I could not find a proof or a counter example for the following claim:

Let $A,B$ be sub-homogeneous $C^*$-algebras and let $C$ be a $C^*$-algebra s.t. there is an exact sequence: $0\to A\to C\to B\to 0$. Does it follow that $C$ is sub-homogeneous?

Thanks

If $U\subseteq\mathbf{R}^n$ is an additive subgroup, discrete with respect to the induced topology, then $U$ is a finitely generated abelian group.

Is it true that if $U\subseteq\mathbf{A}_K^n$ is an additive subgroup, with $\mathbf{A}_K$ the adèle ring of a number field, $U$ discrete with respect to the induced topology, then $U$ is a finitely generated abelian group?

For example, if $U = \mathcal{O}_K$, the ring of integers of a number field $K$, then $U$ is a finitely generated abelian group, and it is discrete both in $\mathbf{A}_K$ and in $\mathbf{R}\otimes_{\mathbf{Z}}\mathcal{O}_K\simeq\mathbf{R}^{[K:\mathbf{Q}]}$. If $\mathcal{O}_K^{\times}$ is the unit group of $\mathcal{O}_K$, then the image $U$ of $\mathcal{O}_K^{\times}$ in the trace-zero hyperplane in $\mathbf{R}^{r_1 + r_2}$, an $r_1+r_2-1$ dimensional $\mathbf{R}$-vector space, is discrete.

Let $(A,<_A,0_A)$ and $(B,<_B,0_B)$ be ordered sets and $\psi$ an increasing surjection from $A$ to $B$ such that $\psi^{-1}(0_B)=\left\{0_A\right\}$. If $P\subset A$ has a lower bound for $<_A$, we denote it by $\inf_A P$

Let $\alpha$ be an ordinal and

$F: \alpha\times \alpha \to A\,;\, (x,y)\mapsto F(.,y)(x)=F(x,.)(y)=F(x,y)$ an application with the following properties :

1)For any $y,z,t\in \alpha$ we have $t\leq z\Rightarrow\psi(F(z,y))<\psi (F(t,y))$

2) $\forall (x,y)\in \alpha^2$, $y\leq x\Rightarrow F(x,y)=F(y,y)$

3)$\forall x\in \alpha,\,\,\inf_A F(x,A)=0_A$

"Questions" :

a) Do we have $\inf_A(\left\{F(y,y),y\in \alpha\right\})=0_A$ ?

b) Same question if $(A,<_A)$ and $(B,<_B)$ are lattices

c) Same question if $A$ is a lattice

"Motivation"

If $u,v\subset \mathbb N$, we define $<_b$ as follow :

$u<_b v$ if and only if there exists $a,b,c,d\in \mathbb{N}$ such that for any $n\in \mathbb N$, $u^*(n+a)+b>v^*(n+c)+d$ where $u^*$ (resp. $v^*$) is the only increasing injection from $u$ to $\mathbb N$ (resp. $v$ to $\mathbb N$)

$<_b$ is a preorder on $\mathcal P(\mathbb N)$ and if we call $b$ the associated equivalent relation, we say that $B=:\mathcal P(\mathbb N)/b$ is canonically ordered by $<_B$ (such that $u<_b v$ if and only if $[u]_b<_B[v]_b$, where $[u]_b$ denote the class of $u$ under $b$, meaning all subsets $u'$ of $\mathbb N$ such that $u<_b u'$ and $u'<_b u$ both hold).

If $A$ is $\mathcal P(\mathbb N)/a$ where $a$ is the equality up to finite subset, and $<_a$ the inclusion "up to finite subset" (meaning for any $u,v\subset \mathbb N$, $u\subset v \cup f$ for some finite $f\subset \mathbb N$) we define $<_A$ to be such that $u<_a v\Leftrightarrow [u]_a<_A [v]_a$

Then if question a) is true at least in this very case, then it should not be difficult* to build from any $F\subset A$ with lower bound $0_A=0_B=0$, $T\subset A$ with lower bound $0$, such that the restriction of $<_A$ to $T$ is a total order. It is a result proved (I think in 2015) by Malliaris ans Shelah, but you need to be aware of set theory to understand the proof, and I'm far to be able to! I'm looking for a simple argument, after I read the Tim Gowers web blog, indeed the great mathematician hoped for a simple argument, but It seams that the problem is so difficult that he left away this hope, so it would really be astonishing if question a) was true with a simple proof. Is it even true in the particular case of the "motivation"?

