TITLE: If $Sx\neq0$. Why $S^{1/2}Sx \neq0$? where $S\geq0$.
QUESTION [1 upvotes]: Let $\mathcal{H}$ be a complex Hilbert space. Let $T\in \mathcal{B}(\mathcal{H})$ and let $S\in \mathcal{B}(\mathcal{H})^+$ (i.e. $S^*=S$ and $\langle Sx\;| \;x\rangle \geq0,\;\forall x\in \mathcal{H}$).

Assume that there exists $x\in \mathcal{H}$ such that $Sx\neq0$. Why $S^{1/2}Sx \neq0\,$ ?

Thank you

REPLY [4 votes]: Assume to the contrary that $ S^{1/2}Sx =0$. Then
$S^2x=S^{1/2}(S^{1/2}Sx)=0$, hence
$||Sx||^2=\langle Sx|Sx\rangle =\langle S^2x|x\rangle=0$ and therefore $Sx=0$, a contradiction.


TITLE: Questions about Rudin's rank theorem
QUESTION [3 upvotes]: I am trying to understand the rank theorem in Rudin's Principles of Mathematical Analysis.
The theorem states:

Theorem Suppose $m,n,r$ are nonnegative integers, $m\ge r, n\ge r$, $F$ is a $C^1$ mapping of an open set $E\subset \mathbb{R}^n$ into $\mathbb{R}^m$, and $F'(x)$ has rank $r$ for every $x\in E$. Fix $a\in E$, put $A = F'(a)$, let $Y_1$ be the range of $A$, and let $P$ be a projection in $\mathbb{R}^m$ whose range is $Y_1$ and let $Y_2$ be the kernel of $P$. Then there are open sets $U$ and $V$ in $\mathbb{R}^n$, with $a\in U\subset E$, and there is a 1-1 $C^1$ mapping $H$ of $V$ onto $U$ (whose inverse is also of $C^1$) such that $$F(H(x)) = Ax+\phi(Ax)\;\;\;\;(x\in V)$$where $\phi$ is a $C^1$ mapping of the open set $A(V)\subset Y_1$ into $Y_2$.

I have quite a few problems trying to understand this theorem, but perhaps mainly this:

What kind of projection does Rudin have in mind? All he says about the projection is that its range is $Y_1$. Is this the projection of the entire  $\mathbb{R}^m$ and the $\text{im}(P)=Y_1$? Can you choose any projection with such range? Does the choice of such projection affect the null space of P?
What does the equation $F(H(x)) = Ax + \phi(Ax)$ tell us intuitively? Why are we mapping with $\phi$ from the range of $P$ to its nullspace?

REPLY [0 votes]: (See comments...) Finally, for the intuition. Think of $H$ as a change of coordinate system in the domain. Let's go ahead and do a change of coordinate system in the range as well. But this one is easier: There is an invertible linear map $B: R^m \to R^m$ such that
$$
(B \circ A) (x_1, x_2, \ldots, x_n) = (x_1, x_2, \ldots, x_r).
$$
So, the claim of the theorem is (you need to replace your $\phi$) that  given any $C^1$ maps that has rank $r$ on an open set, we can do a local $C^1$ change of coordinate in domain and a linear change of coordinate in the range, such that in the new coordinates, the function depends only on the first $r$ coordinates:
$$
(x_1, x_2, \ldots, x_n) \mapsto (x_1, x_2, \ldots, x_r) + \phi(x_1, x_2, \ldots, x_r).
$$
This becomes extra beautiful if $m=r$ in which case we can take $\phi$ to be zero.
If you wan to avoid doing the change of coordinates in the range, then you can interpret the result as saying that "you know where the image of $f$ (after the change of coordinates in domain) is if you know where the image of the (simpler) map $A$ is, i.e. $Ax$ completely determines $f(x)$."


TITLE: Group of order 28 with normal subgroup of order 4 is abelian
QUESTION [1 upvotes]: Herstein ch2.11 q19

Prove that if $G$ of order 28 has normal subgroup of order 4, then $G$ is abelian.

My attempt: The 7-sylow subgroup lies in center. So $\circ(Z)=7, 14$ or $28$.
For $\circ(Z)=14$, $G/Z$ is cyclic. But this argument fails for $\circ(Z)=7$.
I have not utilized the fact $G$ has normal subgroup of order 4.
Please give a hint. Please do not give solution. Thanks!
(This looks problematic. Also in one of the comments, meaning of $[\operatorname{Aut} H :1]$ is unclear)
Edit: Thanks to @DietrichBurde 's comment, this answers this question. So my post is a duplicate.

