# Ellen Eischen, Automorphic Forms on Unitary Groups, Talk 1

:::{.remark}
Overall plan:

- Introduce automorphic forms on unitary groups,
- Techniques to study *algebraic* aspects of \(L\dash \)functions,
- Using unitary groups as a convenient setting -- a large enough class of groups to be interesting, but confined enough to be tractable.

Today:

- Motivations from modular forms,
- Fundamental definitions.

If the previous talk was a "fairy tale", this will flesh out the "based on a true story" part!

:::

## Motivation from modular forms

:::{.example title="?"}
Consider $\zeta(2k)$ for $k\in \ZZ_{\geq 0}$; known to Euler as
\[
\zeta(2k) = (-1)^k \pi^{2k} {2^{2k-1} \over (2k-1)!} \qty{-{ {B_{2k}} \over 2k} }
= (-1)^k \pi^{2k} {2^{2k-1} \over (2k-1)!} \zeta(1-2k)
,\]
where $B_{2k}$ is the $2k$th **Bernoulli number**, whose exponential generating function is
\[
{ze^z\over e^z-1} = \sum_{k\geq 0} B_k {z^k\over k!} \in \QQ\fps{z}
.\]

Proving rationality of $\zeta(2k)$ (up to powers $\pi^n$) involves the normalized Eisenstein series
\[
G_{2k}(q) = \zeta(1-2k) + 2\sum_{k\geq 1} \sigma_{2k-1}(n) q^n, \quad q\da e^{2\pi i z}, \quad \sigma_k(n) \da \sum_{d\divides n} d^k
,\]
and one can use similar techniques to prove rationality for

- Dedekind zeta functions for $K\in \NF$ totally real:
\[
\zeta_K(s) = \sum_{\mfa\normal \OO_K} {1\over N(\mfa)^s}, \qquad N(I) \da \Norm_{K/\QQ}(I)
,\]
  where rationality was proved by realizing it as the constant term of a Fourier expansion of an Eisenstein series and studying spaces of modular forms.

- \(L\dash \)functions $L(\chi, s)$ for $\chi$ a **Hecke character** of a totally real field, and their \(p\dash \)adic analogs.

All of these correspond to **Artin \(L\dash \)functions** $L(s, \rho)$ for $\rho$ a Galois representation.
Dimensions $n=1$ and (partially) $n=2$ are handled class field theory.
Can we generally show special values are algebraic?
And if so, what do these values mean?
:::

:::{.remark}
More generally, one can ask about algebraicity or rationality of special values of \(L\dash \)functions attached to modular forms.
Our first tool for constructing such things will be **Rankin-Selberg convolution**.
Why care about special values: Kummer used congruences for $\zeta$, checking if $p\divides \cl(K)$ is equivalent to checking if $p$ divides numerators of Bernoulli numbers, which can be used to prove special cases of Fermat.
Picked up later for Iwasawa theory, controls behavior of towers of towers of cyclotomic extensions in $\mods{G_K}$.
:::

:::{.remark}
Conjectures

- Meanings of \(L\dash \)function values, e.g. Deligne's conjecture that they come from **motives**.
- Langlands: connections between Galois reps $\rho$ and automorphic reps.

:::

:::{.warnings}
It might seem like $\GL_n$ for $n\geq 3$ is the next step, but this turns out to be too general!
Even $\SL_n$ in these ranges is difficult.
Instead we'll move to **unitary groups**, where we'll have Shimura varieties to work with.
:::

## Unitary groups

:::{.remark}
Fix $K\in \CM\Field$, so $K/K^+/\QQ$ with $K^+/\QQ$ totally real and $K/K^+$ quadratic imaginary, and $V\in \Vect\slice K$ with a nondegenerate Hermitian pairing $\ip{\wait}{\wait}$, which can be extended linearly to $V_R \da V\tensor_{K^+} R$ for any $R\in\alg\slice {K^+}$.
:::

