let ${\vec v:SL_2({\mathbb Z})\rightarrow{\mathbb Z}^2}$ denote the cocycle (with respect to ${\rho_2}$) such that ${\vec v(S)=(0,0)}$ and ${\vec v(T)=(1,-1)}$. This defines an affine action, therefore a morphism to ${Aff_2({\mathbb C}). Up to conjugacy, the ${6}$-dimensional representation ${\sigma}$ is the direct sum of this morphism and ${\rho_3}$.

5.3. Next episode

Tomorrow, we shall solve the homogeneous functional equations (using some classical theory of modular forms), then solve the nonhomogeneous functional equations in the form

$\displaystyle F(\tau,x)=\int_0^\infty K(\tau,z)e^{\pi iz |x|^2}\,dz,$

where ${K}$ is meromorphic on ${H\times H}$, with special requirements on residues at poles (which are the fixed points of ${PSL_2({\mathbb Z})}$). It satisfies

$\displaystyle K(\tau,z)|_{d/2}A=0$

(with respect to the ${\tau}$ variable) for all ${A\in I}$.

6. Solving the functional equations

Definition 14 Let ${J}$ be a right-ideal of the group ring ${R}$. For an even integer ${k\in 2{\mathbb Z}}$, denote by

$\displaystyle Ann_k(J,P)=\{f\in P\,|\,f|_k \gamma=0\,\forall\gamma\in J\}.$

We shall describe ${Ann_k(J,P)}$ in terms of modular forms.

6.1. Brief overview of classical modular forms

Definition 15 Let ${\Gamma}$ be a discrete finite covolume subgroup of ${SL_2({\mathbb R})}$. A (holomorphic) modular form of weight ${k}$ is a function ${f\in P}$ such that ${f|_k \gamma=f}$ for all ${\gamma\in\Gamma}$. They form a finite dimensional space ${M_k(\Gamma)}$.

We also define the infinite dimensional space ${M^!_k(\Gamma)}$ of weakly holomorphic modular forms by admitting poles near cusps.

We are interested in the cases ${\Gamma=SL_2({\mathbb Z})}$ and ${\Gamma=\Gamma(2)}$, the principal congruence subgroup at level ${2}$.

6.2. Modular forms for ${SL_2({\mathbb Z})}$

Eisenstein series are defined for ${k\geq 4}$ by

$\displaystyle E_k(z):=\frac{1}{2\zeta(k)}\sum_{(m,n)\in{\mathbb Z}^2\setminus\{(0,0)\}}(mz+n)^{-k}.$

Its Fourier expansion is

$\displaystyle E_k(z)=1-\frac{2k}{B_k}\sum_{n=1}^\infty G_{k-1}(n)e^{2\pi i nz},$

Then, as an algebra, the direct sum of all ${M_k}$ is freely generated by ${E_4}$ and ${E_6}$.

Ramanujan’s cusp form of weight ${12}$ is

$\displaystyle \Delta=\frac{1}{1728}(E_4^3-E_6^2)=q\prod_{n=1}^\infty (1-q^n)^{24}.$

${\Delta}$ does not vanish on ${H}$, its inverse is a weakly holomorphic modular form of weight ${-12}$. ${\Delta}$ vanishes at all cusps. Another famous weakly modular form is the ${j}$-modular invariant ${j=\Delta^{-1}E_4^3}$. It is a Hauptmodul for the modular surface ${X(1)=PSL_2({\mathbb Z})\setminus H}$.

${M^!(SL_2({\mathbb Z}))}$ is generated by ${E_4,E_6}$ and ${\Delta^{-1}}$.

One can also define Eisenstein series of weight ${2}$, but not by the series, by the following formula instead,

$\displaystyle E_2=1-24\sum_{n=1}^\infty \sigma_1(n)q^n=\frac{\Delta'}{2\pi i\Delta}.$

It satisfies ${E_2(z+1)=E_2(z)}$ and

$\displaystyle E_2(-1/z)=z^2E_2(z)-\frac{6iz}{\pi}.$

It is called a quasimodular form, and more generally, one calls quasimodular form all elements of the algebra generated by ${E_2}$, ${E_4}$ and ${E_6}$.

6.3. Modular forms for ${\Gamma(2)}$

$\displaystyle \theta_{00}(z)=\sum_{n\in{\mathbb Z}}e^{\pi i n^2 z},\quad \theta_{01}(z)=\sum_{n\in{\mathbb Z}}(-1)^n e^{\pi i n^2 z},\quad \theta_{10}(z)=\sum_{n\in{\mathbb Z}}e^{\pi i (n+\frac{1}{2})^2 z}.$

These are modular forms of weight ${1/2}$, we shall merely need their powers

$\displaystyle U:=\theta_{00}^4,\quad V:=\theta_{01}^4,\quad W:=\theta_{10}^4,$

which are honest modular forms. They satisfy the Jacobi identity,

$\displaystyle U=V+W.$

Then the modular ring ${M(\Gamma(2))}$ is generated by ${V}$ and ${W}$, the weakly modular ring ${M^!(\Gamma(2))}$ is generated by ${V,W}$ and ${\Delta^{-1}}$.

