# Aaron Pollack, Talk 3: Examples of (and theorems about) modular forms on $\G_2$


## Degenerate Eisenstein Series

:::{.remark}
Recall $\G_2\contains P$ a Heisenberg parabolic, with $P = MN$ where $M\cong \GL_2$.
Write $\nu$ for the composition $P\to M \mapsvia{\det} \GL_1$.
Suppose $\ell > 0$ is even, and recall that $V_\ell = \Sym^2(\CC^2)\tensor \actsonl K \leq \G_2$ for $K$ a maximal compact.
Let
\[
f_{\ell, \infty} (g; s) = \Ind_{P(\RR)}^{\G_2(\RR)} \abs{\nu}^s \tensor V_\ell
\]
be defined by

\[
f_{\ell, \infty}(pg^j s) &= N(\mu)^s f_{e\ll, \infty} (g; s) \quad \forall p\in P(\RR) \\
f_{\ell,\infty}(gk; s) &= k\inv f_{\ell, \infty}(g) \quad \forall k\in K
.\]

By the Iwasawa decomposition $C_3(\RR) = P(\RR) K$, $f$ is uniquely determined one we set
\[
f_{\ell, \infty}(1) = x^\ell y^\ell \in V_\ell = \gens{x^?, x^?, \cdots, y^{2\ell}}
.\]
Let $f_?$ be a flat section in $\Ind_{P(\AA_f)}^{\G_2(\AA_f)}(\abs{\nu}^3)$, and let $f_g(g, s) = f_?(gf, s) f_{\ell,\infty}(g_?; s) \in \G_2(\AA)$.
Define
\[
E_\ell(g,f,s) = \sum_{\gamma\in \dcosetl{P(\QQ)}{ \G_2(\QQ)} } f_\ell(\gamma g, s)
.\]
If $\Re(s) > 3$, set $E_\ell(g) \da E_\ell(g, f, s=\ell+1)$.
:::

:::{.theorem title="?"}
If $\ell>0$ is even and $\ell\geq 4$, then $E_\ell(g)$ is a quaternionic modular form on $\G_2$ of weight $\ell$.
:::

:::{.proof title="?"}
$f_{\ell, ?}(g, s=\ell+1)$ is annihilated by $D_\ell$, so $E_\ell(g)$ is as well by absolute convergence.
:::

:::{.remark}
If $\pi$ is a cuspidal automorphic representation of $\GL_2 = M$ associated to a holomorphic weight $3\ell$ modular form which is cuspidal,

- $f_\pi\in \Ind_{P(\AA)}^{\G_2(\AA)}(\pi)$
- $E(g, f_\pi) = \sum_{ \gamma\in \dcosetl{P(\QQ)}{\G_2(\QQ)} } f_\pi( \gamma g)$
- If $\ell \geq 6$ this is a weight $\ell$ modular form on $\G_2$.
- If $\ell=4$, one can make sense of $E(g, f_\pi)$ using analytic continuation to produce a weight 4 modular form on $\G_2$ associated to the Ramanujan $\Delta$.

:::

:::{.fact}
If $\phi$ is a level 1 quaternionic modular form on $\G_2$,

- $a_\phi(w) \neq 0 \implies f_w(u, v)= au^3 + \cdots + dv^3$ with $a,\cdots,d\in \ZZ$ (integrality)
- $a_p(w\gamma) = \det(\gamma)^\ell a_\phi(w)$ for $r\in \GL_2(\ZZ)$, so Fourier coefficients are constant on orbits of binary cubic forms.

:::

:::{.fact}
There is a bijection
\[
\ts{\text{Integral binary cubic forms}}/\GL_2(\ZZ)
\mapstofrom
\ts{\text{Cubic rings}}\modiso
,\]
where cubic rings are free rank 3 $\ZZ\dash$algebras.
Thus if $\ell>0$ is even, $\phi$ is a level 1 weight $\ell$ modular form on $\G_2$, and $A$ is a cubic ring, 
there is a well-defined map $a_\phi(A) = a_\phi(w)$ if $A\mapstofrom f_w$.
:::

:::{.remark}
If $f_w$ is nondegenerate, the cubic ring $A(f_w)$ associated to $f_w$ is totally real $\iff f$ is positive semidefinite.
:::

:::{.theorem title="?"}
Suppose $A$ is the maximal order in a totally real cubic etale $\QQ\dash$algebra $E$.
There exists a constant $c_\ell\in \CC$, independent of $A$, such that 
\[
a_{E_\ell}(A) = c_\ell \zeta_E(1-\ell)
,\]
where $\ell$ is even.
The LHS are Fourier coefficients of modular forms on $\G_2$.
:::

:::{.remark}
It is not known that $c_\ell$ is nonzero.
:::

:::{.question}
An open question: $E(g, f_\pi)$ is Eisenstein, can anything be said about its Fourier coefficients.
:::

