TITLE: Complex Analysis Integrals
QUESTION [2 upvotes]: I'm unsure how to apply what I've learned in complex analysis to the following question types:
$$ \int_{-\pi}^\pi \frac 1 {1 + \sin^2(\theta)}\,d\theta $$
and
$$ \int_{-\pi}^\pi \frac {\sin(n\theta)} {\sin(\theta)}\,d\theta. $$
Any help regarding where to look is greatly appreciated

REPLY [1 votes]: Hint: $\displaystyle \sin\theta=\frac{e^{i\theta}-e^{-i\theta}}{2i}\implies \sin\theta=\frac{z-z^{-1}}{2i},d\theta=\dfrac{dz}{iz}$


TITLE: Why heat capacity of an ideal gas varies with temperature? Why can't it remain the constant?
QUESTION [0 upvotes]: Why heat capacity of an ideal gas varies with temperature? Why can't it remain the constant?
Is there any molecular reason why the heat capacity of a gas cannot remain constant with temperature?
How do we understand heat capacity of a gas at the molecular level?

REPLY [1 votes]: In a purely classic world, heat capacity of an ideal gas would be independent on temperature. The reason can be traced back to the fact that the Hamiltonian of a classical perfect gas is purely kinetic (depends only on the momenta, not on positions) and quadratic. The  well known classical statistical mechanics equipartition theorem guarantees that the average energy associated with each quadratic term in the Hamiltonian is $\frac12 k_BT$. So, for instance a monoatomic gas  of $N$ atoms would have an internal energy $U=\frac32 N k_B T$. A gas of diatomic molecules modeled by $N$ rigid rods  would be $U=\frac52 k_B T$ (three quadratic terms corresponding to the momentum of the center of mass of each "molecule", two quadratic terms for rotations around the center of mass). The resulting constant volume heat capacities would be $\frac32 N k_B$ in the first case and $\frac52 N k_B$ in the second, at any temperature. I.e. hat capacity is directly proportional to the number of quadratic terms in the Hamiltonian
However, classical mechanics and classical statistical mechanics are just approximations of a more fundamental quantum theory. It turns out that the classical equipartition theorem breaks down when temperature decreases at the point  typical thermal energy $k_BT$ becomes comparable with the spacing between quantum energy levels. Blow that threshold, internal energy decreases with temperature  faster than linearly. The resulting effect on heat capacity is sometimes referred to as freezing of degrees of freedom because in a classical descritpion it would be equivalent to an effective  reduction of the number of quadratic terms in the Hamiltonian.
Rotational energy level separations of typical molecules are in the range of energis of microwaves, i.e. below $1$ meV. This ensures that at room temperature rotational degrees of freedom behave classically and heat capacity of a diatomic molecule is quite close to the classical value $\frac52 N k_B$. However, by decreasing temperature on can observe  a transition to the value $\frac32 N k_B$.


TITLE: Calculating the integral closure (normalizatrion) of a quotient over a polynomial ring in three variables.
QUESTION [1 upvotes]: Let $A=\mathbb{Q}[x,y,z]/((x+a)^2-z(y+z^b)^2)$ for some $a,b\in\mathbb{N}$. Assume that $A$ is an integral domain (this is easy to show, you can just use Generalised Eisenstein over $\mathbb{Q}[y,z][x+a]$ with prime ideal $(z)$ I think). The question is to compute the integral closure of $A$.
My initial guess was to find an integral element say $t$ not in $A$ and adjoin to find $A[t]$ and hopefully this would be my integral closure. Letting $t=\frac{x+a}{y+z^b}$ seems to be the obvious choice, and belongs to $Frac(A)$.
My attempt: Shift the variables of $A$ to be over $u,v,z$ where $u=x+a$, $v=y+z^b$ (First question, why can I so this? I haven't found a reason I trust enough, though I'm sure its just isomorphisms that allow me to do this? Is there a way to do this without shifting?).
Then I have the string of isomorphisms as below:
\begin{align*}
    A[t] \cong A\left[\frac{u}{v}\right] &\cong \left(\mathbb{Q}[u, v, z] /\left(u^{2}-z v^{2}\right)\right)\left[\frac{u}{v}\right]\\
    &\cong \mathbb{Q}\left[u, v, \frac{u^2}{v^2}\right]\left[\frac{u}{v}\right]\\
    &\cong \mathbb{Q}\left[v,\frac{u}{v} \right]
\end{align*}
Now clearly $A[t]$ is integrally closed if you note that $v, u/v$ is independent over $\mathbb{Q}$ and thus we have it is isomorphic to the UFD $\mathbb{Q}\left[v,\frac{u}{v} \right]$. My second question, is why exactly is this $A[t]$ the integral closure? Is it becasue $A[t]$ is contained in the integral closure of $A$ or is it other way around, i.e. it contains it. And why? It doesn't seem obvious to me.
Lastly, is there an easier/simpler way to tackle this? My course is working through Atiyah and MacDonald's Commutative Algebra.
(I haven't had the most solid basis for ring theory, so even simple explanations are helpful.)