I think the question(s) is (are) interesting independently from the "motivation" - witch is implied by questions a) and c) but not b).

*I will give more details about the link between "questions" and "motivation" in the comments if anyone needs it, but the post is quite long already...

If we consider the AdS-Schwarzschild manifold, defined by $M^n=[s_0,\infty)\times\mathbb{S}^{n-1}$ equipped with the Riemannian metric $$\overline{g}=\frac{1}{1-ms^{2-n}+s^2}ds\otimes ds+s^2g_{\mathbb{S}^{n-1}},$$ where $m>0$ is a fixed positive number, $s_0$ is the unique positive solution of the equation $1+s_0^2-ms_0^{2-n}=0$ and $g_{\mathbb{S}^{n-1}}$ is the standard round metric on the unit sphere $\mathbb{S}^{n-1}$. The scalar curvature of $M$ is equals $-n(n-1)$.

My question is:

There exists some spin structure on $M$ ?

Since the Hyperbolic space is a particular case when $m\to0$ and the hyperbolic space admits a spin structure, is natural asking for a spin structure on the AdS- Schwarschild space?

Let $\mathfrak{g}$ be a complex semisimple Lie algebra. In his paper *Semisimple Subalgebras of Semisimple Lie Algebras*, Dynkin states that by a result of Weyl,

1) Two regular semisimple subalgebras are conjugate if and only if their simple roots are conjugate under the Weyl group.

From this Dynkin concludes that for the classical Lie algebras,

2) Any regular semisimple subalgebra of a given type (e.g. $A_2\oplus B_2$ in $B_5$) is unique up to conjugacy, with some exceptions if $\mathfrak{g}$ is of type $D_n$. The same is true of the exceptional Lie algebras, except $E_7$ and $E_8$ have several exceptions.

Does anyone know where I can find a proof of Statements 1 and 2? The first statement supposedly follows from a result in *Theorie der Darstellung kontinuierlicher halb-einfacher Gruppen durch lineare Transformationen* by Weyl, but I am unable to find an English translation. I am more interested in Statement 2, however, but Dynkin does not cite any source, nor does he hint at a proof.

In “A new infinite family in $_{2}\pi^S_*$" (1976), Mark Mahowald constructs elements $\eta_j \in \pi_{2^j}(S^0)$ for $j \neq 2$ which come from permanent cycles in the Adams Spectral Sequence that are generated by $h_1h_j \in Ext_A^{2, 2^j}(\mathbb{Z}_2, \mathbb{Z}_2)$. Let $H^*$ denote reduced mod-2 cohomology and for $Y$ a CW-complex let $Y_\ell$ denote the $\ell$-skeleton. Mahowald actually constructs a certain map from a stable complex $f_j: X_j \to S^0$ where $X_j$ has dimension $2^j-1$, as well as a map $g_j: S^{2^j} \to X_j$, so that $X_j/(X_j)_{2^j-2} \simeq S^{2^j-1}$, the composition of $g_j$ with the quotient $X_j \to X_j/(X_j)_{2^j-2}$ is the Hopf map, $H^{< 2^j - 2^{j-3}}(X_j) = 0$, and $Sq^{2^j}$ is nonzero in the mapping cone of $f_j$. Then he defines $\eta_j$ be the composition $f_j \circ g_j$ and concludes that by ``standard arguments'', $h_1h_j$ is a permanent cycle, etc.

What are these standard arguments? I know that a good way to show that maps are nonzero in the $\pi^S_*$ is to show that they are detected by a primary or secondary stable cohomology operation. I know that secondary cohomology operations come from relations in the Steenrod algebra; I know that relations in the Steenrod algebra give rise to the second column in the Adams Spectral Sequence. Unfortunately, I can't quite put things together to see why, for example $f_j \circ g_j$ ``represents $h_1h_j$'' (as Mahowald says)! In particular, I have no idea why the product on the ASS should be related to composition of maps of complexes.

