For the above Axiomatization of the Example in Garrett's main vignette on the subject (the rank one case derivation, from Bernstein's general theory), you will have to add the declaration

\usepackage{color}

to your preamble. Otherwise it should compile in AMS enabled Latex.

\textbf{Axiomatization of the ``Example" of $\mathrm{SL}(2)$

given at the conclusion of the ``vignette".}

Preliminaries: notation.

\[

L^2(\Gamma\backslash G/K,\ell):=\left\{f\;\text{is locally integrable

on $\Gamma\backslash G/K$ so that}\; \int_{\Gamma\backslash G}

|f(x)|^2a_x^{-2\ell}\intd x<+\infty\right\}.

\]

We will also call $L^2(\Gamma\backslash G/K)$ $``V"$ or $V_{\ell}$, because

it is the domain Banach space for the system $X_s$.

\vspace*{0.2cm}

\noindent\textbf{Define}

\[

E_s(g)=E(a_g^s).

\]

The series expression for $E_s$ converges absolutely and uniformly

on compacta in $g\in G$ and $s\in\mathbf{C}$ for $\mathrm{Re}(s)

\geq \sigma_0>1$.\vspace*{0.2cm}

\noindent {\color{red} First input to theory:} $E_s$

is a function of $\textbf{moderate growth}$ meaning that for

large $N$,

\[

a^{-N}E_s\;\text{is bounded on every Siegel set $S_t$.}

\]\vspace*{0.2cm}

\noindent {\color{red} Second input:} Computation (rough!) of the constant

term--consisting of the

\noindent \textbf{Proposition} The constant term $c_PE_s$ of $E_s$

is of the form

\[

c_PE_s=a^s+ba^{1-s}

\]

for a constant $b$ depending on $s$. (We would like to

redo this proof without using adeles.)\vspace*{0.2cm}

\noindent {\color{red} Third input:} By the \textit{Theory

of the Constant Term} (as exposed in another Garrett ``vignette")

$E_s\in L^2(\Gamma\backslash G/K,\ell)$ if $\ell$ is large enough. Specifically,

what is needed is proved in the Proposition and Corollary on

P. 6 of the ``Constant Term" vignette.\vspace*{0.2cm}

\noindent{\color{red} Fourth input} The eigenvalue relation

\[

\eta*E_s=\lambda_s E_s,

\]

where

\begin{itemize}

\item $\eta$ is (any) $K$-conjugation invariant test function on $G$.

\item $\lambda_s$ is a function of $s$, depending on $\eta$, of course.

\item $\lambda_s$ can be shown to be an \textit{entire} function of

$s\in\mathbf{C}$, and $\eta$ can be chosen so that $\lambda_s$

is \textit{non-constant}, and in particular not identically $0$,

and given so there is a choice of $\eta$ so that $\lambda_{s_0}\neq 0$.

\end{itemize}\vspace*{0.2cm}

\noindent{\color{red}Important remarks.} In our version we will

want to avoid all this abstraction by doing computations with a specific

(Gaussian) $\eta$. In EGM's version they appear to use the ``Selberg

transform" introduced in Chapter 3 to make the relation between

$\eta$ and $\lambda_s$ more explicit. So in any case

this part of the argument can be bypassed.\vspace*{0.2cm}

\noindent\textbf{Definition.} $X_s$ is the holomorphically

parametrized system of equations in $L^2(\Gamma\backslash G/K,\ell)$

\begin{itemize}

\item[(1)] $\left(a\frac{\partial}{\partial a}-(1-s)\right)\cdot

c_Pv_s=(2s-1)a^s$, as distributions.

\item[(2)] $(\eta-\lambda_s)v_s=0$, for all $\eta\in C_c^{\infty}(G)^

{\rm inv}$.

\end{itemize}\vspace*{0.2cm}

\noindent{\color{red} Important remarks.} Presumably (2)

will be replaced by the condition that Eisenstein series

are eigenfunctions of convolution with a specific family of Gaussians;

(1) must be determined in an \textit{ad hoc} fashion following

the ``rough" computation of the constant term.\vspace*{0.2cm}

\noindent{\color{red} Combination of Second, Third, Fourth

inputs gives:} For $\mathrm{Re}(s)\gg 0$, $X_s$ has at least

the solution $E_s$.\vspace*{0.2cm}

\noindent{\color{red} This range of convergence, \textit{i.e.} the region

$\mathrm{Re}s\gg 0$, is precisely where (as will be shown)

