It is a consequence of Langlands functoriality conjecture that the Rankin-Selberg convolution of two automorphic representations gives rise to a third one, which should be seen through their L-functions. Defining the "Kernel affine space" of such an automorphic L-function as the affine space of minimal dimension containing all of its non trivial zeros, one can associate to this space a linear space of same dimension over the field of real numbers. Requiring that the so-obtained kernel linear space of the Rankin-Selberg convolution of $\pi $ and $ \pi' $ is the tensor product of the kernel linear spaces associated respectively to $ \pi$ and $ \pi' $ , one should get that the dimension of all kernel linear spaces are of dimension 1 over the field of real numbers, which is a reformulation of GRH.

So is GRH actually equivalent to the closure of the set of automorphic representations under Rankin-Selberg convolution ?

Suppose $G$ is a **Topological group** then classification theorem of Principal $G$ bundles says that

there is a Principal $G$ bundle $EG\rightarrow BG$ such that any principal $G$ bundle over a decent **topological space** $X$ has to be pullback of a continuous map $f:X\rightarrow BG$.

Can we replace Topological group by Lie group and Topolgical space by Smooth manifold. Do we get all Principal $G$ bundles over smooth manifold in this case? Is $BG$ a smooth manifold??

In his book Fiber bundles, Dale Husemoller does not say anything (I could not see anything) about smooth version of that classification result. Now I have a doubt if that Milnor constriction $BG$ for a Lie group $G$ gives a smooth manifold or is this classification only for **topological Principal $G$ bundles**.

In similar way, when doing classification of vector bundles we construct what is called Grassmannian $G_n$ for each $n$ and a **topological vector bundle** $E_n\rightarrow G_n$ and say that for a decent **topological space** $X$, any rank $n$ vector bundle (in Topological sense, not smooth sense) over $X$ should be pullback of a continuous map $X\rightarrow G_n$. Here also we are classifying only topological vector bundles, not smooth vector bundles, right? I was thinking $G_n$ is a manifold and it classifies all smooth vector bundles but then realise I am thinking wrong.

Is the classification only restricted to Topological set up?

Is there similar classification in smooth set up? Like classifying smooth Principal $G$ bundles and classifying smooth vector bundles?

I am going through a book (Roberts and Schmidt, *Local Newforms for GSp(4)*), which states the following group decomposition. Let $F$ be a non-archimedean field, $\mathfrak{o}$ its ring of integers and $\mathfrak{p}$ its maximal ideal. Then, in terms of subgroups of $\mathrm{GSp}(4)$,
$$\left(
\begin{array}{cccc}
\mathfrak{o}&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{o}
\end{array}
\right)
=
\left(
\begin{array}{cccc}
1& & & \\
\mathfrak{p}^n&1& & \\
\mathfrak{p}^n& &1& \\
\mathfrak{p}^n&\mathfrak{p}^n&\mathfrak{p}^n&1
\end{array}
\right)
\left(
\begin{array}{cccc}
\mathfrak{o}^\times&&&\\
&\mathfrak{o}&\mathfrak{o}&\\
&\mathfrak{o}&\mathfrak{o}& \\
& & &\mathfrak{o}^\times
\end{array}
\right)
\left(
\begin{array}{cccc}
\mathfrak{1}&\mathfrak{o}&\mathfrak{o}&\mathfrak{o}\\
&\mathfrak{1}& &\mathfrak{o}\\
& &\mathfrak{1}&\mathfrak{o}\\
& & &\mathfrak{1}
\end{array}
\right)
$$

where missing entries are zeros. I would like to understand this decomposition more generally, for I would like to apply that for other groups. Is there any general setting for this Iwahori factorization? (it seems similar to an $LU$ factorization, however is it always true and what are exactly the three groups appearing on the right?)

I got only partial answers on MSE hence I post the question here.

I'm looking for functions $f\in L^{\frac{2n}{n+1}}$ such that $\hat{f}=\infty$ on $S^{n-1}$. Is there any explicit expression of such kind of examples?