:::{.definition title="General Unitary Groups"}
The **general unitary group** is the algebraic group $G \da \GU(V, \inp{\wait}{\wait})$ which is defined for each $R\in \alg\slice {K^+}$ as 
\[
R \mapsto \ts{g\in \GL_{K_R}(V_R) \st \inp{gv}{gw} = \nu \inp v w \text{ for some } \nu\in R}
.\]
The **unitary group** is the subgroup for which $\nu = 1$ is enforced.
:::

:::{.remark}
If $R=\RR$, choose an ordered basis for $B$ to define the **signature**
\[
\inp v w = vA \transp{w}, \qquad A = \matt{\one_a}{0}{0}{-\one_b},\qquad
\signature(A) \da (a, b)
.\]

For the remainder of today, assume $K^+ = \QQ$.
:::

## Automorphic forms on unitary groups, connections to modular forms
  
:::{.remark}
On the modular form side:

1. $f: \mfh \to \CC, f(z) = (cz+d)^{-k}f(\gamma z)$, holomorphic at cusps, etc
2. \[
\phi_f: \SL_2(\RR)\to \CC, \SL_2(\RR)\actson \mfh
\]
transitively fixing $i$, 
\[
\phi_f(g) = j(g, i)^{-k}f(gi)
,\] and
\[
\phi_f: \dcosetl{\Gamma}{G(\RR)} &\to \CC \\
\phi_f(g(\rot(\theta)) &= e^{ki\theta} \phi_f(g) \qquad \rot(\theta) \da 
\left[\begin{array}{cc}
\cos \theta & -\sin \theta \\
\sin \theta & \cos \theta
\end{array}\right]
,\] 
extend to 
\[
\phi: \dcosetl{\Gamma Z(G)}{G(\RR)} \to \CC
\]
for $G=\GL_2, \SL_2, \GL_2^+$, etc
3. Adelic interpretation: $\GL_2(\AA) = \GL_2(\QQ) \GL_2^+(\RR) \tilde K$ where $\tilde K \da \prod_p K_p$is a compact open subgroups of $\GL_2(\QQpadic)$ with determinant $\ZZpadic\units$ and equal to $\GL_2(\QQpadic)$ for all but finitely many places.
  - Can recover $\Gamma \da \GL_2(\QQ) \intersect (\GL_2^+(\RR) \times \mck)$
  - Match up
  \[
\dcosetl{\Gamma}{\GL_2(\RR)} \mapstofrom \dcoset{\GL_2(\QQ)}{\GL_2(\AA)}{\mck}
  .\]
  - Get functions 
  \[
\phi_f: \dcosetl{\GL_2(\QQ)}{\GL_2(\AA)} \to \CC
  .\]

On the automorphic side:

1. Replace $\mfh$ with $G/\mck_\infty = \U_{n, m}(\RR)/ \U_n\cartpower{2}$, a quotient by a compact.
2. Writing $G\da \GU(n, m)$ for a form of signature $(n,m)$, replace with $\dcosetl{\Gamma Z(G)}{G}$ where $G\contains \mck_\infty \da U(n)\cartpower{2}$, and analogously $\dcosetl{\Gamma Z(G)}{G(\RR)} \to \CC$.
3. For $G(\AA_f) = \disjoint_i G(\RR)\inv ? \mck$.

:::

:::{.remark}
An **automorphic form** on $\U_{n, n}$ is a holomorphic function $f\in \mfh_n\to V$ where $\rho\actson V$ is a representation of $\GL_n(\CC)\cartpower{2}$ where
\[
f(z) = \rho(cz+d, \transp{\bar c} z + \bar d)\inv f( \gamma z)\qquad \forall \gamma \in \Gamma = \matt a b c d\in \U_{n, n}(\OO_K)
\]
where 
\[
\gamma z = (az+b)(cz+d)\inv, \qquad \mfh_n \da \ts{z\in \Mat_n(\CC) \st i(\transp{\bar z} - z) > 0}
.\]
:::

:::{.remark}
One thing we haven't mentioned yet: modular forms as sections of line bundles over modular curves, so moduli of elliptic curves with level structure, and the generalized setup will be vector bundles over (unitary) Shimura varieties.

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