REPLY [1 votes]: You can shift the variables in such a way by what is called the universal property of polynomial rings: a ring homomorphism from a polynomial ring $R[X_i:i\in I]$ is completely determined on $R$ and $X_i$.
We have two morphisms $\Psi, \Phi$ between polynomial rings $\mathbb{Q}[x,y,z] \rightleftarrows \mathbb{Q}[u,v,w]$ such that
$$\Psi(x)=u-a, \Psi(y)=v-w^b, \Psi(z)=w,$$
$$\Phi(u)=x-a, \Phi(v)=y+z^b, \Phi(w)=z.$$
Since $\Psi\circ\Phi$ and $\Phi\circ\Psi$ map indeterminates to themselves, they must be identity, i.e. $\Psi$ is an isomorphism. It follows that $\Psi$ maps the ideal $((x+a)^2-z(y+z^b)^2)$ to $(u^2-v^2w)$ and by quotienting we get $A\simeq \mathbb{Q}[u,v,w]/(u^2-v^2w)=\mathbb{Q}[u,v,u^2/v^2]$. We can see that $B=\mathbb{Q}[u,u/v]$ is integral over $A$ since it is obtained by adjoining an integral element $\in \operatorname{Frac}A$ so that $B
$ is contained in the integral closure, and it is integrally closed since it is UFD. As in the case of fields, if we have a tower of ring extensions $A\subset B\subset \operatorname{Frac}A=\operatorname{Frac} B$, where $A\subset B$ is integral extension, and an element $a\in \operatorname{Frac}A$ integral over $A$, then $a$ is also integral over $B$ so that $a\in B$. This shows $B$ contains eny element $\in \operatorname{Frac}A$ integral over $A$, i.e. contains the integral closure.
Lastly, this is the most usual and common procedure to find the integral clousure of an integral domain as far as I know.


TITLE: When do elements in the braid group $B_n$ commute?
QUESTION [8 upvotes]: I have been looking around for an answer to this question, but I have not been able to find anything. My question is:
Is it known how to tell whether two elements $b_1, b_2 \in B_n$ commute?
EDIT: Is there perhaps a nicer/cleaner rule for $B_3$ and $B_4$?

REPLY [4 votes]: You are asking: "Suppose that a and b are braids, with inverses A and B.  Is the element abAB trivial?"
This is a special case of the word problem for the braid group, which has many solutions.  Perhaps Artin's 1947 paper (Theorems 17 and 18) is the first.  Many of the papers mentioned by other answers in the thread also "quickly" give solutions to the word problem (eg for linear groups over $\mathbb{Z}$, for groups acting on free groups, etc).


TITLE: How to solve a convoluted absolute value inequality?
QUESTION [4 upvotes]: $$ \lvert \lvert x-2\rvert -3\rvert \lt 5 $$
How can I attack this the best way? I see that both sides are positive.
Squaring yields:
$$ \lvert x-2\rvert ^2 -6 \lvert x-2\rvert +9\lt 25 $$
$$ x^2-4x -6\lvert x-2\rvert \lt 12 $$
This is where I'm stuck. How to proceed?