This seems to be a well-known result, but I can not find it in standard references such as Stein's Harmonic Analysis and Grafakos's classical Fourier Analysis.

Thanks in advance.

Few days I asked this question (https://math.stackexchange.com/questions/2907733/simple-ordinal-question) on MSE. Summary of the question is that I defined a certain function over ordinals $x \mapsto \beta_x$ (where $x<\omega_1$) and asked about its relation to the function $x \mapsto \omega^{CK}_x$. And it seems they are "nearly" the same (I mean the values, not the definitions) with the former function being continuous while the latter function is discontinuous (it does obey continuity condition on some limit values). In particular, $\omega^{CK}_1=\beta_0$ and $\omega^{CK}_2=\beta_1$.

Since asking that question, another question has come to my mind. Before getting into detail I will just briefly state the statement of question: **"Can we rigorously define an ordinal $\gamma$ which (intuitively) is to $\omega_1$ what $\omega^{CK}_2$ is to $\omega^{CK}_1$?"**

Here is the main motivation for the question. The (fictional) idea is that suppose completely "fictitiously" that we indeed have some well-order (of $\mathbb{N}$) with order-type $\omega_1$. Now suppose someone asked me whether this given $\gamma$ (which we are trying to describe rigorously) is greater than say $\omega{_1}^{\omega_1}$? My answer would be yes. And my "reasoning" would be that if somehow one just had access to the fictitious well-order just described, then using that hypothetical orcale function there also exists an ordinary program that describes the well-order (of $\mathbb{N}$) with order-type $\omega{_1}^{\omega_1}$. By the same reasoning $\gamma$ has to be greater than, say, the first fixed point of $x \mapsto (\omega_1)^x$. And so on...

I hope the drift of the question is clear at this point. The question is that what would be a reasonable rigorous definition of $\gamma$ that also corresponds well with this intuition.

I also have a related side question. If we consider an ORM then it seems it would easily go beyond points such as $\beta_0$,$\beta_1$ or $\beta_{\omega_{CK}}$ etc. and probably also points far beyond it (I don't know the exact limit). And I suppose one of reasons is that it could easily decide whether a given (ordinary) program (with some oracle possibly) codes a well-order or not. Because corresponding to every well-order (of $\mathbb{N}$) with a certain order-type one can describe a "tree of all possible descents". The most important property of such a tree would be that it wouldn't contain any infinite path/branch. It seems to me that if one just narrows a few more properties for this "descent tree" well-enough, one can make them correspond exactly with well-orders (of $\mathbb{N}$). [I haven't thought about this in a fully thorough way so I would be happy to be corrected if I have made some mistake in this paragraph.]

Now my side question is that if we add an extra instruction of the form to $u=u+\omega_1$ (where $u$ is a variable) to the definition of ORMs, then what is the smallest point $p$ that such programs can't reach? My main reason for asking this side question it seems to me that $p \ge \gamma$ ($\gamma$ being the answer to the first part of this question). It also seems that luxury of using well-orders of $\mathbb{N}$ (described in previous paragraph) is unavailable to these programs.

My side question is **"Is the inequality $p \ge \gamma$ strict or not?"**

The Weyl character formula and the denominator identity play important roles in the representation theory of classical simple Lie algebras and Kac-Moody Lie algebras over $\mathbb{C}$

Can you suggest any reference for similar formulas for Lie super-algebras? I suppose there are such formulas, at least for classes of (say classical simple) Lie super-algebras.

Let $f$ be a newform of weight $k \geq 2$ and level $N \geq 1$ without complex multiplication. A prime $p$ is said to be ordinary for $f$ if the $p$-th Fourier coefficient $a_p(f)$ is a $p$-adic unit (to make sense of this in general, one needs to choose a prime ideal above $p$ in the field $K_f$ of Fourier coefficients of $f$, equivalently an embedding of $K_f$ into $\overline{\mathbf{Q}_p}$).