The modular ${\lambda}$-function is ${\lambda:=\frac{V}{U}}$. It takes its values in ${{\mathbb C}\setminus\{0,1\}}$.

It is a Hauptmodul of the modular curve ${X(2)=\Gamma(2)\setminus H}$, i.e. it generates its function field over ${{\mathbb C}}$.

The fundamental domain of ${X(2)}$ is the union of two adjacent fundamental domains of ${X(1)}$. The function ${\lambda}$ descends to an isomorphism of ${X(2)}$ with ${{\mathbb C}\setminus\{0,1\}}$, extending to a compactification by ${\lambda(\infty)=\infty}$, ${\lambda(0)=1}$ and ${\lambda(1)=\infty}$.

One denotes by ${\mathcal{L}:=\log\lambda}$ and ${\mathcal{L}_S=\log(\lambda(-1/z))}$.

6.4. Role of ideal ${I}$

We introduce two other ideals

$\displaystyle I_+=(S+1)R+(T-1)^2R,\quad I_-=(S-1)R+(T-1)^2R.$

They have codimension ${3}$ in ${R}$, their sum is ${R}$ and their intersection is ${I}$. An alternate description is

$\displaystyle I_\pm=\{r\in R\,|\,(S-\pm 1)r\in T\}.$

It follows that

$\displaystyle Ann_k(I,P)=Ann_k(I_+,P)\oplus Ann_k(I_-,P).$

We shall treat both pieces in different manners, the first in terms of quasimodular forms, the second in terms of logarithmic derivatives of the ${\lambda}$-function.

Proposition 16 Let ${k}$ be an even integer. Then

$\displaystyle Ann_k(I_+,P)=\phi_2 M_{k-2}(SL_2({\mathbb Z}))+\phi_0 M_k(SL_2({\mathbb Z})) +\phi_{-2}M_{k+2}(SL_2({\mathbb Z})),$

where

$\displaystyle \phi_2(\tau)=\tau E_2^2-\frac{6i}{\pi}E_2,\quad \phi_0=\tau E_2-\frac{3i}{\pi},\quad \phi_{-2}=\tau.$

It follows that

$\displaystyle \mathrm{dim}Ann_k(I_+,P)=\max\{0,\lceil \frac{k}{4}\rceil+1\}.$

Proposition 17 Let ${k}$ be an even integer. Then

$\displaystyle Ann_k(I_-,P)=\psi_4 M_{k-4}(SL_2({\mathbb Z}))+\psi_2 M_{k-2}(SL_2({\mathbb Z})) +\phi_{0}M_{k}(SL_2({\mathbb Z})),$

where

$\displaystyle \psi_4(\tau)=\xi_4\mathcal{L}+(\xi_4|_4 S)\mathcal{L}_S,\quad \psi_2=\xi_2\mathcal{L}+(\xi_2|_2 S)\mathcal{L}_S,\quad \psi_0=1,$

and ${\xi_4=U^2+W^2-2V^2}$, ${\xi_2=U+W}$.

It follows that

$\displaystyle \mathrm{dim}Ann_k(I_-,P)=\max\{0,\lceil \frac{k-2}{4}\rceil+1\}.$

Comment. It is fortunate that solutions can be expressed in modular terms. It is folklore that the symmetric square representation of ${SL_2({\mathbb Z})}$ is related to modular forms. It is not so folklore for the affine representation. However, it is not that dramatic: we knew a priori that the annihilators were finite dimensional. In case of need, we could have used numerical calculations to describe them.

6.5. Solving the inhomogeneous equation

We shall use again the fundamental domain for ${\Gamma(2)}$,

$\displaystyle D:=\{z\in h\,|\,-1<\Re e(z)<1,\,|z-\frac{1}{2}|>\frac{1}{2},\,|z+\frac{1}{2}|>\frac{1}{2}\}.$

Here are premiminary technical points.

Proposition 18 Let ${h_1,h_2}$ be continuous functions on ${H}$.

1. Analytic continuation. Assume ${h_1,h_2}$ are holomorphic. Let ${O\subset H}$ be an open neighborhood of ${\bar D}$. Let ${f:O\rightarrow{\mathbb C}}$ be a holomorphic function which satisfies the following transformation laws:
1. ${f|_k (T-1)^2=h_1}$,
2. ${f|_k S(T-1)^2=h_2}$

whenever both sides are defined. Then ${f}$ extends to a holomorphic function on ${H}$ which satisfies the same equations (a) and (b).

2. Propagation of moderate growth bounds. Suppose that ${f}$ is a continuous function on ${H}$ which satisfies equations (a) and (b). Assume that ${f}$ has moderate growth on ${D}$, and that ${h_1}$ and ${h_2}$ have moderate growth. Then ${f}$ has moderate growth on ${H}$.