(Why I am asking this question: I am not much of a homotopy theorist, but for some reason I had to read a later paper of Mahowald's that was based on observations of this one in which he shows that certain Eilenberg-Maclane spectra are Thom spectra. This paper seemed interesting.)

By a large power of R is meant a topological vector space which is the product of infinitely many copies of the real line. Is every closed subspace of such a TVS linearly homeomorphic to a power of R?

Let $s \in \Sigma^*$ be a string and suppose we have a single-string grammar $g = \{ S\to aA^2BbcA; \ \ A \to BB; \ \ B \to abc\}$

Then $P(AB | g) = 1/3$ or the probability of seeing the symbol $B$ right after $A$ is $1/3$ given the grammar.

Notice that if $g$ is minimized (we cannot compress it further, that is), then we don't have to consider the probability of say $AA$ preceeding $B$ since $AA$ doesn't repeat enough to have a significant probability.

Thus this model considers probabilities arising from many lengths of preceeding symbols in an efficient manner.

Now if $|s| \gg |\Sigma|$ then the grammar is highly compressed relative to $|s|$. Thus we have a highly compressed probablistic model for prediction of a stream of data in the alphabet $\Sigma$.

To add new incoming data to the "stream" $s$, you just append to the end of the start rule of $g$ and perform a minimization routine. Since you only receive a small string of symbols at a time, this should run in $|g|^2$ time.

**Have you seen this probablistic model before? Do you think it's a smart idea? :)**

Note, by minimized we don't mean "smallest grammar" for which there is no known efficient algorithm.

Also note that we that when computing probability, referring to the grammar given above, we don't count the occurences of $Aa$ that would arise by expanding $B$ (there is only 1 actually) since in general the goal with prediction theory is to predict the longest data sequence you can, and that is found via the nature of the grammar (ie. only count $Ax$ when you see it in the grammar itself, and not the expanded string).

Now as explained above, since we are compressing symbols, we only need to consider transitions between any two grammar symbols (and not say $P(ABC | g)$) and so there is a matrix representing the symbol transition probabilities.

I learned that for stochastic ODE dX = F(X)dt + dB where B is Brownian motion, if F is locally Lipschitz, then the solution exists and is unique over [0,T] where T is "almost surely positive".

My question is, can we find a measure 1 set and a deterministic constant C, such that on this measure 1 set T>C? This seems stronger than saying "T being almost surely positive".

Given an unweighted graph $G = (V,E)$. I am interested in enumerating all pairs of disconnected cliques, i.e. there should not exist any edge between two cliques. (For example, there would not exist any such pair in a complete graph). A clique could be of any size $\geq 1$.

Are there any standard approaches to solve this problem? If so, could someone please highlight the sources or suggest some approaches?

The collection of all self-equivalences of a category $C$ constitutes a $2$-group, which is a categorification of the notion of a group. My question is about what happens when one replaces equivalences by general adjoint functors.

The n-lab page on adjunctions says:

a morphism in an adjunction need not be invertible, but it has in some sense a left inverse from below and a right inverse from above.

I would like to know if this can be formalised in an interesting way.

In order to ask a specific question, I want to take a single adjoint pair of functors $(L,R):C\to C$, and let $M$ be the monoid of endofunctors of $C$ generated by $L$ and $R$ (together with some suitably chosen family of natural transformations).

**Q.** What kind of ($2$-)algebraic structure is $M$?

I apologise that this is a somewhat vague question. An ideal answer would have the form "$M$ is a (lax/colax/...) $2$-blah", where "blah" is an interesting algebraic structure that arises in other contexts not a priori having anything to do with adjoint functors. If it makes more sense to change the setting a little---e.g., to consider instead of $M$ the collection of all functors on $C$ admitting a two-sided adjoint---then please go ahead and do so.

Let $(X_n)_{n\geq1}$ be a Markov chain over a finite state space $\Omega$, that is not time-homogeneous.