REPLY [1 votes]: Let $H$  be the subgroup of order 4. It is well known that $H$ must be abelian.
Now, the map that send $g \in G$ to the conjugation by $g$ define an omomorphism from $G$ to $Aut H$.
What can you say about the image and the kernel of this omomorphism?
Can you see how to conclude?


text:
\section{Quaternionic Shimura varieties and the main theorem}
\label{sec:qsv-mt}

\subsection{Quaternionic Shimura varieties}
Let $F$ be a totally real field. Let $B$ be a {\it non-split} quaternion algebra over $F$
and $\Sigma$ the set of infinite places of $F$ where $B$ is split. We fix an isomorphism
\[
B\otimes_F \R \simeq \M_2(\R)^d \times \H^{n-d},
\]
which gives an identification
\[
G_B (\R) \simeq \GL_2(\R)^d \times (\H^\times)^{n-d},
\]
where $G_B:=\Res_{F/\Q} B^\times$.
For $v\in \Sigma$, let $\tau_v$ denote the composite
\[
B \otimes_{F,\sigma_v} \R \simeq \M_2 (\R) \hookrightarrow \M_2 (\C),
\]
and for $v\in \Sigma_\infty\smallsetminus \Sigma$, the composite map
\[
B \otimes_{F,\sigma_v} \R \simeq \H \hookrightarrow \M_2 (\C)
\]
Then $\tau_v$ may be viewed as giving a two-dimensional complex representation of
$G_B$.
We identify $\C^\times$ with a subgroup of $\GL_2 (\R)$ via
\begin{equation}
\label{eqn:iota}
z=a+bi \mapsto \iota(z) := \begin{bmatrix} a& b \\
-b & a \end{bmatrix}.
\end{equation}
Let $X$ denote the $G_B(\R)$-conjugacy class of
\[
h: \Ss \rightarrow G_{B,\R}, \quad h(z) = \left(\iota(z),\cdots, \iota(z), 1, \cdots ,1\right),
\]
so that $h_v(z):=h(z)_v=\iota(z)$ for $v \in \Sigma$ and $h_v (z) = 1$ for the infinite places not in $\Sigma$. We write either $\Sh_{G_B}$ (or
for ease of notation, simply $\Sh_B$) for the associated Shimura variety.
The variety $\Sh_B$ admits a canonical model over the reflex field $F_\Sigma$.
The Hecke algebra $\cH (G_B (\A_f), \cK)$ acts on $\Sh_{B,\cK}$ via correspondences. Moreover, the
inverse limit
\[
\Sh_B = \varprojlim_{\cK} \Sh_{B,\cK}
\]
admits a right $G_B( \A_f)$-action. We refer the reader to
\cite{periods1}, \S 1 for a more detailed discussion of the
Shimura varieties $\Sh_B$.

\subsection{Local systems}
Let $\pi$ be an automorphic representation of $\GL_2 (\A_F)$ attached to a holomorphic
Hilbert modular form of weight $(\uk,r)$, where $\uk=(k_1, \ldots, k_d)$ is a collection of integers of the same parity and $r$ is an integer with
$k_i\equiv r \bmod 2$. (If $v$ is the $i$th place in the ordering, we
will often denote $k_i$ by $k_v$.) We suppose that $\pi$ admits a Jacquet-Langlands transfer $\pi_B$ to $G_B(\A)$. We also assume that  $k_i\ge 2$ for all $i$. This implies that
the representation $\pi_{B,\infty}$ is cohomological, namely $\pi_B$ contributes to the cohomology of a
local system on $\Sh (G_B)$. The local system is attached to the representation $\tau_{\uk,r}^\vee$, where
\begin{align*}
\tau_{\uk,r} & := \bigotimes_{v} (\tau_v \circ \nu)^{(r-k_v+2)/2}\Sym^{k_i-2} (\tau_v ) \\ & = \bigotimes_v ({\det} \circ \tau_v )^{(r-k_v+2)/2} \Sym^{k_v-2} (\tau_v )
\end{align*}
of $G_B(\C)$. By \cite{waldspurger-rationality} (Prop. I.3 and \S II.2) the restriction of the representation $\tau$ to the group $G_B$ is defined over (any field $L$ containing)
 $\Q(\uk)$, where $\Q(\uk)$ is the fixed field of the subgroup
 \[
 \{ \sigma \in \Aut(\C/\Q): \ \sigma \circ (\sum_v k_v \tau_v) = \sum_v k_v \tau_v \}.
 \]
More precisely, this representation contains an $L$-structure invariant by $G_B$ and that is unique up to homothety.

\begin{rem}
 In this paper, we are only concerned with the case when $\pi$ has trivial central character up to twisting by a power of the reduced norm.
 This implies that the weights $k_i$ must all be even.
 Then, by twisting $\pi$ by a power of the norm character, we may assume that $r=0$. For simplicity, we will thus make this assumption for the rest of the paper and drop $r$ from the notation. Thus we will just write $\tau_{\uk}$ below. (For the more general case of nontrivial central characters, see \S \ref{sec:non-self-dual}.)
 \end{rem}