REPLY [0 votes]: A systematic approach is to use the rule
$$x\le0\implies |x|=-x,\\
x\ge0\implies |x|=x$$ to get rid of the absolute value.
So we have
$$x-2\le0\to|2-x-3|=|-x-1|<5,\\
x-2\ge0\to|x-2-3|=|x-5|<5$$
which we further subdivide
$$x-2\le0,-x-1\le0\to x+1<5,\\
x-2\le0,-x-1\ge0\to-x-1<5,\\
x-2\ge0,x\le5\to5-x<5\\
x-2\ge0,x\ge5\to x-5<5$$
Remains to cleanup a little.


text:
\section{Effects of Curvatures on One-dimensional Operators}

On curved spaces, the Hessian of the distance function depends on sectional curvatures. The well known Hessian and Laplacian comparison theorems (see for example \cite{Sakai96}) provide sharp comparison with distance functions on spaces of constant sectional curvature $\k \in \R$, which are spheres ($\k >0$), Enclidean spaces ($\k=0$), and hyperbolic spaces ($\k <0$). To control the full Hessian of the distance function, one needs bounds on the sectional curvatures. However, if we only need to estimate the  Laplacian of the distance function from above, then lower bounds on Ricci curvature suffice.

We extract an example from \cite{AC13} to show how the curvatures effect the one-dimensional operators.

\begin{prop}\label{prop identify 1D mfd}
Assume the Ricci curvature of $M$ is bounded from below by $(n-1)\k$ for some $\k\in \R$. Then the following pair of $f$ and $F$ satisfy the structure condition \eqref{eq 6.3 structure mfd}.

    \begin{equation*}\label{}
        F(p,X)=-\tr \left[ \left(\a(|p|)\frac{p \otimes p}{|p|^2} +\b(|p|) \left(I- \frac{p \otimes p}{|p|^2}\right) \right) X \right],
\end{equation*}
where $\a$ and $\b$ are nonnegative functions.
    \begin{equation*}
        f(\vp',\vp'')=-\a(\vp')\vp''+(n-1)\b(\vp')T_{\k},
    \end{equation*}
where the function $T_k$ is defined for $\k \in \R$ by
\begin{equation}\label{def T}
     T_\k(t)=\begin{cases}
   \sqrt{\k} \tan{(\sqrt{\k}t)}, & \k>0, \\
   0, & \k=0, \\
   -\sqrt{-\k}\tanh{(\sqrt{-\k}t)}, & \k<0.
    \end{cases}
\end{equation}

\end{prop}

\begin{proof}[Proof of Proposition \ref{prop identify 1D mfd}]
The proof is essentially the same as in \cite{AC13}.
Let $x,y\in M$ be such that $d(x,y)=2s>0$ and $\gamma:[0,1] \to M$ be a length-minimizing geodesic connecting $x$ and $y$ with $|\gamma'|=2s$. Choose an orthonormal frame $\{e_i(0)\}_{i=1}^n$ for $T_{x_0}M$ with $e_n(0)=\gamma'(0)$ and parallel translate it along $\gamma$ to produce orthonormal frame $\{e_i(s)\}_{i=1}^n$ for $T_{\gamma(s)}M$ with $e_n(s)=\gamma'(s)$ for all $s\in [0,1]$.

