All Questions
Tagged with cohomology nt.number-theory
54
questions
0
votes
0
answers
563
views
Cocycles and the Collatz problem?
Let $T(n) = n+R(n)$, where $R(n) = -n/2 $ if $n\equiv 0 \mod 2$ else $R(n) = \frac{n+1}{2}$.
$R(n)$ is the Cantor ordering of the integers:
https://oeis.org/A001057
In the Collatz problem, one is ...
- 2,608
3
votes
0
answers
105
views
A question on the averages of Kloosterman sums
Sorry to disturb. Recently, I encountered a puzzle on the sums involving two Kloosterman sums. That is,
For any $h, q_1,q_2\in \mathbb{N}$ with $(q_1,q_2)=1$ and $Q>1$, how two get a bound
$$\sum_{...
- 553
2
votes
0
answers
68
views
Framed bordism and string bordism in 3-dimensions vs topological modular form
In simple colloquial terms, how are the framed bordism and string bordism in 3-dimensions related to the study of the theory of topological modular form TMF? I want to know some simple derivable ...
- 10.4k
11
votes
1
answer
1k
views
Hodge conjecture as the equality of arithmetic and algebraic weights of motivic L-functions
Recently I became aware of the following statement given on page 13 of this paper. First, let us recall the following definitions:
Definition 4.1. Suppose $L(s)$ is an analytic $L$-function with ...
- 533
1
vote
1
answer
227
views
On the estimate for a double exponential sum
I encounter a hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,s \in \mathbb{N}^+$(which may not be necessarily co-prime with each other), how to bound the sum:
$$...
- 553
0
votes
0
answers
90
views
A question on the evaluations of certain three-dimensional hyper-Kloostermans
There is a basic question regrading the 3-dimensional hyper-Kloosterman sum which needs some help from the experts here:
For any integers $q,h \in \mathbb{N}$, how to estimate the sum:
$$\sideset{_{}^{...
- 553
2
votes
1
answer
149
views
On the estimate for the mixed 3-dimensional hyper-Kloosterman sum
There is a basic question regrading the mixed 3-dimensional hyper-Kloosterman sum:
For any positive integer $n$ not divisible by $p$, how to prove
$$\sideset{_{}^{}}{^{\ast}_{}}\sum _{x,y ,z\bmod p}
\...
- 553
12
votes
1
answer
876
views
Comparing singular cohomology with algebraic de Rham cohomology
Let $X$ be a smooth projective variety over a number field $K$. Then there are two cohomology groups we can attach to $X$: the algebraic de Rham cohomology group
$H^k_{\text{dR}}(X/K), $
which is a ...
- 1,889
2
votes
1
answer
188
views
On the upper-bound for a type of quintuple Kloosterman sums
Sorry to disturb, dear experts here. I have a question involving the quintuple Kloosterman sum, and expect some hints to show the upper-bound.
My question is, for any $x,y,z,w,\delta \in \mathbb{Z}$ ...
- 553
3
votes
1
answer
334
views
Estimates for certain double-Kloosterman sums
Sorry to disturb. I encounter a double-Kloosterman sum, which needs some help from the experts here.
For any $q\in \mathbb{N}^+$, how can we estimate the type of sum
$$ \sideset{_{}^{}}{^{\ast}_{}}\...
- 553
1
vote
1
answer
225
views
A question involving the three-dimensional Kloosterman sum
Sorry to disturb. I have a question involving the three-dimensional Kloosterman sum, which needs some help from the experts here.
For any $\alpha, \beta, \gamma \in \mathbb{Z}$ and $q\in \mathbb{N}^+$,...
- 553
6
votes
0
answers
286
views
Eichler-Shimura isomorphism as simple $(k-1)$-fold integration?
I am currently trying to learn about the Eichler-Shimura isomorphism, and all the definitions seems to be somewhat cumbersome and hard to commit to memory as intuitive. On the other hand, the ...
- 2,837
4
votes
1
answer
333
views
A Tate-Sen theorem mod $p$
Let $p$ be a prime number, $G=\textrm{Gal}(\overline{\mathbb{Q}}_p/\mathbb{Q}_p)$, and $\chi:G\rightarrow\mathbb{Z}_p^\times$ the cyclotomic character. Let $\mathbb{C}_p$ denote the completion of the ...
- 2,824
2
votes
0
answers
207
views
Are Milnor K-groups algebraic groups?
Let $k$ be a field, $K$ a finite extension of $k$, and $K_{n}^{M}(K)$ the $n$-th Milnor K-group of $K$, that is,
$$
K_{n}^{M}(K)=K^{\times}\otimes_{\mathbb{Z}}\cdots\otimes_{\mathbb{Z}} K^{\times}/I,
$...
- 479
4
votes
1
answer
254
views
Reference request: Long exact sequence in profinite Galois cohomology up through $H^2$
I'm looking for a reference of the following statement. Let $G$ be the Galois group of a Galois extension $L/K$, not necessarily finite. Let $A,B,C$ be groups with a continuous $G$-action, and let
$$1\...
- 7,313