Let $L=\Q(\pi)$ be the field of rationality of $\pi$, as defined in \cite{waldspurger-rationality}, \S 1.8. By {\it loc. cit.} Cor.  I.8.3 and Lemma I.2.3, this field contains $\Q(\uk)$ and also agrees with the field generated by (all but finitely many, in particular the unramified) Hecke eigenvalues of $\pi$.
Thus we may view $\tau_{\uk}$ as being defined over $L$, and then
we get an associated local system of $L$-vector spaces $\V_{\uk}(L)$ on $\Sh_B (\C)$ and
for every finite prime $\ell$, an \'{e}tale $L\otimes \Q_\ell$-sheaf $\V_{\uk}(L)_{\ell}$ on $\Sh_B$. (See also \cite{carayol-hmf}, \S 2.1.)
Let
\begin{align*}
V^{\cK}_B (L) &:=H^* (\Sh_{B, \cK}(\C), \V_{\uk}(L)), \\ V^{\cK}_B(\C)&:=H^* (\Sh_{B, \cK}(\C), \V_{\uk}(\C) ), \\
  V_{B}^{\cK}(L)_\ell&:= H^*_{\et} (\Sh_{B, \cK} \otimes_{F_\Sigma} \Qbar, \V_{\uk,\ell}),
\end{align*}
so that there are canonical isomorphisms
\[
V^{\cK}_B (L)  \otimes_L \C \simeq V^{\cK}_B(\C)
\]
and
\[
V^{\cK}_B (L)  \otimes_{\Q} \Q_\ell \simeq  V_{B}^{\cK}(L)_\ell.
\]

We fix an isomorphism
\[
B \otimes \A_F^S \simeq \M_2(\A_F^S),
\]
where $\A_F^S$ denotes the adeles of $F$ outside a finite set of places $S\supset \Sigma_\infty$. This gives an isomorphism
\begin{equation}
\label{eqn:identification-at-finite-places}
B^\times (\A_F^S) \simeq \GL_2(\A_F^S).
\end{equation}
We assume that $\pi$ transfers to $B^\times(\A_F)$, i.e., there exists an
automorphic representation $\pi_B=\pi_B^f \otimes \pi_{B,\infty}$ of $B^\times(\A_F)$ (necessarily unique by strong multiplicity one) such that
$\pi_B^S \simeq \pi^S$ via the identification \eqref{eqn:identification-at-finite-places} above.

For the cohomology with complex coefficients, we can define the $\pi_B$-isotypic component by
\[
V_{B,\pi_B} (\C):= \Hom_{\cH_{\C} (G_B(\A_f), \cK)} ((\pi_B^f)^{\cK}, V_B^\cK (\C)),
\]
for $\cK$ small enough. This
is concentrated in degree $2d$
and is independent of the choice of $\cK$. To work over the field of rationality, we note that
by \cite{waldspurger-rationality}, Lemma 1.2.2 and Sec. II.1, the Hecke module $(\pi_B^f)^{\cK}$ is also defined over $L$.
More precisely, it contains an $L$-structure $(\pi_B^f)^{\cK} (L)$ that is invariant by the Hecke algebra with $\Q$-coefficients, $\cH_{\Q} (G_B(\A_f), \cK)$,
and that is unique up to homothety.
This allows us to define the $\pi_B$-isotypic components
\begin{align*}
V_{B,\pi_B} & := \Hom_{\cH_{\Q} (G_B(\A_f), \cK)} ((\pi_B^f)^{\cK}(L), V_B^\cK(L)), \\ V_{B,\pi_B, \ell} & := \Hom_{\cH_{\Q} (G_B(\A_f), \cK)} ((\pi_B^f)^{\cK} (L),  V_{B}^\cK(L)_\ell),
\end{align*}
for $\cK$ small enough, these being independent of the choice of $\cK$.
Moreover, there are canonical isomorphisms of free $L\otimes \Q_\ell$-modules:
\[
V_{B,\pi_B} \otimes_{\Q} \Q_\ell \simeq  V_{B,\pi_B, \ell}.
\]

\subsection{The main theorem in the general case}

We can now state the main theorem in the case of general local
systems.
Let $B_1$ and $B_2$ be two quaternion algebras that are split at the same set of archimedean places $\Sigma \subset \Sigma_\infty$,
such that $\pi$ transfers to both $B_1^\times(\A_F)$ and $B_2^\times (\A_F)$. For ease of notation we write the transfers
as $\pi_1$ and $\pi_2$ instead of $\pi_{B_1}$ and $\pi_{B_2}$ respectively.

\begin{thm}
Let $L$ be the coefficient field of $\pi$.
There is an isomorphism of $L$-Hodge structures
\[
\iota: V_{B_1,\pi_1} \simeq V_{B_2,\pi_2},
\]
such that for all finite primes $\ell$, the maps $\iota_\ell$, defined by
requiring the diagram
\begin{equation}
\label{eqn:compat-hd-general}
\xymatrix{
V_{B_1,\pi_1} \otimes_{\Q} \Q_\ell \ar[r]^{\iota\otimes 1} \ar[d]^{\simeq}  & V_{B_2,\pi_2}\otimes_\Q \Q_\ell \ar[d]^{\simeq} \\
V_{B_1,\pi_2,\ell} \ar[r]^{\iota_\ell} & V_{B_2,\pi_2,\ell}
}
\end{equation}
be commutative, are $\Gal(\Qbar/ F_\Sigma)$-isomorphisms. \end{thm}

This theorem will be proved in \S \ref{sec:proof-of-main-thm}.

config: arxiv