# Aaron Pollack, Talk 4: Beyond $\G_2$

:::{.remark}
Upshot for today: there exist groups $\G_3, \F_4, \E_{n, 4}$ for $n=6,7,8$ where $\G_2$ is split and the $\E$ groups are rank 4 over $\RR$.
These admit modular forms with Fourier expansions and coefficients similar to the $\G_2$ story.
We'll define these exceptional groups today.
:::

## Exceptional Algebras

:::{.remark}
Let be a composition algebra over $k$, where $\characteristic k = 0$, with a multiplication $C\tensorpower{k}{2}\to C$ which is not necessarily commutative or associative.
There exists a norm map $n_C: C\to K$ given by a nondegenerate quadratic form with $n_C(xy) = n_C(x) n_C(y)$.
:::

:::{.example title="?"}
\envlist

- $C = k$ and
\[
n_c: k &\to k \\
x&\mapsto x^2
.\]

- $C = E/k$ for $E$ a quadratic etale extension, $n_C = \Norm_{E/k}$.
- $C = B/k$ for $B$ a quaternion algebra with $n_C = n_{B, \red}$
- $C = \Theta$ an octonion algebra, with $\Theta = B \bigoplus B$.
  There is an involution $C\to C$ with
  - $x + x^* = \Trace_C(x) 1\in k1$, 
  - $xx^* = n_C(x)1\in k1$

:::

:::{.definition title="?"}
Let $J_C = H_3(C)$ be Hermitian $3\times 3$ matrices with coefficients in $C$, so
\[
J_C = \ts{ 
\begin{bmatrix}
 c_1 & x_3 & x_2^* \\
 x_3^* & c_2 & x_1 \\
 x_2 & x_1^* & c_3
\end{bmatrix}
 \st c_i\in k, c_i\in C
}
.\]
This has dimension $3+3C$ over $k$.
:::

:::{.example title="?"}
For $C=k, H_3(K)$ are symmetric $3\times 3$ matrices and there is a determinant map
\[
\det: J_C &\to k \\
X&\mapsto c_1c_2 c_3 - \sum c_i n_C(x_i) + \Trace_C(x_1(x_2 x_3))
.\]

If $C=k$ this is the usual determinant.
Note that $M_J' = \ts{g\in \GL(J_C) \st \det(gX) = \det(X) \forall X\in J_C}$ has positive dimension, and is thus infinite, making it an interesting algebra.
:::

:::{.remark}
Idea: there exists a group $G_{J_C}$ such that

- $C = \QQ \leadsto \F_4$,
- $C = K$ quadratic imaginary $\leadsto \E_{6,4}$,
- $C= B$ a quaternionic algebra $\leadsto \E_{7,4}$,
- $C = \Theta$ an octonionic algebra $\leadsto \E_{8, 4}$.

All have a good notion of quaternionic modular forms and Fourier expansions/coefficients.

A degree map $J_C \mapsvia{\deg} k$ commuting with $x\mapsto x^3$ and $\det$ recovers $G_{J_C = k} = \G_2$.
Recall 
\[
\lieg_2 = \liesl_{2, \ell}[0] + \liesl_{2, s}[0] + V_2 \tensor W[1]
\]
which is $C_2\dash$graded; we'll mimic this to construct $\lieg_{J_C}$.
:::

## Freudenthal Construction

:::{.definition title="Quaternionic exceptional groups"}
Let $J = J_C/\QQ$ and $k=\QQ$ and
\[
W_J = \QQ \bigoplus J \oplus J\dual \oplus \QQ
.\]
There is a symplectic form
\[
\inp{\tv{a,b,c,d}}{\tv{a,b,c,d}} = ad' - (b,c') - (c, b') - dc'
.\]
There is a degree 4 polynomial map $q: W_J\to \QQ$.
Define
\[
H_J^1 = \ts{g\in \GL(W_J) \st \inp{gw}{gw'} = \inp{w}{w'} \,\forall w,w'\in W_J,\, q(gw) = q(w)}
.\]
This recovers:

| $C$      	| $H_J^1$ 	| $\lieg_J$ 	|
|----------	|---------	|-----------	|
| $\QQ$    	| $\operatorname{C}_3$  	| $\F_4$    	|
| $K$      	| $\operatorname{A}_5$  	| $\E_6$    	|
| $B$      	| $\D_6$  	| $\E_7$    	|
| $\Theta$ 	| $E_7$   	| $\E_8$    	|
|          	|         	|           	|