Let $\rho$ be a smooth function touching $d$ from above at $(x,y)$ and write $\psi(x,y,t)=2\vp\left(\frac{\rho(x,y)}{2},t \right)$. Direct calculation shows that
\begin{align*}
    D_x\psi(x,y,t) &=\vp'(s,t)\gamma'(0), \\
    D_y\psi(x,y,t) &=-\vp'(s,t)\gamma'(1).
\end{align*}
Therefore, we have
\begin{eqnarray*}
&& F(-D_y\psi, Y)-F(D_x \psi, X) \\
&=&  -\a(\vp') Y(e_n(1),e_n(1)) -\b(\vp')\sum_{i=1}^{n-1}Y(e_i(1),e_i(1)) \\
&& +\a(\vp') X(e_n(0),e_n(0)) +\b(\vp')\sum_{i=1}^{n-1}X(e_i(0),e_i(0)) \\
&=& \a(\vp') \begin{pmatrix}
   X & 0 \\
   0 & -Y
   \end{pmatrix} \left((e_n(0),-e_n(1)), (e_n(0),-e_n(1)) \right) \\
  && +\b(\vp') \sum_{i=1}^{n-1}\begin{pmatrix}
   X & 0 \\
   0 & -Y
   \end{pmatrix} \left((e_i(0),e_i(1)), (e_i(0),e_1(1)) \right)\\
   &\leq& \a(\vp') D^2 \psi \left((e_n(0),-e_n(1)), (e_n(0),-e_n(1)) \right) \\
  && +\b(\vp') \sum_{i=1}^{n-1} D^2 \psi \left((e_i(0),e_i(1)), (e_i(0),e_1(1)) \right).\\
\end{eqnarray*}
It is easy to calculate that
\begin{eqnarray*}
&& D^2 \psi \left((e_n(0),-e_n(1)), (e_n(0),-e_n(1)) \right) \\
&=& \vp'' D\rho \otimes D\rho \left((e_n(0),-e_n(1)), (e_n(0),-e_n(1)) \right) \\
&& +\vp' D^2 \rho \left((e_n(0),-e_n(1)), (e_n(0),-e_n(1)) \right) \\
&=& 2\vp''
\end{eqnarray*}
and
\begin{eqnarray*}
&& \sum_{i=1}^{n-1} D^2 \psi \left((e_i(0),e_i(1)), (e_i(0),e_1(1)) \right) \\
&=&\vp'' \sum_{i=1}^{n-1} D \rho \otimes D \rho \left((e_i(0),-e_i(1)), (e_i(0),-e_i(1)) \right) \\
&& +\vp'\sum_{i=1}^{n-1}  D^2 \rho \left((e_i(0),-e_i(1)), (e_i(0),-e_i(1)) \right) \\
&\leq& -2 (n-1)T_\k \vp',
\end{eqnarray*}
where we have used the Laplacian comparison theorem in the last inequality (see for example \cite[page 1018]{AC13} or \cite[page 564]{LW17}).
Combining the above estimates, we obtain
\begin{equation*}
    F(-D_y\psi, Y)-F(D_x \psi, X) \leq -2\left(\a(\vp')\vp'' -(n-1)T_\k \vp' \b(\vp') \right).
\end{equation*}
Thus $f= -\a(\vp')\vp'' +(n-1)T_\k \vp' \b(\vp')$ satisfies \eqref{eq 6.3 structure mfd}.

\end{proof}

Analogous results hold for other examples given in Proposition \ref{prop identify 1D} and \ref{prop 1D identify} (Part (4) of Proposition \ref{prop identify 1D} requires nonnegative sectional curvature). As applications, we obtain the same Lipschitz bounds and gradient estimates as in Section \ref{Sec Lip} provided that $M$ has nonnegative Ricci curvature.

config: arxiv


TITLE: Weakest theory equi-consistent to ZFC
QUESTION [0 upvotes]: I've recently read that ZF is equi-consistent to ZFC. From what I understand, to establish this we transform a formal proof of a contradiction in ZFC into a formal proof of a contradiction in ZF.
We do this by constraining all quantifications to the constructible universe : $\forall x,\phi$ is replaced by $\forall x, \text{Constructible}(x)\Rightarrow \phi$ and $\exists x,\phi$ is replaced by $\exists x, \text{Constructible}(x)\land\phi$. Because the constructible universe is an inner model, this syntactic transformation produces a valid formal proof, that respects inference rules.
And the invocations of the axiom of choice become
$$\forall x, \text{Constructible}(x)\Rightarrow \exists f, \text{Constructible}(f)\land \text{ChoiceFunction}(f,x)$$
But that is no longer an axiom, it is a theorem of ZF. Therefore we have a formal proof of a contradiction in ZF. Is this correct?
I am wondering how far we can continue these equi-consistency proofs in ZF. Can we remove the axiom of foundation ? Even the axiom of infinity ? I doubt it because, if we could remove all axioms, ZFC would be equi-consistent with the empty theory, which is consistent.

REPLY [3 votes]: Yes, you can remove the axiom of foundations by the same argument: move from a universe of $\sf ZF-Fnd$ to the von Neumann universe, which is the largest well-founded and transitive class.
When you remove Infinity, Power set, or Replacement, you get strictly weaker theories. To see why, note that $V_\omega,\mathrm{HC}$ and $V_{\omega+\omega}$ are models of $\sf ZF-Infinity, ZF-Power, ZF-Replacement$ respectively. Therefore $\sf ZF$ proves these theories have a model, so they are consistent. In particular by Gödel's theorem none of them can prove that $\sf ZF$ itself is consistent.


text:
\section{Shimura varieties, local systems and motives}

\subsection{Realizations of motives}
\label{sec:realizations}

Some of our definitions below may be somewhat nonstandard.