$E_s$ is the unique solution for $X_s$.}\vspace*{0.2cm}

\noindent{\color{red} Fifth input.} The system $X_s$ has the

structural property that if $f$ is a difference

of solutions to $X_s$ then $f$ is a left $(\Gamma\cap N)$-invariant

function of moderate growth on $(\Gamma\cap N)\backslash G$

with $\eta f=f$ for some $\eta\in C_c^{\infty}(G)^{\rm inv}$.\vspace*{0.2cm}

\noindent {\color{red}The fifth input and the theory of the constant term}

imply that if $v_1$ and $v_2$ are solutions to $X_s$ in the range of

convergence, then $v_1-v_2\in L^2(\Gamma\backslash G/K)\cap

L^2(\Gamma\backslash G)$. {\color{red} The fourth input} implies

that $v_1-v_2$ is an eigenvalue of convolution with the heat

gaussian with nonconstant

eigenfunction, holomorphic in $s$ (which dependence can

be determined explicitly). Since $v_1-v_2\in L^2(\Gamma\backslash G/K)$,

we can apply a Hilbert space argument to show that $v_1-v_2\equiv 0$.

{\color{blue} This completes the proof of uniqueness

(in the range of convergence $\mathrm{Re}(s)\gg0$)}.\vspace*{0.2cm}

\noindent

{\color{red}Sixth input} There exists an operator

on the Banach space $V$, called \texttt{trunc} such that

for any $\eta\in C_c^{\infty}(K\backslash G)$,

\begin{itemize}

\item[(a)] The operator $\eta\circ\texttt{trunc}$ on $V$ is compact.

\item[(b)] $\texttt{trunc}(\eta*h)=\eta*(\texttt{trunc}\eta)$

\end{itemize}

\vspace*{0.2cm}

\noindent

{\color{red} Seventh input} The system $X_s$ has the property that

for $v\in V_{\ell}$ a solution for the $s$ close to some fixed $s_0$,

\[

c_Pv=c_1a^s+c_2a^{1-s},\;\text{for some constants $c_1, c_2$

poss. depending on $s$.}

\]

In this context, ``constant" means not depending on the $a$, in other words the

$G$-variable. Further, the above inequality must hold

for $s$ \textit{away from finitely many points}.

\vspace*{0.2cm}

\textbf{Remark.} The proof of the seventh input is elementary; it

follows from the observation that $a^s$ and $a^{1-s}$ are two

linearly independent solutions of the differential equation

required of the constant term.\vspace*{0.2cm}

\noindent

{\color{red} Eighth input} The system $X_s$ and the operator

$\texttt{trunc}$ together have the property that

for $v\in V_{\ell}$ a solution for the $s$ close to some fixed $s_0$,

\[

\texttt{trunc}(v)\;\text{is rapidly decreasing}.

\]

\vspace*{0.2cm}

\noindent

{\color{red} Ninth input} The holomorphy of the function

\[

s\mapsto \texttt{trunc}(a^s),

\]

which is in fact entire by a direct computation.

\vspace*{0.2cm}

\noindent

\textbf{Definitions relating to the dominant system.}

\begin{itemize}

\item

$L^2_a(\Gamma\backslash G/K,\ell)=\{f\in V_{\ell}\;|\; \texttt{trunc}f=f\}$;

\item

$V'=\mathbf{C}\oplus\mathbf{C}\oplus L^2_a(\Gamma\backslash G/K,\ell)$;

\item

$T_s: V'\rightarrow V_{\ell}$ given by $T_s(b,c,h)=\left(\Id_{V_{\ell}}-

\texttt{trunc}\right)(ba^s+ca^{1-s})+h$;

\item

$T_s': V'\rightarrow V_{\ell}$ given by $T_s'=(\eta-\lambda_s)\circ T_s$.

\item

$A: V_{\ell}\rightarrow V'$ given by $Av=(0,0,\texttt{trunc}v)$.

\end{itemize}

\noindent {\color{blue} Assembly of proof from these inputs.} We wish

to use the proposition on Dominance of linear systems of equations

found on the top of page 7 in Garrett's notes.