Suppose there exists $\epsilon>0$ such that for all $n\geq1$ and all $x,x'\in\Omega$ either $P(X_{n+1}=x'|X_n=x)\geq \epsilon>0$ or $P(X_{n+1}=x'|X_n=x)=0$.

I would like to know how to define recurrent classes in this context.

It is known that for free $k$-step nilpotent group on $r$ generators $N(r, k)$ its integral homology is torsion-free in degrees $\leq 3$ (obvious for 1 and 2, Igusa&Orr computations for 3). However, there's a nontrivial 3-torsion in $H_4(N(\geq 4, 2), \Bbb Z)$ — which can be computed more or less directly using Kuz'min theorem (that integral homology coincide for two-step free nilpotent Lie algebras and groups) and equals $\Lambda^4(N(r, 2)_{ab} \otimes \Bbb Z/3)$.

- Is there any 2-torsion ever in $H_i(N(r, k), \Bbb Z)$? Is there anything besides 3?

Also we can assemble all $N(r, k)$'s in one compound via canonical projections $$\phi_{k}: N(r, k+1) \to N(r, k)$$ and obvious simplicial maps

$$\delta^{r}: N(r+1, k) \to N(r, k), \,\sigma^{r}: N(r, k) \to N(r+1, k)$$

Can we see some interesting structure in this bigraded thing (some sort of "nilpotent integral Steenrod algebra")? For example:

- $H_1(\phi)$ is always identity, $H_2(\phi)$ is always zero, $H_3(\phi)$ go nontrivially from $2k$ to $k$ and then vanish — what about higher homology?
- simplicial groups $H_1(N(\cdot, k))$ are contractible; computation of homotopy type of $H_2$ and $H_3$ seems interesting and doable

Further, there's a conjecture of Millionscshikov that integral jet groups $J_k := t + \sum_{i = 2}^{k+1}a_it^i, a_i \in \Bbb Z$ with composition modulo $t^{k+2}$ as operation have stable homology with $\textrm{rk} \, H_s(J_k) = s+2$th Fibonacci number for $k$ sufficiently large (I think that it's now proven for degree $\leq 3$ and there's numerical evidence that it's true always).

- Should we expect any kind of stability in free case — a variation of Church-Ellenberg-Farb representation stability, for example?

Let $H$ be a infinite dimensional, separable Hilbert space over $\mathbb{C}$

Let $B$ a subset of $H$ such that $B$ is linearly independent and such that exists a homeomorphism $f : [0,1] \to B$ between the unit interval $[0,1]$ and $B$

I would like to know if is it true that:

For every fixed $x \in (0,1)$ $$ \overline{ \operatorname{span} f([0,x]) } \cap \overline{ \operatorname{span} f((x,1]) } = \operatorname{span} f(x) $$

Let $R$ be the set of all real numbers. For $g: R\to R\cup \{\pm\infty \}$ and $x\in R$, we define $\limsup_{y\to x}g(y)\in R\cup \{\pm\infty \}$ in the usual way. Namely,

$\limsup_{y\to x}g(y):=\inf_{\delta>0}\sup_{0<|y-x|<\delta}g(y)$.

Next, for any $f: R\to R$, we define $B_f\subset R$ as

$B_f := \{ x\in R\mid \limsup_{y\to x} \bigg| \frac{f(y)-f(x)}{y-x} \bigg| <+\infty \}.$

Recently, I have realized that the following interesting claim (probably) holds:

Claim 1: Let $f:R\to R$ and $F_i\subset R~(i=1,2,3,\cdots)$ satisfy the following three conditions:

- Each $F_i$ is a closed set.
- Each $F_i$ has no interior points.
- $R-B_f\subset \cup_i F_i$.

(In other words, we assume that $R-B_f$ is a set of first category.) Then there exists an open interval $(a,b)\subset R$ and $N\geq 1$ such that

$\forall x,y\in (a,b)~~[~~|f(y)-f(x)|\leq N|y-x|~~]$.

My proof is as follows. This proof is based on the Baire category theorem. Could anyone judge the correctness of this proof?

Proof of Claim 1: First, we can prove the following claim.