\subsubsection{Hodge structures}

Let $L$ be a number field given with a fixed embedding in $\C$. An $L$-Hodge structure pure of weight $n$ will be an $L$-vector space $V$ equipped with a descending filtration $F^\cdot V_\C$ on $V_\C = V\otimes_L  \C$ such that for $p+q=n+1$, we have
\[
V_\C = F^p V_\C \oplus \overline{F^q V_\C}.
\]
For any pair $(p,q)$ with $p+q=n$, we set $V^{p,q} = F^p V_\C \cap \overline{F^q V_\C}$.

\subsubsection{Realizations of motives with coefficients}

Let $k$ and $L$ be number fields. Let $\Mot_k^L$ denote the category of motives over $k$ with coefficients in $L$. (For the moment it is
not very important what equivalence relation we use on algebraic cycles.) We are particularly interested in certain
realization functors on $\Mot_k^L$, assuming we are given embeddings $k\hookrightarrow \Qbar \subset \C$ and $L\subset \C$.

\begin{itemize}
\item {\it The Betti realization.} The Betti realization $H_B (M)$ which is
 an $L$-Hodge structure.

\item {\it The $\ell$-adic realizations.} For each rational prime $\ell$, $H_\ell (M)$ is a free $L\otimes \Q_\ell$-module, equipped with a continuous ($L\otimes \Q_\ell$-linear) action of
$G_k:=\Gal (\Qbar/k)$.

\end{itemize}

We also have a natural comparison isomorphism\[
H_B (M) \otimes_{\Q} \Q_\ell \simeq H_\ell (M)
\]
of (free) $L\otimes \Q_\ell$-modules.

There are other realizations which will not concern us in this paper.
Thus we define a category $\cM_{k}^L$ as follows. The objects in this category are collections
\[
(V, V_\ell)
\]
as $\ell$ varies over the primes, where $V$ is an $L$-vector space equipped with an $L$-Hodge structure and $V_\ell$ is a free $L \otimes \Q_\ell$-module with a
continuous ($L\otimes \Q_\ell$-linear) action of $G_k$, along with isomorphisms
\[
i_\ell: V \otimes \Q_\ell \simeq V_\ell.
\]
A morphism between two such objects $(V,V_\ell, i_\ell)$ and $(V',V'_\ell, i'_\ell)$ is an $L$-linear map $j: V \rightarrow V'$ that is a morphism of $L$-Hodge structures such that  the $L \otimes \Q_\ell$-linear maps $j_\ell: V_\ell \rightarrow V'_\ell$ defined
by the commutative diagram below
\[
\xymatrix{
V \otimes \Q_\ell  \ar[r]^{j\otimes 1} \ar[d]^{i_\ell} & V' \otimes \Q_\ell \ar[d]^{i'_\ell} \\
V_\ell \ar[r]^{j_\ell} & V'_\ell
}
\]
are $G_k$-equivariant.

If $L=\Q$, we omit the superscript and simply write $\cM_k$. Note that to any proper smooth variety $X$ over $k$, we can attach
objects
\[
\cH^n (X) = \left(H^n (X (\C), \Q), H^n_{\et} (X_{\Qbar}, \Q_\ell), i_\ell \right)
\]
in the category $\cM_k$.

If $L\subseteq L'\subset \C$, there is a natural functor $\cM_k^L \rightarrow \cM_k^{L'}$, sending $(V,V_\ell)$ to $(V\otimes_L L', V_\ell \otimes_L L' = V_\ell \otimes_{L\otimes \Q_\ell} (L'\otimes \Q_\ell)).$