Everything is obvious or follows from Bernstein's general theory

(first three sections of Garrett notes)

from the definitions except that a) $X'$

is itself a \textit{holomorphically} parametrized system of equations

and b) that $X'$ is locally finite. For a) we use

the {\color{red}ninth input} to show that the family of linear maps

\[

\mathbf{C}^2\rightarrow L^2(\Gamma\backslash G/K,\ell),\;

\text{defined by}\; (b,c)\mapsto E(\mathrm{tail}(ba^s+ca^{1-s}))

\]

is holomorphic in $s$. Since the restriction to $L^2_a(\Gamma\backslash

G/K)$ of $T_s$ is just the inclusion map, and does not depend on $s$,

we can now conclude that $s\mapsto T_s$ is holomorphic. Then

we use the {\color{red}fourth input}, specifically the last

point about $\lambda_s$ being holomorphic, to conclude that

$T_s'$ is holomorphic.

\vspace*{0.2cm}

\noindent In order to confirm b), that $X'$ is locally finite,

we wish to apply Bernstein's {\color{blue}compact operator

criterion}, the Corollary in the upper part of page 8 of

Garrett's notes. With $A$ as above, we calculate that

for $h\in L^2_a(\Gamma\backslash G/K,\ell)$,

\[

\begin{aligned}

A\circ T'_{s_0}(h)&=&&

\texttt{trunc}(\eta_{s_0}*h)-\texttt{trunc}(\lambda_{s_0}\cdot h)\&=&&\texttt{trunc}(\eta_{s_0}*h)-\lambda_{s_0} \texttt{trunc} (h)\&=&&\eta_{s_0}\circ\texttt{trunc}(h)-\lambda s_0 (h),

\end{aligned}

\]

where in the second line we have used simply that $\lambda_{s_0}$ is a

constant (\textit{i.e.}, independent of the $G$-variable),

while in the third line we have used {\color{red}part (b) of the sixth

input}. So we have shown that $(-1/\lambda_{s_0})A$ is an inverse for $T_{s_0}'$

on $L^2_a(\Gamma\backslash G/K,\ell$), modulo a scalar multiple of

the operator $\eta\circ\texttt{trunc}$, and we now apply {\color{red}part (a) of the

sixth input} and use the fact that $L^2_a(\Gamma\backslash G/K,\ell)$

is of finite codimension in $V'$ to conclude that $A$ is an inverse for $T_{s_0}'$

modulo a compact operator. This allows the use of the compact operator criterion

and finishes the proof of completeness.

NB: this exposition is still incomplete because we haven't

mentioned the source of the fact that $\lambda_{s_0}$ is a \textit{nonzero} scalar,

which is used in the last stage of the argument for the application

of the Compactness Criterion. Maybe some other things...?

\usepackage{color}

to your preamble. Otherwise it should compile in AMS enabled Latex.

\textbf{Axiomatization of the ``Example" of $\mathrm{SL}(2)$

given at the conclusion of the ``vignette".}

Preliminaries: notation.

\[

L^2(\Gamma\backslash G/K,\ell):=\left\{f\;\text{is locally integrable

on $\Gamma\backslash G/K$ so that}\; \int_{\Gamma\backslash G}

|f(x)|^2a_x^{-2\ell}\intd x<+\infty\right\}.

\]

We will also call $L^2(\Gamma\backslash G/K)$ $``V"$ or $V_{\ell}$, because

it is the domain Banach space for the system $X_s$.

\vspace*{0.2cm}

\noindent\textbf{Define}

\[

E_s(g)=E(a_g^s).

\]

The series expression for $E_s$ converges absolutely and uniformly

on compacta in $g\in G$ and $s\in\mathbf{C}$ for $\mathrm{Re}(s)

\geq \sigma_0>1$.\vspace*{0.2cm}

\noindent {\color{red} First input to theory:} $E_s$

is a function of $\textbf{moderate growth}$ meaning that for

large $N$,

\[

a^{-N}E_s\;\text{is bounded on every Siegel set $S_t$.}

\]\vspace*{0.2cm}

\noindent {\color{red} Second input:} Computation (rough!) of the constant

term--consisting of the

\noindent \textbf{Proposition} The constant term $c_PE_s$ of $E_s$

is of the form

\[

c_PE_s=a^s+ba^{1-s}

\]

for a constant $b$ depending on $s$. (We would like to

redo this proof without using adeles.)\vspace*{0.2cm}

\noindent {\color{red} Third input:} By the \textit{Theory

of the Constant Term} (as exposed in another Garrett ``vignette")

$E_s\in L^2(\Gamma\backslash G/K,\ell)$ if $\ell$ is large enough. Specifically,

what is needed is proved in the Proposition and Corollary on

P. 6 of the ``Constant Term" vignette.\vspace*{0.2cm}

\noindent{\color{red} Fourth input} The eigenvalue relation

\[

\eta*E_s=\lambda_s E_s,

\]

where

\begin{itemize}

\item $\eta$ is (any) $K$-conjugation invariant test function on $G$.

