# Ellen Eischen, Talk 4: Revisiting the Doubling Method for $n=1$

## Reducing to Finite Sums

:::{.remark}
Goal: see what happens if we do the doubling method in the following setup.
Let 

- $n=1$
- $K\in \NF$ be an imaginary quadratic field 
- $V\in \Vect^{\dim = 1}\slice K$
- $W = V\sumpower{2}$
- $G\da \U(V, \inp{\wait}{\wait}_V) \cong \U_1 = \ts{g\in \GL_2 \st g\bar g = 1}$.
  Note that $G \subseteq \GU(V, \inp\wait\wait _V)\cong \GU_1 \cong \GL_1$

- $H\da \U(W, \inp{\wait}{\wait}_V) \cong \U_{1, 1}$.
  Note that $H \subseteq \GU(W, \inp\wait\wait _V)\cong \GU_{1, 1}$
:::

:::{.remark}
Spoiler: we'll get an expression for the \(L\dash \)function $L(s, \chi)$ for a Hecke character $\chi: \dcosetl{K\units}{\AA_K\units} \to CC\units$ as a finite sum of values $E_\chi(A) \chi(A)$ for some elliptic curves $A$ with CM by $\OO_K$, and we'll obtain an algebraicity result.
:::

:::{.remark}
Note that
\[
\GU_{1, 1} \cong \dcosetr{\GL_2\times \Res_{K/\QQ} \GG_m }{\GG_m}
,\]
and the associated symmetric space consists of copies of the upper half plane $\mfh_1$.
The associated modular form is a modular form, possibly with mild additional conditions on each component.
:::

:::{.remark}
Reminder of the doubling method: we had an integral
\[
Z(s,\chi,\phi,\tilde\phi) = \int_{\dcosetl{G\cartpower 2 (\QQ)} {\GG\cartpower 2(\AA)} } E_{f_S, \chi}(g, h) \phi(g) \tilde \phi(h) \chi\inv (\det h) \, dg\, dh
.\]

Some properties:

- $Z$ is an automorphic form on $\GU_1 = \GL_1$, and thus a Hecke character.
- If one chooses $\phi=\chi\inv$ so $\phi\inv = \chi$ (plus some compatibility conditions), $Z$ collapses to a finite sum:
\[
Z(s,\chi,\phi\tilde\phi) = \sum_{\scriptscriptstyle \dcoset{ G\cartpower 2 (\QQ) } { G\cartpower 2 (\AA)} { \mck }} E_{S,\chi}(g,h)\chi\inv(g)
.\]
:::

:::{.remark}
There is a diagram:
\begin{tikzcd}
{G(\U_1\cartpower{2})} && {\GU_{1,1}} & {} \\
\\
{\U_1\cartpower{2}} && {\U_{1,1}}
\arrow[hook, from=3-1, to=3-3]
\arrow[hook, from=1-1, to=1-3]
\arrow[hook, from=3-3, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsNSxbMiwwLCJcXEdVX3sxLDF9Il0sWzIsMiwiXFxVX3sxLDF9Il0sWzAsMCwiRyhcXFVfMVxcY2FydHBvd2VyezJ9KSJdLFswLDIsIlxcVV8xXFxjYXJ0cG93ZXJ7Mn0iXSxbMywwXSxbMywxLCIiLDAseyJzdHlsZSI6eyJ0YWlsIjp7Im5hbWUiOiJob29rIiwic2lkZSI6InRvcCJ9fX1dLFsyLDAsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoidG9wIn19fV0sWzEsMCwiIiwxLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=)

Moreover there is an embedding:
\begin{tikzcd}
{\GU(V)\times\GU(V)} && {\GU(W)} \\
\\
& {\G(\U(V) \times \U(-V)) = \ts{(g,h) \in \GU\cartpower{2} \st \nu(g) = \nu(h)}}
\arrow[hook', from=3-2, to=1-1]
\arrow[hook, from=3-2, to=1-3]
\end{tikzcd}

> [Link to Diagram](https://q.uiver.app/?q=WzAsMyxbMSwyLCJcXEcoXFxVKFYpIFxcdGltZXMgXFxVKC1WKSkgPSBcXHRzeyhnLGgpIFxcaW4gXFxHVVxcY2FydHBvd2VyezJ9IFxcc3QgXFxudShnKSA9IFxcbnUoaCl9Il0sWzAsMCwiXFxHVShWKVxcdGltZXNcXEdVKFYpIl0sWzIsMCwiXFxHVShXKSJdLFswLDEsIiIsMCx7InN0eWxlIjp7InRhaWwiOnsibmFtZSI6Imhvb2siLCJzaWRlIjoiYm90dG9tIn19fV0sWzAsMiwiIiwyLHsic3R5bGUiOnsidGFpbCI6eyJuYW1lIjoiaG9vayIsInNpZGUiOiJ0b3AifX19XV0=)

These induce embeddings of corresponding Shimura varieties:

- $\mcm_{G(\U_1\cartpower 2)} \to \mcm_{\GU_1\cartpower{2}}$ which classifies produces $A_1\times A_2$ 1-dimensional AVs with PEL structures, so elliptic curves with CM by $\OO_K$,
- $\mcm_{G(\U_1\cartpower 2)} \to \mcm_{\GU_{1, 1}}$ which classifies *certain* 2-dimensional AVs with PEL structures.