Claim 2: Let $f:R\to R$ and $x\in R$ satisfy $\limsup_{y\to x} \bigg| \frac{f(y)-f(x)}{y-x} \bigg| <+\infty$. Then there exist positive integers $N,M\geq 1$ such that

$\forall y,z\in R~~[~~x-\frac{1}{M}<y<x<z<x+\frac{1}{M}\Rightarrow |f(z)-f(y)|\leq N(z-y)~~]$.

The proof of this claim 2 is basically the same as that of "straddle lemma" (so we omit the proof). Next, let $f:R\to R$ and $F_i\subset R~(i=1,2,3,\cdots)$ satisfy the three conditions of claim 1. For any positive integers $N,M\geq 1$, we define $B_{N,M}\subset R$ as

$B_{N,M}:=$ $\{ x\in R\mid \forall y,z\in R~~[~~x-\frac{1}{M}<y<x<z<x+\frac{1}{M} \Rightarrow |f(z)-f(y)|\leq N(z-y)~~]~~\}$.

By claim 2, we have $B_f\subset \cup_{N,M\geq 1}B_{N,M}$. Keeping in mind $R-B_f\subset \cup_i F_i$, we have

$R\subset (\cup_{N,M\geq 1}B_{N,M})\cup (\cup_i F_i).~~~~$ (1)

Next, we show that each $B_{N,M}$ is closed. Let $x\in R$ and $x_i\in B_{N,M}~(i\geq 1)$ satisfy $x_i\to x~(i\to +\infty)$. We have only to show that $x\in B_{N,M}$. Let $y,z\in R$ satisfy

$x-\frac{1}{M}<y<x<z<x+\frac{1}{M}$.

Since $x_i\to x$, we have

$x_i-\frac{1}{M}<y<x_i<z<x_i+\frac{1}{M}~$ (for sufficiently large $i$).$~~~~$ (2)

Let $i$ be one of them. Then it follows from (2) and $x_i\in B_{N,M}$ that $|f(z)-f(y)|\leq N(z-y)$. Thus we conclude that

$\forall y,z\in R~~[~~x-\frac{1}{M}<y<x<z<x+\frac{1}{M}\Rightarrow |f(z)-f(y)|\leq N(z-y)~~]$.

This implies $x\in B_{N,M}$, so $B_{N,M}$ is closed. Then the right hand side of (1) is a countable union of closed sets. Applying Baire category theorem, there exists $F_i$ or $B_{N,M}$ which has interior points. Since $F_i$ has no interior points, it follows that there exists $B_{N,M}$ which has interior points. So there exists an open interval $(a,b)$ such that $(a,b)\subset B_{N,M}$. Without loss of generality, we may assume $b-a<1/M$ and $(a,b)\subset B_{N,M}$. Next, let $x,y\in (a,b)$. We would like to show that $|f(y)-f(x)|\leq N|y-x|$. We may assume $x\leq y$. If $x=y$, then we have $|f(y)-f(x)|\leq N|y-x|$. If $x<y$, then let $c:=(x+y)/2$. Since $b-a<1/M$, we have

$c-1/M<x<c<y<c+1/M.~~~~$ (3)

Moreover, we have $c\in (x,y)\subset (a,b)\subset B_{N,M}$, i.e. $c\in B_{N,M}$. Then it follows from (3) and $c\in B_{N,M}$ that $|f(y)-f(x)|\leq N(y-x)$. Thus we complete the proof.$~~$Q.E.D.

In addition, since the above proof proceeds in a conventional way, it is most likely that claim 1 is already known (if correct). So I am looking for the references which refer to the above claim 1. Does anyone know?

Let $\Lambda$ be the von Mangoldt function and $\chi$ a primitive character mod $q$, then we have the explicit formula $$ \sum_{n \leq X} \Lambda(n) \chi(n) = \delta_{\chi} X - \sum_{ |Im \ \rho| \leq T} \frac{X^{\rho}}{\rho} + O((\frac{X}{T}+1) \log^2(qXT)), $$ where $\delta_{\chi}$ is $1$ if $\chi$ is the principal character and $0$ otherwise, and $\rho$'s are the non-trivial zeros of the $L$ function $L(s, \chi)$. Let us take $T = X^{a}$ where $0< a < 1$. From this formula we can easily deduce that $$ | \sum_{ |Im \ \rho| \leq T} \frac{X^{\rho}}{\rho} | \ll X. $$ I was wondering does the bound still hold if I put the absolute value inside the sum?, i.e. do we have $$ \sum_{ |Im \ \rho| \leq T} | \frac{X^{\rho}}{\rho} | \ll X. $$ My guess is that it is true but I was not sure how to see this. Any comments would be appreciated. Thank you very much.