\item $\lambda_s$ is a function of $s$, depending on $\eta$, of course.

\item $\lambda_s$ can be shown to be an \textit{entire} function of

$s\in\mathbf{C}$, and $\eta$ can be chosen so that $\lambda_s$

is \textit{non-constant}, and in particular not identically $0$,

and given so there is a choice of $\eta$ so that $\lambda_{s_0}\neq 0$.

\end{itemize}\vspace*{0.2cm}

\noindent{\color{red}Important remarks.} In our version we will

want to avoid all this abstraction by doing computations with a specific

(Gaussian) $\eta$. In EGM's version they appear to use the ``Selberg

transform" introduced in Chapter 3 to make the relation between

$\eta$ and $\lambda_s$ more explicit. So in any case

this part of the argument can be bypassed.\vspace*{0.2cm}

\noindent\textbf{Definition.} $X_s$ is the holomorphically

parametrized system of equations in $L^2(\Gamma\backslash G/K,\ell)$

\begin{itemize}

\item[(1)] $\left(a\frac{\partial}{\partial a}-(1-s)\right)\cdot

c_Pv_s=(2s-1)a^s$, as distributions.

\item[(2)] $(\eta-\lambda_s)v_s=0$, for all $\eta\in C_c^{\infty}(G)^

{\rm inv}$.

\end{itemize}\vspace*{0.2cm}

\noindent{\color{red} Important remarks.} Presumably (2)

will be replaced by the condition that Eisenstein series

are eigenfunctions of convolution with a specific family of Gaussians;

(1) must be determined in an \textit{ad hoc} fashion following

the ``rough" computation of the constant term.\vspace*{0.2cm}

\noindent{\color{red} Combination of Second, Third, Fourth

inputs gives:} For $\mathrm{Re}(s)\gg 0$, $X_s$ has at least

the solution $E_s$.\vspace*{0.2cm}

\noindent{\color{red} This range of convergence, \textit{i.e.} the region

$\mathrm{Re}s\gg 0$, is precisely where (as will be shown)

$E_s$ is the unique solution for $X_s$.}\vspace*{0.2cm}

\noindent{\color{red} Fifth input.} The system $X_s$ has the

structural property that if $f$ is a difference

of solutions to $X_s$ then $f$ is a left $(\Gamma\cap N)$-invariant

function of moderate growth on $(\Gamma\cap N)\backslash G$

with $\eta f=f$ for some $\eta\in C_c^{\infty}(G)^{\rm inv}$.\vspace*{0.2cm}

\noindent {\color{red}The fifth input and the theory of the constant term}

imply that if $v_1$ and $v_2$ are solutions to $X_s$ in the range of

convergence, then $v_1-v_2\in L^2(\Gamma\backslash G/K)\cap

L^2(\Gamma\backslash G)$. {\color{red} The fourth input} implies

that $v_1-v_2$ is an eigenvalue of convolution with the heat

gaussian with nonconstant

eigenfunction, holomorphic in $s$ (which dependence can

be determined explicitly). Since $v_1-v_2\in L^2(\Gamma\backslash G/K)$,

we can apply a Hilbert space argument to show that $v_1-v_2\equiv 0$.

{\color{blue} This completes the proof of uniqueness

(in the range of convergence $\mathrm{Re}(s)\gg0$)}.\vspace*{0.2cm}

\noindent

{\color{red}Sixth input} There exists an operator

on the Banach space $V$, called \texttt{trunc} such that

for any $\eta\in C_c^{\infty}(K\backslash G)$,

\begin{itemize}

\item[(a)] The operator $\eta\circ\texttt{trunc}$ on $V$ is compact.

\item[(b)] $\texttt{trunc}(\eta*h)=\eta*(\texttt{trunc}\eta)$

\end{itemize}

\vspace*{0.2cm}

\noindent

{\color{red} Seventh input} The system $X_s$ has the property that

for $v\in V_{\ell}$ a solution for the $s$ close to some fixed $s_0$,

\[

c_Pv=c_1a^s+c_2a^{1-s},\;\text{for some constants $c_1, c_2$

poss. depending on $s$.}

\]

In this context, ``constant" means not depending on the $a$, in other words the

$G$-variable. Further, the above inequality must hold

for $s$ \textit{away from finitely many points}.