\subsection{Shimura varieties and local systems}

\subsubsection{Shimura varieties}
Let $\Ss = \Res_{\C/\R} \GG_m$ denote the Deligne torus.
As usual, a Shimura datum is a pair $(G,X)$ consisting of a reductive algebraic group $G$ over $\Q$
and a $G(\R)$-conjugacy class $X$ of homomorphisms $h:\Ss \rar G_\R$ satisfying the following conditions:
\begin{enumerate}
\item For $h$ in $X$, the Hodge structure on the Lie algebra $\fg$ of $G_\R$ given by $\Ad \circ h$ is of type $(0,0) + (-1,1) + (1,-1)$. (In particular, the restriction of such an $h$ to $\GG_{m,\R} \subset \Ss$ is trivial.)
\item For $h$ in $X$, $\Ad \ h(i)$ is a Cartan involution on $G_{\R}^{\ad}$, where $G^{\ad}$ is the adjoint group of $G$.
\item $G^\ad$ has no factor defined over $\Q$ whose real points form a compact group.
\end{enumerate}

These conditions imply that $X$ has the natural structure of a disjoint union of Hermitian
symmetric domains.
The group $G(\R)$ acts on $X$ on the left by
\[
(g \cdot h)(z) = g\cdot h(z) \cdot g^{-1}.
\]

Let $\A$ and $\A_f$ denote respectively the ring of ad\`{e}les and finite ad\`{e}les of $\Q$.
Let $\K$ be an open compact subgroup of $G(\A_f)$. The Shimura variety associated to $(G,X,\K)$ is the quotient
\[
\Sh_\K (G,X) =  G(\Q) \backslash X \times G(\A_f)/ \K.
\]
For $\cK$ small enough, this has the natural structure of a smooth variety over $\C$. The inverse limit
\[
\Sh (G,X) = \underleftarrow{\lim}_\K \ \Sh_\K (G,X)
\]
is a pro-algebraic variety that has a canonical model over a number field $E(G,X)$, the reflex field of the Shimura datum $(G,X)$. In particular, each $\Sh_\K (G,X)$ has a canonical model over $E(G,X)$. For brevity of notation, we will often write simply
$\Sh_G$ or $\Sh_{G,\cK}$ since $X$ will be understood from context.

We recall the definition of $E(G,X)$.
This field is defined to be the field of definition of the conjugacy class of co-characters
\[
\mu_h: \GG_{m,\C} \rightarrow \Ss_\C \rar G_\C,
\]
where the first map is $z\mapsto (z,1)$ and the second is the one induced by $h$.

\begin{rem}
The field $E(G,X)$ is given as a subfield of $\C$, and as such
has by definition a canonical embedding into $\C$. When not specified below,
any embedding of $E(G,X)$ in $\C$ will always be this canonical embedding. Indeed, we will not
have use for any other embedding.
\end{rem}

{\it  All Shimura varieties occurring in this paper will be compact, so we will assume this to be the case in the rest of this chapter.}

\subsubsection{Local systems and cohomology}

Let $(\rho,V)$ be a finite-dimensional representation of $G$ defined over a number field $L\subset \C$. To the data $(G,X,\rho)$, we can associate the following:
\begin{enumerate}
\item A local system $\V$ of $L$-vector spaces on $\Sh_G$.
\item For each prime $\ell$, an $\ell$-adic local system $\V_{\ell}$ (of $L\otimes \Q_\ell$-vector spaces) on $\Sh_G$.
\end{enumerate}
Then $H^i (\Sh_{G,\cK} (\C) , \V)$ is an $L$-vector space (in fact, an $L$-Hodge structure) and there are natural isomorphisms of free $L\otimes \Q_\ell$-modules:
\begin{equation}
\label{eqn:betti-etale-comparison-K}
H^i (\Sh_{G,\cK} ( \C), \V) \otimes_{\Q} \Q_\ell \simeq H^i_{\et}  (\Sh_{G,\cK} \otimes_{E(G,X)} \Qbar , \V_{\ell}).
\end{equation}
Note that we are using the given embedding $E(G,X) \hookrightarrow \Qbar \subset \C$ on both sides of the isomorphism above.
The Hecke algebra $\cH (G(\A_f), \cK)$ acts on both sides of \eqref{eqn:betti-etale-comparison-K} and the isomorphism is Hecke equivariant.
Taking the direct limit over $\cK$, we get
an isomorphism:
\[
H^i (\Sh_{G} ( \C), \V) \otimes_{\Q} \Q_\ell \simeq H^i_{\et}  (\Sh_{G} \otimes_{E(G,X)} \Qbar , \V_{\ell}).
\]