Non-ordinary primes seem to be rare. For example, if $f$ has coefficients in $\mathbf{Z}$, then a very rough guess is that $a_p(f)$ modulo $p$ is randomly distributed, so we may expect that

\begin{equation*} \# \{p \leq x : a_p \equiv 0 \textrm{ mod } p \} \stackrel{?}{\approx} \sum_{p \leq x} \frac{1}{p} \sim \log \log x. \end{equation*} On the other hand, for the modular form $\Delta = \sum_{n \geq 1} \tau(n) q^n$, it seems not to be known that there are infinitely many primes $p$ such that $\tau(p) \not\equiv 0$ mod $p$, so estimating the number of ordinary or non-ordinary primes is difficult in general.

If we consider elliptic curves, we may ask whether every prime $p$ is non-ordinary for some elliptic curve $E$ over $ \mathbf{ Q } $, which means that the reduction of $E$ mod $p$ is supersingular. In fact, Deuring has shown that given any integer $a$ such that $|a|<2\sqrt{p}$, there exists an elliptic curve $E_p$ over the finite field $\mathbf{F}_p$ with exactly $p+1-a$ points over $\mathbf{F}_p$. Now take any elliptic curve $E$ over $\mathbf{Q}$ whose reduction mod $p$ is $E_p$ and use the modularity theorem. We get a modular form $f$ of weight $2$ with integral coefficients satisfying $a_p(f)=a$, and we may choose it to be non-CM (in fact, we get infinitely many such modular forms).

My question is whether this kind of result is known or even expected in higher weight. Certainly, if you have a finite set of newforms without CM, the above heuristics suggest that there exist infinitely many primes which are ordinary for all these newforms. But I don't know what happens for an infinite set of newforms, even if this set is "thin" in some sense.

Here is another example of question which arises.

Fix a weight $k \geq 3$. Is every prime $p$ non-ordinary for some non-CM newform of weight $k$ and level not dividing $p$? At least, are there infinitely many such primes?

Hida's theory implies that if $p \geq 5$ and $k$ is equal to $\{3,4,5,\ldots,10,14\}$ modulo $p-1$, then all newforms of weight $k$ and level $1$ are non-ordinary at $p$. This gives a positive answer to the question for some primes $p$ which are small with respect to $k$ (in particular, these primes satisfy $p \leq k-2$).

One may try to use the theory of congruences between modular forms, like the theory of Hida families, but usually one starts with a modular form which is ordinary at $p$, and I don't know to which extent the theory has been generalized to the non-ordinary case.

Let $p$ be an odd prime and let $S_p$ denote the determinant $$\det\left[\left(\frac{i^2+j^2}p\right)\right]_{1\le i,j\le (p-1)/2}$$ with $(\frac{\cdot}p)$ the Legendre symbol. By Theorem 1.2 of my paper arXiv:1308.2900 available from http://arxiv.org/abs/1308.2900, $-S_p$ is a quadratic residue modulo $p$. Here I ask a further question.

QUESTION. Is it true that for each prime $p\equiv3\pmod4$ the number $-S_p$ is always a positive square divisible by $2^{(p-3)/2}$?

Define $a_p=\sqrt{-S_p}/2^{(p-3)/4}$ for any prime $p\equiv3\pmod4$. Then \begin{gather*}a_3=a_7=a_{11}=1,\ a_{19}=2,\ a_{23}=1,\ a_{31}=29,\ a_{43}=254, \\a_{47}=367,\ a_{59}=9743,\ a_{67}=305092,\ a_{71}=29,\ a_{79}=1916927. \end{gather*} I have computed the values of $a_p$ for all primes $p\equiv3\pmod4$ with $p<2000$. Based on the numerical data, I conjecture that the above question has an affirmative answer but I'm unable to prove this.

Any ideas towards the solution? Your comments are welcome!