Now I describe our ansatz. We look for solutions in the form

$\displaystyle F(\tau,x)=e^{\pi i|x|^2\tau} +4\sin(\pi|x|^2/2)^2\int_{0}^{\infty} K(\tau,it)e^{-\pi|x|^2 t}\,dt.$

Theorem 19 For dimensions ${d=8,24}$, there exist unique meromorphic functions ${K=K^{(d)}}$ on ${H\times H}$ satisfying the following properties:

1. For a fixed ${z\in H}$, the poles of ${K(\tau,z)}$ in ${\tau}$ are all simple and contained in the ${SL_2({\mathbb Z})}$-orbit of ${z}$.
2. The functional equations hold,

$\displaystyle K(\tau,z)|_{d/2} (T-1)^2=0,$

$\displaystyle K(\tau,z)|_{d/2} S(T-1)^2=0,$

3. For ${z\in H}$ and ${A\in R}$,

$\displaystyle Res_{\tau=z}(K|_{d/2} A)=-\frac{1}{2\pi}\Phi(A),$

where ${\Phi:R/I\rightarrow {\mathbb C}}$ is the linear map defined by its values on the basis ${\{1,T,TS,S,ST,STS\}}$ as follows,

$\displaystyle \Phi(1)=0,\quad \Phi(T)=1,\quad\Phi(TS)=0,\quad \Phi(S)=0,\quad\Phi(ST)=0,\quad\Phi(STS)=0.$

4. The functions

$\displaystyle K^{(8)}(\tau,z)(j(\tau)-j(z))\Delta(\tau)\Delta(z),\quad K^{(24)}(\tau,z)(j(\tau)-j(z))\Delta(\tau)\Delta^2(z)$

are in the class ${P}$ both as functions of ${\tau}$ and ${z}$. Moreover

$\displaystyle K^{(8)}(\tau,z)=O(|\tau e^{2\pi i\tau}|),\quad \tau^{-4} K^{(8)}(-1/\tau,z)=O(|\tau e^{2\pi i\tau}|),$

$\displaystyle K^{(24)}(\tau,z)=O(|\tau e^{4\pi i\tau}|),\quad \tau^{-12} K^{(24)}(-1/\tau,z)=O(|\tau e^{4\pi i\tau}|),$

as ${\Im m(\tau)}$ tends to infinity.

The residue constraint is found by plugging in the answatz, switching contours and fitting residues.

6.6. Idea of the proof of Theorem 19

The proof consists in providing explicit formulae for these functions. Assume that a solution ${K}$ exists.

The group ring ${R}$ comes equipped with an involution ${\ast}$ such that ${(\gamma)^\ast=\gamma^{-1}}$ for group elements. The slash operation, when performed on the ${z}$ variable, relates to the usual slash operation.

Lemma 20 Given matrices ${M,N\in SL_2({\mathbb Z})}$,

$\displaystyle res_{\tau=Nz} K|^z_{2-d/2} M=res_{\tau=z} K|^\tau_{d/2} NM^\ast.$

We introduce one more ideal,

$\displaystyle \tilde I:=\{M\in R\,|\,\phi(NM^\ast)=0\,\forall N\in R\}.$

Fix ${M\in \tilde I}$. For all ${\in SL_2({\mathbb Z})}$,

$\displaystyle res_{\tau=Nz} K|^z_{2-d/2} M=res_{\tau=z} K|^\tau_{d/2} NM^\ast=\frac{1}{2\pi}\Phi(NM^\ast)=0.$

Therefore ${K|^z_{2-d/2} M}$ is holomorphic in the ${\tau}$ variable. Growth conditions imply that ${K|^z_{2-d/2} M}$ belongs to ${P}$, hence to ${Ann_{d/2}(I,P)}$ as a function of ${\tau}$. We also knowthat this function vanishes at cusps. This implies that ${K|^z_{2-d/2} M=0}$ for all ${M\in\tilde I}$.

Now we can describe the spaces ${Ann_k(\tilde I,P)}$. We use the fact that

$\displaystyle K^{(d)}(\tau,z)(j(\tau)-j(z))\Delta(\tau)\Delta(z)^{m(d)}\in Ann_k(I,P)\otimes Ann_k(\tilde I,P)$

for suitable ${k}$, this gives us an explicit formula for ${K^{(d)}}$ in terms of Eisenstein series.

For ${\tau\in D}$ and ${x\in {\mathbb R}^d}$, ${|x|>\sqrt{2n_0-2}}$ (this is required for the integral to converge), remember we set

$\displaystyle F(\tau,x)=e^{\pi i|x|^2\tau} +4\sin(\pi|x|^2/2)^2\int_{0}^{\infty} K(\tau,it)e^{-\pi|x|^2 t}\,dt.$

Lemma 21 The function ${\tau\mapsto F(\tau,x)}$ extends to a holomorphic function on an open neighborhood of ${D}$ in ${H}$, and satisfies

$\displaystyle F|_k (T-1)^2=0,\quad F|_k S(T-1)^2=e^{\pi\tau|x|^2}|_{d/2}S(T-1)^2.$