What is the difference between the alternating product of the Hasse-Weil $L$-functions of the generic fiber of an arithmetic scheme $X\to\text{Spec}(\mathbf{Z})$ and the zeta function of $X$? (each with its own gamma factor)

**In the end, this question is about understanding the BSD conjecture for abelian varieties over $\mathbf{Q}$ a bit better.**

If $X$ is an abelian variety over $\mathbf{Q}$, is there a Birch and Swinnerton-Dyer conjecture for the **zeta function** of its Néron model? Is it related to the Birch and Swinnerton-Dyer conjecture for the Hasse-Weil $L$-function of $X$?

**EXAMPLE:** let $E$ be an elliptic curve over $\mathbf{Q}$, $\mathcal{E}$ its Néron model over $\text{Spec}(\mathbf{Z})$.

The **zeta function** of $\mathcal{E}$ is, by definition:
$$Z(\mathcal{E}/\mathbf{Z},s) := \prod_{(p)\in\text{Spec}(\mathbf{Z})^0}Z(\mathcal{E}_p,p^{-s}),$$
where $\text{Spec}(\mathbf{Z})^0$ is the set of closed points of $\text{Spec}(\mathbf{Z})$, $\mathcal{E}_p$ is the mod $p$ fiber of $\mathcal{E}$, a variety over $\mathbf{F}_p$, and

$$Z(\mathcal{E}_p, T) = \frac{1 - a_pT + pT^2}{(1-T)(1-pT)},$$
while for **the Hasse-Weil $L$-function**
$$L(E/\mathbf{Q},s) := \prod_{p\ {\rm good}}\frac{1}{1-a_pp^{-s} + p^{1-2s}}\cdot\prod_{p\ {\rm split\ mult.}}\frac{1}{1-p^{-s}}\cdot\prod_{p\ {\rm non-split\ mult.}}\frac{1}{1+p^{-s}}\cdot\prod_{p\ {\rm additive}}1.$$
where $a_p$ is the size of $\mathcal{E}_p(\mathbf{F}_p)$.

It seems: $$(*)\ \ \ Z(\mathcal{E}/\mathbf{Z},s) = \frac{\zeta(s)\zeta(s-1)}{L(E/\mathbf{Q},s)}.$$

We can introduce gamma factors and get

$$\hat{Z}(\mathcal{E}/\mathbf{Z},s) = \frac{\hat{\zeta}(s)\hat{\zeta}(s-1)}{\hat{L}(E/\mathbf{Q},s)}$$ where $\hat{\zeta}(s)$ is the completed Riemann zeta function, and $\hat{L}(E/\mathbf{Q},s)$ is the Hasse-Weil $L$-function of $E$ multiplied by its gamma factor as described by Serre.

**PRECISE QUESTIONS:**

**(1.1):** Is the BSD conjecture about $\hat{L}(E/\mathbf{Q},s)$?
Ie. Is it true that $\text{ord}_{s = 1}\hat{L}(E/\mathbf{Q},s) = \text{ord}_{s=1}L(E/\mathbf{Q},s)$? (It looks the answer is yes because the gamma factor doesn't vanish at $s=1$). Is the BSD formula for $L$ and $\hat{L}$ the same? (ie. do we have $\text{Res}_{s=1}L(E/\mathbf{Q},s) = \text{Res}_{s=1}\hat{L}(E/\mathbf{Q},s)$?)