Define
\[
\lieg_J =\liesl_2[0] + \lieh_J^0[0] + (V_2\tensor W_J)[1]
,\]
where $\lieh_J^0 = \Lie(H_J^1)$, and define
\[
G_J \da \Aut^0(\lieg_J)
.\]

If $n_C: C\tensor\RR\to \RR$ is positive definite, we say $\G_J$ is a **quaternionic exceptional group**.
:::

:::{.fact}
If $K_J \subseteq \G_J(\RR)$ is a maximal compact, then
\[
K_J = {\SU_2 \times L_J' \over \mu_2}
\]
where $L_J^1$ is a compact form of $H_J^1$.

There is a Cartan involution $\theta: \lieg_J\to \lieg_J$, which over $\CC$ yields
\[
\lieg_J^{\theta = \id} &= k_0\tensor \CC \cong \liesl_2 + \lieh_J^0  \\
\lieg_J^{\theta = -\id} &= p_0\tensor \CC \cong V_2 + W_J  
.\]
:::

:::{.remark}
There is an action
$K_J\actson V_\ell = \Sym^{2\ell}(\CC^2)\tensor \one$
:::

:::{.definition title="Modular forms"}
A **modular form** on $\G_3$ of weight $\ell$ is an automorphic form
\[
\phi: \dcosetl{G_J(\RR)}{G_J(\AA)} \to V_\ell
\]
such that 

1. $\phi(gk) = k\inv \phi(g)$ for all $k\in K_J$
2. $D_\ell \phi \equiv 0$

Here $D_\ell$ is defined as in the $\G_2$ case, replacing $\Sym^3(V_2) = W$ with $W_J$.
:::

:::{.remark}
There is a Heisenberg parabolic $P = MN \leq G_J$ with $M = H_J$, and $N \contains Z$ a two-step filtration with $Z$ 1-dimensional and $N/Z \cong W_J$ abelian.
:::

:::{.theorem title="?"}
Modular forms on $G_J$ of weight $\ell$

1. Have Fourier coefficients and expansions along $N/Z$:
\[
\phi_Z(g) = \phi_N(g) + \sum_{w\in W_J(\QQ), w\geq 0} a_\phi(w) W_w(g)
,\]
  where $a_\phi(w)\in \CC$ are the Fourier coefficients of $\phi$ and $W_w$ are completely explicit.

2. Under appropriate embeddings
\[
\G_2 \embeds \F_4 \embeds \E_{6,4}\embeds E_{7,4} \embeds E_{8,4}
,\]
for a modular form $\phi$ of weight $\ell$ of one groups, the pullbacks $i^* \phi$ to a smaller group are again modular forms of weight $\ell$ whose Fourier coefficients are sums of Fourier coefficients of $\phi$.
:::

:::{.theorem title="?"}
\envlist

1. There exists a nonzero weight 4 modular form $\theta_{\min}$ on $\E_{8, 4}$ with rational Fourier coefficients.
2. There exists a nonzero weight 8 modular form $\tilde\theta_{\min}$ on $\E_{8, 4}$ with rational Fourier coefficients.

:::

:::{.proof title="Sketch"}
\envlist

- Construct $\tilde\theta_{\min}$ using Eisenstein series
- Savim: most of the Fourier coefficients are zero, particularly the ones that are harder to compute explicitly
- By explicit computations, the remaining coefficients are rational.

:::

:::{.definition title="?"}
Say a modular form $\phi$ on $\G_3$ is **distinguished** iff

- There exists a $w_0\in W_J(\QQ)$ such that $q(W_0)\neq 0$ and $a_\phi(w_0) \neq 0$
- If $w\in W_J(\QQ), a_\phi(w)\neq 0$, then $q(w)\equiv q(w_0) \mod (\QQ\units)\cartpower{2}$.

:::

:::{.theorem title="?"}
Suppose $K/\QQ$ is quadratic imaginary, then there exists a distinguished modular form of weight 4 $\Theta_K$ on $G_{J_K} = \E_{6,4}$.
:::

:::{.proof title="?"}
Set $\Theta_K i^*(\Theta_{\min})$, pullback to $\E_{6,4}$.
By arithmetic invariant theory, one shows it is distinguished.
:::