## Cusp Forms

:::{.theorem title="?"}
Suppose $\ell\geq 16$ is even.
There exist nonzero cusp forms on $\G_2$ of weight $\ell$, all of whose Fourier coefficients are algebraic integers.
:::

:::{.proof title="of theorem"}
Steps:

1. Start with a holomorphic Siegel modular form $f$ on $\SP_4$ of weight $\ell$, so $f$ has Fourier coefficients in $\ZZbar$
2. Take a $G\dash$lift of $f$ to $\SO_{4,4}$ to obtain $G(f)$, and define
\[
\Theta(f)(g) = \int_{[\SP_4]} \theta(g, h) \bar{f(h)} \, dh
,\]
then $\Theta$ on $\SO_{4,4}\times \SP_4$ is a $\theta$ function.

3. There is a good theory of quaternionic modular forms on $\SO_{4, n}$, so choose $\theta(g,h)$ such that $\Theta(f)$ is a one of weight $\ell$ (and cuspidal).

4. Express the Fourier coefficients of $\Theta(f)$ in terms of classical Fourier coefficients of $f$, showing that the Fourier coefficients of $G(f)$ are in $\ZZbar$.

5. Use $\G_2 \embedsvia{\iota}\SO_{4,4}$ and pullback to obtain $i^*(\Theta(f))$, which is still cuspidal and has Fourier coefficients that are sums of the original coefficients, so still in $\ZZbar$.

:::

:::{.theorem title="R. Dalal"}
There is an explicit dimension formula for the level 1 cuspidal quaternionic modular forms of weight $\ell$.
In particular, the smallest is a level 1 cusp form of weight 6.
:::

:::{.theorem title="Cicek-Dadivdoff,Dijok,Hammonds, P, Roy"}
Suppose $\phi$ is a level 1 cuspidal quaternionic modular form on $\G_2$ associated to a cuspidal automorphic representation $\pi$ on $\G_2(\AA)$.
Suppose that the Fourier coefficient $a_\phi(\ZZ\cartpower 3) \neq 0$, then

1. The complete standard \(L\dash \)function of $\pi$ has a functional equation:
\[
\Lambda(\pi, \std, s) = \Lambda(\pi, \std, 1-s)
.\]

2. There exists a Dirichlet series for this \(L\dash \)function expressing the Fourier coefficients in terms of an \(L\dash \)function:
\[
\sum_{T \subseteq \ZZ\cartpower{3},n\geq 1} {a_\phi(\ZZ + nT) \over [\ZZ\cartpower{3} : T]^{s-\ell+1}  }n^{-s}
=
a_p(\ZZ\cartpower{3}) {L(\pi, \std, s-z\ell+1) \over \zeta(s-2\ell + 2)^2 \zeta(2s-4\ell+2) }
.\]
:::

:::{.proof title="?"}
Carry out a refined analysis of a Rankin-Selberg integral (due to Gurevich-Segal).
:::

## A Theorem

:::{.remark}
There is a theory of *half*-integral weight modular forms on $\G_2$.
These have a good notion of Fourier coefficients taking values in $\CC/\ts{\pm 1}$.

Suppose $R \subseteq E$ is a cubic ring in a totally real cubic field.
Let $\del_R$ be the different, and let $Q_R$ be the square roots of $\del_R\inv$ in the narrow class group of $E$.
Say $(I, \mu)$ is balanced

- $I$ is a fractional ideal in $R$.
- $\mu\in E_{>0}\units$ is totally positive
- $I \mu^2 \subseteq \partial_R\inv$
- $N(I)^2 N(\mu) \disc(R) = 1$.

Note that if $R$ is the maximal order, $(I, \mu)$ is balanced iff $I^2\mu = \del_R\inv$.
Define an equivalence relation by
\[
(I, \mu) \sim (I', \mu') \iff \exists \beta\in E\units, I' = \beta I, \mu' = \beta^{-2}\mu
\]
and set $Q_R$ to be the balanced pairs mod equivalence.

:::

:::{.remark}
\envlist

- $Q_R$ can be empty
- For $Q_R$ nonempty and $R$ a maximal order in $E$,
\[
\abs{Q_R} = \size \Cl E\units[2]
.\]

:::

:::{.theorem title="Leslie-P"}
There exists a weight $1/2$ modular form $\theta'$ on $\G_2$ whose Fourier coefficients include the numbers $\pm \abs{Q_R}$ for $R$ *even monogenic*, i.e. 
\[
R = \ZZ[y]/ \gens{y^3 + cy^2 + by+a, a,b,c\in 2\ZZ}
.\]
:::

:::{.proof title="?"}
Define $\Theta$ on $\tilde \F_4$? Then let $\Theta'$ be a pullback along $\G_2\to \F_3$.
:::