**(1.2):** As a baby example of **(1.1)**. For $\zeta_K(s)$ the Dedekind zeta function of a number field $K$, and $\hat{\zeta}_K(s)$ the completed Dedekind zeta function, $\text{Res}_{s=1}\zeta_K(s)$ is predicted by the class number formula. What's the relation with $\text{Res}_{s=1}\hat{\zeta}_K(s)$? The former is
$$\text{Res}_{s=1}\zeta_K(s) = \frac{2^{r_1}(2\pi)^{r_2} h_K R_K}{\omega_K\sqrt{|\Delta_K|}},$$
with usual meanings.

If we call $$\zeta_{K,\infty}(s) := |\Delta_K|^{s/2}\Gamma_{\mathbf{R}}(s)^{r_1}\Gamma_{\mathbf{C}}(s)^{r_2}$$ then the latter is $$\text{Res}_{s=1}\hat{\zeta}_K(s) = \zeta_{K,\infty}(1)\cdot\text{Res}_{s=1}\zeta_K(s) = 2^{r_1-r_2}\pi^{-r_1/2}\Gamma(1/2)^{r_1}\Gamma(1)^{r_2}\cdot \frac{ h_K R_K}{\omega_K}.$$

Does this expression mean anything? I was expecting to get a cleaner expression for the completed zeta function.

It looks the answer to **(1.1)**, then, should be
$$\text{Res}_{s=1}\hat{L}(E/\mathbf{Q},s) = L_{\infty}(1)\cdot \text{Res}_{s=1}L(E/\mathbf{Q},s).$$

**(2): (the question I am most interested in)**

Is there a BSD conjecture for $\hat{Z}(\mathcal{E}/\mathbf{Z},s)$ (call it BSD$(\hat{Z})$)? Is it related to the BSD conjecture for $\hat{L}(E/\mathbf{Q},s)$ (call it BSD$(L)$)?

(specifically, do we have BSD$(\hat{Z})\Rightarrow$ BSD$(L)$?)

**(3):** What is a formula analogous to $(*)$, for higher dimensional abelian varieties and their zeta vs Hasse-Weil $L$-functions?

**========================================**

**EDIT: an equivalent question I would be very interested to know the answer of.**

In the function field case and for elliptic curves, Tate proved the BSD conjecture (for each elliptic curve over a function field) is equivalent to the Artin-Tate conjecture for a canonical elliptic surface (over a finite field). I think this has been generalized to abelian varieties. Is it possible to prove something analogous over number fields? ie. is it possible to deduce the BSD conjecture for the Hasse-Weil $L$-function of an abelian variety from a number field analog of the Artin-Tate conjecture (where probably the leading coefficient formula involves torsion and volumes on higher Chow groups) for the **zeta** function of an appropriate arithmetic scheme?

(without making use of Weil-êtale cohomologies, etc)

I am looking for a reference on continuity of (proximal) subdifferentials. For a continuous function $F: \mathbb R^n \rightarrow \mathbb R$, a vector $v$ is called **proximal subgradient** at $x$ if there exist $r>0, \delta >0$ s. t.

$$\forall y \; \text{s. t.} \; \|y-x\| \le r \quad F(y) \le F(x) + \langle v, y-x \rangle - \delta \| y-x \|^2$$

The set of all proximal subgradients is the **proximal subdifferential** denoted by $\partial F(x)$.

**Proximal $\varepsilon$-subgradient** at $x$ is a vector $v_{\varepsilon}$ s. t.

$$\exists r_{\varepsilon}>0, \delta_{\varepsilon} >0 \; \forall y \; \text{s. t.} \; \|y-x\| \le r_{\varepsilon} \quad F(y) \le F(x) + \langle v_{\varepsilon}, y-x \rangle - \delta_{\varepsilon} \| y-x \|^2 - \varepsilon$$

The set of all such vectors is called **proximal $\varepsilon$-subdifferential** and denoted by $\partial_{\varepsilon} F(x)$.

I am looking for a bound, depending on $\varepsilon$, on the following:

$$\sup_{v_{\varepsilon} \in \partial_{\varepsilon} F(x)} \| v_{\varepsilon} - \partial F(x) \|$$

In other words, I'd like to find, for a given proximal $\varepsilon$-subgradient $v_{\varepsilon}$ a proximal subgradient $v$ s. t. $\|v - v_{\varepsilon} \| \le M(\varepsilon)$ where $M(\varepsilon)$ should be (ideally) some multiple of $\varepsilon$.