Given a complex simple Lie algebra $\mathfrak{g}$ of rank $n\in\mathbb{N}$ with $n$ sufficiently large (say $n\ge10$), is there a way to determine whether $\mathfrak{g}$ contains a simple subalgebra of a prescribed type with rank "close" to $n$? For example, if $\mathfrak{g}$ is of type $B_n$ with $n\ge10$, does $\mathfrak{g}$ contain a subalgebra of type $C_{n-2}$?

Can the theory of Galois categories (as developed in SGA1) be modified to produce the usual fundamental group of a topological space (maybe assumed to be path connected and locally path connected)?

Recall from SGA 1 that the theory of Galois categories is developed to construct the étale fundamental group of a connected locally noetherian scheme. Given a Galois category $\mathcal{C}$, the main result is that a choice of fiber functor $F$ determines an equivalence between $\mathcal{C}$ and the category of finite $\pi := \operatorname{Aut}(F)$-sets.

Can one prove an analogous result that applies to a path connected+locally path connected topological space? Except in special cases, one can't literally use Galois categories since a topological space can certainly admit connected covers of infinite degree. But maybe we can modify the construct by removing some finiteness conditions?

Ideally, if the answer is "yes", I'd be great to have a written reference developing this idea.

Given an $n$-dimensional vector $\mathbf{c}\in [0,1]^n$, let $\Delta_{\mathbf{c}}$ be the set of points $\{\mathbf{x}\in [0,1]^n: \langle \mathbf{c},\mathbf{x} \rangle \le 1\}$, where $\langle \mathbf{c},\mathbf{x} \rangle$ is the inner product between $\mathbf{c}$ and $\mathbf{x}$.

**Question**: Given $\mathbf{c}$ and a vector $\mathbf{z}\in [0,1]^n$, how can we *efficiently* compute the projection $P(\mathbf{z}, \Delta_{\mathbf{c}})$ of $\mathbf{z}$ onto $\Delta_{\mathbf{c}}$?

By writing $P(\mathbf{z}, \Delta_{\mathbf{c}})$, we mean $\arg\min_{\mathbf{z'}\in\Delta_{\mathbf{c}}} \Vert \mathbf{z'}-\mathbf{z} \Vert$, where $\Vert \cdot \Vert$ denotes the regular Euclidean norm.

For an odd prime $p$, a *Tarski monster group* is an infinite group $G$ such that every proper, non-trivial subgroup $H < G$ is a cyclic group of order $p$. It is known that for every prime $p > 10^{75}$ a Tarski monster exists. These groups are counter-examples to some famous problems in group theory, including von Neumann's conjecture.

However, that they can only be proven to exist with $p$ so large is very strange to me. What are some properties of small primes that prevent such groups from existing? That is, what is the exact 'law of small numbers' that force such groups to be finite?

I'm interested in the number of permutations for a specified number of fixed points and cycles.

Suppose we are in $S_n$. For any permutation in $S_n$, let $h$ be the number of changed points (the complement of fixed points) and $N$ be the number of cycles of the permutation with length no less than 2 (number of cycles for the changed points). Therefore, for a given $h$, the number of cycles $N$ can range from $1$ to $\lfloor h/2\rfloor$.

So my question is how many permutations are there in $S_n$ for a given $h$ and $N$? If the exact expression is complicated, can we derive a concise and sharp upper bound of the number in terms of $n,h,N$?

I’m coding a puzzle-solver, and I need to be able to generate each game state. To do so, I need an algorithm related in some way to combinations but I can’t figure it out. Here is a sample problem that the algorithm needs to be able to solve:

“There are 4 balls and 3 baskets. Each ball must go into a basket and each basket may hold a maximum of 2 balls. List all possible combinations of baskets holding balls. Order does not matter.”

Possibility 1: 2 baskets with 2 balls, 2 baskets with 0 balls or baskets = [2, 2, 0, 0] *(order does not matter)

Possibility 2: 1 basket with 2 balls, 2 baskets with 1 ball or baskets = [2, 1, 1, 0] *(order does not matter)

Possibility N-1: ...