Let $\Pi$ be an irreducible cohomological automorphic representation of $G(\A)$. The $\Pi$-isotypic component of $H^i(\Sh_{G} ( \C), \V_\C)$ is defined to be
\begin{align*}
H^i (\Sh_{G} ( \C), \V_\C)_\Pi & := \Hom_{\cH(G(\A_f),\cK)} (\Pi_f^{\cK}, H^i (\Sh_{G} ( \C), \V_\C )^{\cK} ) \\ & = \Hom_{\cH(G(\A_f),\cK)} (\Pi_f^{\cK}, H^i (\Sh_{G,\cK} ( \C), \V_\C ) )
\end{align*}
for $\cK$ small enough, this being independent of the choice of $\cK$.
By Matsushima's formula \cite{borel-wallach},
\[
H^i (\Sh_{G,\cK} ( \C), \V_\C) \simeq \bigoplus_{\pi} m(\pi) H^i (\fg, K; \pi_\infty \otimes \V_\C) \otimes \pi_f^{\cK}
\]
where the sum is over automorphic representations $\pi= \pi_\infty \otimes \pi_f$ of $G(\A)$ and $m(\pi)$ is the multiplicity of $\pi$ in the discrete spectrum of $G$. It follows that
\[
H^i (\Sh_{G} ( \C), \V_\C)_\Pi \simeq \bigoplus_{\pi, \pi_f^{\cK} \simeq \Pi_f^{\cK}}  m(\pi) H^i (\fg, K; \pi_\infty \otimes \V_\C),
\]
where the sum is over those $\pi$ such that $\pi_f^{\cK} \simeq \Pi_f^{\cK}$ as $\cH(G(\A_f),\cK)$-modules.

\subsubsection{Pullback and pushforward}

Let $f:(h_1,G) \rightarrow (h_2,H)$ be a morphism of Shimura data. We
assume that the reflex fields of $(h_1,G)$ and $(h_2,H)$ are the same
subfield $E$ of $\C$. Let $\rho$ be a finite-dimensional representation of $H$ defined over a number field $L\subset \C$ (which we can also view as a representation of $G$ via the map $G\rightarrow H$) and denote by $\V$ the associated local systems on $\Sh_H$, $\Sh_G$. (Thus the local system on $\Sh_G$ is just obtained by pull back from $\Sh_H$.) Then there are functorial maps
\[
f^*: H^i(\Sh_H, \V) \rightarrow H^i(\Sh_G, \V),
\]
defined both in Betti and $\ell$-adic cohomology, which may be viewed as giving a morphism in the category $\cM_E^L$. Suppose in addition that:
\begin{enumerate}
\item \label{GHsurj} $G\rightarrow H$ is surjective with kernel $Z$ contained in the center of $G$.
\item \label{Zcohtriv} $Z$ is cohomologically trivial so that $G(\A) \rightarrow H(\A)$ is surjective as well.
\end{enumerate}
Then there is a bijection between automorphic representations $\Pi_H$ of $H(\A)$ and $\Pi_G$ of $G(\A)$ on which $Z$ acts trivially.
Assuming that $L$ contains the (common) field of definition of $\Pi_H$ and $\Pi_G$,  the map
$f^*$ induces an isomorphism
\[
H^i(\Sh_H, \V)_{\Pi_H}  \simeq H^i(\Sh_G, \V)_{\Pi_G}
\]
in $\cM^L_E$.

We will also need to consider the case when the map $G\rightarrow H$ satisfies \eqref{GHsurj} but not \eqref{Zcohtriv}. Typically, in such cases, we will be interested in maps in the opposite direction. Indeed, there is a natural pushforward map
\[
f_*: H^i(\Sh_{G}, \V) \rightarrow H^i(\Sh_{H}, \V).
\]
If $\cK_1$ and $\cK_2$ are (small enough) open compact subgroups of $G(\A_f)$ and $H(\A_f)$ respectively, the induced map
\[
\Sh_{G,\cK_1} \rightarrow \Sh_{H,\cK_2}
\]
is finite etale onto its image which is a union of components of
$\Sh_{H,\cK_2}$. Thus $f_*$ can be defined by taking the trace to the
image, and then extending by zero outside the image.