Something in this direction goes along with 1, 2 and 3. But the bounds there are given for convex functions and usual (not proximal) subdifferentials.

Let $\mathcal{T}$ be the trivalent directed tree, with two parents and one child for each vertex (see below). Let $\mathcal{V}$ be the set of vertices of $\mathcal{T}$ and $H$ be the Hilbert space $\ell^2(\mathcal{V})$. Let $S$ be the shift operator on $H$ defined so that $S(v)$ is the child of the vertex $v$. Note that $S$ is bounded and its adjoint $S^{\star}$ is given by reversing the direction of each edge.

**Question:** What is the von Neumann algebra generated by $S \in B(H)$ and denoted $W^{\star}(S)$?

Is it hyperfinite? Is it a factor? of each type?

Note that $W^{\star}(S)$ is not abelian because the operator $S$ is not normal, because for any vertex $v$, $$(SS^{\star}-S^{\star}S) \cdot v = v-v'$$ with $v'$ the other parent of the child of $v$.

Suppose $A$ is a DGA algebra and $C$ a DGA coalgebra. An $(A,C)$-bimodule is an object $M$ that is both a right $A$-module and a left $C$-comodule in the evident compatible way. An $(A,C)$-bundle is an $(A,C)$-bimodule with a differential whose underlying $(A,C)$-bimodule is $C\otimes A$. Thus, for example, a twisted tensor product $C\otimes_t A$ is an $(A,C)$-bundle.

In the article *Differential homological algebra* by Husemoller, Moore and Stasheff, there is a Remark 1.7 (page 26 here). This says that in the article *On a theorem of E.H. Brown* by Gugenheim (available here), it is proven that every $(A,C)$-bundle is a twisted tensor product. I scanned the paper by Gugenheim but couldn't pin down where this is done.

I would highly appreciate if someone that is familiar with this article or the result cited in Remark 1.7 can pin down where this is proven and, if possible, provide other comprehensive references.

**Context** (Update 24/12) I would like to show that every minimal free resolution $C\otimes A$ of the trivial module $k$ of a graded connected $k$-algebra $A$ arises as a twisted tensor product $C\otimes_t A$ where $t :C\longrightarrow A$ is a $\gamma$-cochain (in the sense of Alain Proute's PhD thesis) between the minimal $A_\infty$-coalgebra $C=\text{Tor}$ and $A$.

I have a candidate for this $t$, obtained as follows. Choose a homotopy retraction $(i,p,h)$ of $BA\otimes_\beta A$ onto $C\otimes A$ satisfying the side conditions where $i$ is a honest inclusion of cycles, and produce a minimal $A_\infty$-coalgebra structure on $C$ along with an $A_\infty$-quasi-isomorphism $f : C\longrightarrow BA$ following say the formulas of Markl here, which greatly simplify since $BA$ is a honest coalgebra. Now consider the composition of $\Omega f: \Omega C \longrightarrow \Omega BA$ with the counit $\Omega BA\longrightarrow A$, which are both quasi-isomorphism. This corresponds to a cochain $t = \beta f : C\longrightarrow A$, following Proute. In this way one obtains a quasi-isomorphism $f\otimes_\beta 1 : C\otimes_t A\longrightarrow BA\otimes_\beta A$ which one can check is $i$ because of the side conditions and certain compatibility relations between $\beta$, $h$ and $\Delta_{BA}$.

Note that the first step produces a map $\Phi$ from minimal free resolutions of $k$ with an homotopy retraction data from $BA\otimes_\beta A$ to minimal $A_\infty$-coalgebra structures on $\text{Tor}$ along with the data of an $A_\infty$-quasi-isomorphism to $BA$, while the second step gives a map $\Psi$ from minimal structures along with such an $A_\infty$-quasi-isomorphism to $BA$ to minimal free resolutions of $k$ along with a quasi-isomorphism to $BA\otimes_\beta A$. The above says that $\Psi\Phi$ is the 'identity', modulo the fact I have to figure out how to recover the homotopy data and not only $i$.