Possibility N: ...

Note: [2, 2, 0, 0] and [0, 2, 0, 2] are counted as equal and only one should be produced as a possibility.

To recap, I need a way to find all possibilities for a situation like above; the number of balls and baskets may vary. Please let me know if anything needs to be clarified or if I left something out. Thanks!

I know from Hudson that if $X_1, \ldots, X_n$ are iid random vectors with operator-stable distributions, then $A_n\sum \limits_{i=1}^n X_i$'s are also operator-stable, where $A_n$'s are non-singular, real matrices. Can I derive that $\sum \limits_{i=1}^n A^i X_i$'s are operator-stable? Here $A$ is a non-singular real matrix and $A^n=\underbrace{A\cdot A\cdots A}_{n ~\text{times}}$.

Note: I am not from the math background, so I will really appreciate answers with not many technical terms.

I'm looking for the residues of the following function $$s \mapsto\sum^\infty_{m,n =1} (m+n) \left[ amn + (m-n)^2 \right]^{-s}$$ at $s=\frac{1}{2}$ and $s=\frac{3}{2}$, where $a$ is some real positive number.

Yet, I have literally no idea how to precedure here. I tried to rearrange the terms in order to get some well known zeta function but it failed. Is there any useful integral representation or any simple trick I'm not aware of?

Thanks in advance!

I am trying to understand the difference between Cohen Macaulay and Locally Cohen Macaulay curves.

The stacks project https://stacks.math.columbia.edu/tag/02IN says that a scheme (a curve in particular) $X$ is Cohen Macaulay if for every $x \in X$, there is an open subset $U$ of $X$ containing $x$ such that the module $\mathcal{O}_{X}(U)$ is Cohen Macaulay.

On the other hand, although there is a lot of papers on locally Cohen Macaulay curves, I was not able to find the precise definition for it. It seems that a curve $C$ is locally Cohen Macaulay if it has no embedded nor isolated points. (As Harthorne says in his paper "Stable reflexive sheaves" (https://link.springer.com/article/10.1007%2FBF01467074) for instance).

So my questions are: 1) Where can I find the precise definition of locally Cohen Macaulay curve? 2) If the definition of locally Cohen Macaulay curve that I stated is correct, how can I see that Cohen Macaulay implies locally Cohen Macaulay? (Which I believe that should be the case because of their names)

Thanks in advance.

need Help

Let x be a vector in a three-dimensional space R ^ 3 and c is constant vector and let A an operator acting on R ^ 3 with values in R ^ 3, then i m looking for the form of the operator A such that A (x + c) = c + A^2 (x) Thank you for your reply

I have the following basic question. Everything is over $\mathbb{C}$.

Let $X$ be a hyperkähler (irreducible holomorphic symplectic) variety and we consider a small contraction $f\colon X \rightarrow Y$. By this I mean that f is birational and surjective (e.g. induced by a big and semiample line bundle) and the exceptional locus $B$, i.e. the locus where $f$ is not an isomorphism, has codimension $\ge$ 2. Is $B$ uniruled?

If B is an irreducible divisor, this is true and follows from Section 1 of Huybrechts - Compact hyper-Kähler manifolds.

However, I could not find a statement in the case that $B$ is not a divisor. The only statement I was able to find is that each irreducible component of $B$ is algebraically coisotropic (Theorem A in https://arxiv.org/abs/math/0111089).

Is more known about the structure of $B$?

If $f:X\rightarrow Y$, $g:Y\rightarrow Y$ are functions and $g$ is monotone increasing function then $$ \operatorname{argmin}_{x \in X} f(x) = \operatorname{argmin}_{x \in X} g\circ f(x) . $$ X and Y are partially ordered sets.

So monotonicity is sufficient for preserving extrema. Are necessary and sufficient conditions known for $g$ to preserve minima? If so does anyone know a good reference?