\begin{rem}
To make the definition of $f_*$ independent of the choice of $\cK_1$ and
$\cK_2$, we need to normalize it by multiplying by the factor $\vol
(\cK_1)/\vol(\cK_2)$
for some choice of Haar measure on $G(\A_f)$ and $H(\A_f)$. In our application, we will
impicitly make such a choice in \S \ref{ss:form-construction} and \S
\ref{sec:proof-of-main-thm}, but the
exact choice is unimportant.
\end{rem}


TITLE: Find all postive $m$ such $\gcd(m,2n!+1)=1,\forall n\in N^{+}$
QUESTION [3 upvotes]: Find all postive integer $m$,such
  $$\gcd(m,2n!+1)=1,\forall n\in N^{+}$$

when $n=1$  then we have
$$\gcd(m,3)=1$$
when $n=2$,then we have
$$\gcd(m,5)=1$$
when
$n=3$,then we have
$$\gcd(m,13)=1$$
when $n=4$, then we have
$$\gcd(m,49)=1$$
$$\gcd(m,241)=1$$
$$\gcd(m,1441)=1$$
I guess the $m=2^a?$ is such it,and other form? in other words,if this answer $2^a$ is only form,How to prove?

REPLY [3 votes]: As you note in your question, we must have $\gcd(m,3) = 1$.
If $p > 3$ is prime, let $n = p - 3$. We have
$$2n! + 1 \equiv -(p-2)! + 1 \equiv (p-1)! + 1 \equiv 0 \pmod{p},$$
where the last step uses Wilson's theorem.
Thus $\gcd(m,p) = 1$ for all odd primes $p$. Hence $m$ must be a power of $2$.
Conversely, since $2n! + 1$ is odd for all $n$, any power of $2$ answers the question.


TITLE: Dimension of the set of singular hypersurfaces
QUESTION [0 upvotes]: Let $N$ be the number of degree $d$ monomials in $n$ variables. We can then view each non-zero point in $\mathbb{A}^N_k$ as a degree $d$ homogeneous form, $k$ an algebraically closed field. Let $X$ be the union of $\mathbf{0}$ and the set of points where the corresponding form is not smooth. I have heard that smoothness is a generic condition. This implies that $X$ is a Zariski closed set, I have not been able to find a proof of this fact. Any reference or explanation how to prove this along with what $\dim X$ is would be highly appreciated.

REPLY [2 votes]: Given a smooth projective variety $X\subset\mathbb{P}^{N-1}$, let $I(X)\subset X\times(\mathbb{P}^{N-1})^*$ denote the locus of pairs $(x,H)$ where $x\in H$ and $T_x(X)\subset T_x(H)$; here we are thinking of $(\mathbb{P}^{N-1})^{*}$ as the collection of hyperplanes in $\mathbb{P}^{N-1}$.
One can show (using the Euler sequence which is the "global" version of the Jacobian criterion mentioned by Mohan) that $I(X)\to X$ is the projective bundle $\mathbb{P}_X(K_{X/\mathbb{P}^{N-1}})$, where $K_{X/\mathbb{P}^{N-1}}$ is the co-normal bundle of $X$ in $\mathbb{P}^{N-1}$. This can be used to show that $I(X)$ is smooth of dimension $N-2$ in $(\mathbb{P}^{n})^{*}$.
Since $X\times(\mathbb{P}^{N-1})^{*}\to(\mathbb{P}^{N-1})^{*}$ is proper, the image $D(X)$ of $I(X)$ is a closed subvariety of dimension $\leq (N-2)$.
This is the closed subvariety the question is looking for when $X\subset\mathbb{P}^{N-1}$ is the embedding of $X=\mathbb{P}^{n-1}$ given by the monomials mentioned in the question. (In other words, $X$ is the Veronese embedding of $\mathbb{P}^{n-1}$ in $\mathbb{P}^{N-1}$.)
There are examples where $D(X)$ is not a of dimension $N-2$. However, "in general" one finds that $D(X)$ is a hypersurface called the dual hypersurface of $X$. It appears to be rather difficult to write the equation of $D(X)$ in general.
(Ref: "The topology of complex projective varieties after S. Lefschetz" by Klaus Lamotke.)