:::

:::{.remark}
Recall that the adelic points of our quotients are $\CC\dash$points of unitary Shimura varieties, and $\mcm_{\GU_{1,1}}(\CC) = \disjoint \dcosetl{\Gamma_K}{\mfh_1}$ where we mod out by some level.
Any $z\in \mfh = \mfh_1$ corresponds to some $\CC\cartpower{2}/\gens{(z\bar a + \bar b, za + b)}$ where $a,b$ are in some $\OO_K$ lattice.
Note the similarity to $\CC/\gens{\ZZ+ \tau \ZZ}$ for elliptic curves.
:::

:::{.remark}
An upshot is that there are three special things in this case:

- Integral is a finite sum,
- There are only characters,
- We're evaluating at special points!
:::

:::{.remark}
Let $Z(s, \chi) \da Z(s, \chi, \phi,\tilde\phi)$.
We can choose $f_{S,\chi}$ such that 
\[
Z(s,\chi) = c L(s, \chi)
\]
for $c$ a scalar, i.e. they differ by a multiple.
This expresses $L(s, \chi)$ as a finite sum of values of $E(s,\chi) \cdot \chi(\wait)$ for $E$ an automorphic form on $\U_{1, 1}$, so a special kind of modular form.
There is a variant of *Damerell's formula*, which expresses $L(s, \chi)$ as such a finite sum where $E$ is an Eisenstein series in a space of Hilbert modular forms.
:::

## Rationality Properties for Eisenstein Series

:::{.remark}
We can obtain an Eisenstein series on $\mfh = \mfh_1$ of the form
\[
\sum_{(c,d)\in \Lambda} {\chi(d) \over (cz+d)^k (cz+d)^s}
\]
where $\Lambda$ is an appropriate $\OO_K$ lattice, and for certain characters will converge for $\Re(s) + k > 2 = 2n$.
This will have rational Fourier coefficients, and is holomorphic for $s=0$.
As in the case of modular forms, there is a $q\dash$expansion (or more generally in other signatures, a Fourier-Jacobi expansion) principle: 


:::{.slogan}
Automorphic forms on $\U_{n,n}$ are determined by their $q\dash$expansions.
:::

In particular, if the coefficients of the $q\dash$expansion are contained in $R$, then $f$ is in fact defined over $R$.
Kai-Wen Lan proved a more general version of this principle for $\U_{a,b}$ with any signature, and showed that algebraic $q\dash$expansions and analytic (i.e. Fourier) expansions agree.

So things look good for $s=0$!
:::

:::{.question}
What about $s\neq 0$, i.e. when the Eisenstein series is not holomorphic?
:::

:::{.answer}
We use Mass-Shimura differential operators $\delta^{(r)}_K$ to relate $E$ at $s\neq 0$ to $E$ at $s=0$, where here $\delta$ raises weights by $2r$.
For $F$ a modular form defined over $\QQbar$, Shimura proved the following:
\[
{ (\delta_k^{(r)} F)(A) \over \Omega^{k+2r} } \in \QQbar
\]
for each CM point $A$.
These operators have incarnations in $\U_{n, m}$ and there are analogous algebraicity results.
In fact,
\[
E(z, -r, \chi) = c (-4\pi y)^r \delta_k^{(r)} E(z, 0,\chi)
.\]
where $c$ is a nice rational factor.
Combining these results yields
\[
{L(r, \chi) \over \Omega^{k+2r}}\in \QQ
.\]
:::

:::{.remark}
A word about this operator:
\[
\delta_k f = {1\over 2\pi i} \qty{{k\over 2iy} + \dd{}{z} } f = {1\over 2\pi i}y^{-k} \dd{}{z} (y^k f), \qquad \delta_k^{(r)} = \delta_k \circ \delta_k \circ \cdots \circ \delta_k
.\]
Katz's idea: reexpress this operator geometrically over a moduli space of elliptic curves, or more generally AVs, in terms of the **Gauss-Manin connection** and the **Kodaira morphism**, and a splitting 
\[
H^1_{\dR} = \omega \oplus H^{0, 1}
\]
which preserves algebraicity at CM points.
:::