\vspace*{0.2cm}

\textbf{Remark.} The proof of the seventh input is elementary; it

follows from the observation that $a^s$ and $a^{1-s}$ are two

linearly independent solutions of the differential equation

required of the constant term.\vspace*{0.2cm}

\noindent

{\color{red} Eighth input} The system $X_s$ and the operator

$\texttt{trunc}$ together have the property that

for $v\in V_{\ell}$ a solution for the $s$ close to some fixed $s_0$,

\[

\texttt{trunc}(v)\;\text{is rapidly decreasing}.

\]

\vspace*{0.2cm}

\noindent

{\color{red} Ninth input} The holomorphy of the function

\[

s\mapsto \texttt{trunc}(a^s),

\]

which is in fact entire by a direct computation.

\vspace*{0.2cm}

\noindent

\textbf{Definitions relating to the dominant system.}

\begin{itemize}

\item

$L^2_a(\Gamma\backslash G/K,\ell)=\{f\in V_{\ell}\;|\; \texttt{trunc}f=f\}$;

\item

$V'=\mathbf{C}\oplus\mathbf{C}\oplus L^2_a(\Gamma\backslash G/K,\ell)$;

\item

$T_s: V'\rightarrow V_{\ell}$ given by $T_s(b,c,h)=\left(\Id_{V_{\ell}}-

\texttt{trunc}\right)(ba^s+ca^{1-s})+h$;

\item

$T_s': V'\rightarrow V_{\ell}$ given by $T_s'=(\eta-\lambda_s)\circ T_s$.

\item

$A: V_{\ell}\rightarrow V'$ given by $Av=(0,0,\texttt{trunc}v)$.

\end{itemize}

\noindent {\color{blue} Assembly of proof from these inputs.} We wish

to use the proposition on Dominance of linear systems of equations

found on the top of page 7 in Garrett's notes.

Everything is obvious or follows from Bernstein's general theory

(first three sections of Garrett notes)

from the definitions except that a) $X'$

is itself a \textit{holomorphically} parametrized system of equations

and b) that $X'$ is locally finite. For a) we use

the {\color{red}ninth input} to show that the family of linear maps

\[

\mathbf{C}^2\rightarrow L^2(\Gamma\backslash G/K,\ell),\;

\text{defined by}\; (b,c)\mapsto E(\mathrm{tail}(ba^s+ca^{1-s}))

\]

is holomorphic in $s$. Since the restriction to $L^2_a(\Gamma\backslash

G/K)$ of $T_s$ is just the inclusion map, and does not depend on $s$,

we can now conclude that $s\mapsto T_s$ is holomorphic. Then

we use the {\color{red}fourth input}, specifically the last

point about $\lambda_s$ being holomorphic, to conclude that

$T_s'$ is holomorphic.

\vspace*{0.2cm}

\noindent In order to confirm b), that $X'$ is locally finite,

we wish to apply Bernstein's {\color{blue}compact operator

criterion}, the Corollary in the upper part of page 8 of

Garrett's notes. With $A$ as above, we calculate that

for $h\in L^2_a(\Gamma\backslash G/K,\ell)$,

\[

\begin{aligned}

A\circ T'_{s_0}(h)&=&&

\texttt{trunc}(\eta_{s_0}*h)-\texttt{trunc}(\lambda_{s_0}\cdot h)\&=&&\texttt{trunc}(\eta_{s_0}*h)-\lambda_{s_0} \texttt{trunc} (h)\&=&&\eta_{s_0}\circ\texttt{trunc}(h)-\lambda s_0 (h),

\end{aligned}

\]

where in the second line we have used simply that $\lambda_{s_0}$ is a

constant (\textit{i.e.}, independent of the $G$-variable),

while in the third line we have used {\color{red}part (b) of the sixth

input}. So we have shown that $(-1/\lambda_{s_0})A$ is an inverse for $T_{s_0}'$

on $L^2_a(\Gamma\backslash G/K,\ell$), modulo a scalar multiple of

the operator $\eta\circ\texttt{trunc}$, and we now apply {\color{red}part (a) of the

sixth input} and use the fact that $L^2_a(\Gamma\backslash G/K,\ell)$

is of finite codimension in $V'$ to conclude that $A$ is an inverse for $T_{s_0}'$

modulo a compact operator. This allows the use of the compact operator criterion

and finishes the proof of completeness.

NB: this exposition is still incomplete because we haven't

mentioned the source of the fact that $\lambda_{s_0}$ is a \textit{nonzero} scalar,

which is used in the last stage of the argument for the application

of the Compactness Criterion. Maybe some other things...?

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