All Questions
Tagged with algebraic-k-theory nt.number-theory
24 questions
57
votes
2
answers
7k
views
What arithmetic information is contained in the algebraic K-theory of the integers
I'm always looking for applications of homotopy theory to other fields, mostly as a way to make my talks more interesting or to motivate the field to non-specialists. It seems like most talks about ...
- 30.3k
23
votes
3
answers
1k
views
References for $K_{4k}(\mathbb{Z})$
Weibel's "Algebraic K-theory of rings of integers in local and global fields" says $K_{4k}(\mathbb{Z})$ are known to have odd order, with no prime factors less than $10^7$, but are conjectured to be ...
- 17.6k
20
votes
0
answers
890
views
Epsilon factors - a la Beilinson - What is it?
I understand, to some extent, Tate's thesis. Could somebody explain perhaps what are the epsilon factors in Beilinson's works, such as "$\epsilon$-factors for Gauss-Manin determinants", or "...
- 5,562
19
votes
1
answer
3k
views
What is a path in K-theory space?
In a comment on Tom Goodwillie's question about relating the Alexander polynomial and the Iwasawa polynomial, Minhyong Kim makes the cryptic but tantalizing statement:
In brief, the current view is ...
- 22.1k
17
votes
2
answers
2k
views
Who first noticed that the Hilbert symbol is a Steinberg symbol ?
Hilbert reformulated the quadratic reciprocity law of Gauß as a product formula
$$
\prod_v(a,b)_v=1
$$
for the various local Hilbert symbols. For each place $v$ of $\bf Q$, the Hilbert symbol $(\ ,\ ...
- 18.7k
14
votes
3
answers
1k
views
Stable homology of arithmetic groups
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\Sp{Sp}$Suppose that $F/Q$ is a number field.
Using automorphic forms, Borel computed the ($R$-) stable cohomology of $\SL_n(O_F)$, and as a result, ...
user631
12
votes
1
answer
1k
views
Rationalised K-theory of number fields
Let $A$ be the ring of integers in a number field, and consider the rationalised algebraic $K$-theory groups $\mathbb{Q}\otimes K_*(A)$. A theorem of Borel calculates the ranks of these groups; the ...
- 56.9k
12
votes
1
answer
795
views
Status of the extended form of the Lichtenbaum conjecture
The extended Lichtenbaum conjecture concerns the relationship of special values of $L$-functions of number fields $K$, to the algebraic $K$-theory and etale cohomology of the ring of integers $O_K$.
...
- 8,167
9
votes
1
answer
425
views
Abelianization of $\mathrm{GL}_2(R)$
$\DeclareMathOperator\GL{GL}$Let $R$ be a number ring. Are there known lower bounds for $H_1(\GL_2(R);\mathbb Q)$ or $H_1(\GL_2(R),\GL_1(R);\mathbb Q)$ in terms of properties of $R$ (class number, ...
- 965
8
votes
1
answer
321
views
rationalized K-Theory of the group ring of finite cyclic groups
I am interested in calculating the rationalized algebraic K-Theory groups of the group ring of $\mathbb Z/n$, that is $K_i(\mathbb Z[\mathbb Z/n])\otimes \mathbb{Q}$ for any natural number $n\geq 2$. ...
- 175
8
votes
2
answers
833
views
is there a p-adic Borel theorem?
Let $F$ be a number field. Denote, as usual, $\mathcal{O}_F$ the ring of integers and $r_1$, $r_2$ the number of real and complex embeddings. Let $\zeta_F(s)$ be the Dedekind zeta function of $F$. The ...
- 81
8
votes
1
answer
1k
views
Galois descent for K-groups (or for étale cohomology groups)
Let $F/K$ be a Galois extension of number fields with Galois group $G$. Let $\mathcal{O}_F$ and $\mathcal{O}_K$ be the associated rings of integers, and let $n\geq 1$.
When is
$$
K_{2n-1}(\...
- 13k
6
votes
1
answer
1k
views
A weak version of Bass' conjecture
Let $A$ be a finitely generated $\mathbb{Z}$-algebra which is a UFD. Then (a special case of) the Bass conjecture states that $K_0(A)$ is a finitely generated abelian group. As far as I am aware, this ...
- 21.3k
6
votes
1
answer
411
views
Obtaining the Hilbert symbol from cup product on motivic cohomology
Let F be a number field. Does the Hilbert symbols at the various places of F arise from the cup product on the motivic cohomology groups of Spec(F)? And if so, is it possible to interpret Moore's ...
- 61
6
votes
0
answers
487
views
Inverse Galois Problem...and parallelizable vector fields?
Usual approaches to the inverse Galois problem start with realizations of a group $G$ over a larger field, and then try to specialize to ${\Bbb Q}$.
One could also start by building suitable objects ...
- 17.6k
5
votes
1
answer
353
views
Is there a simple proof that Milnor $K_2$ of a number field is torsion?
This is a theorem of Garland. I had a look at the original paper which looks pretty complicated. I was wondering if the proof has been simplified over the years or if a different approach is nowadays ...
- 71
4
votes
0
answers
94
views
How to describe a concrete generator of $\widetilde{K_0(\mathbb{Z}[C_{23}])} \cong \widetilde{K_0(\mathbb{Z}[\zeta_{23}])}$
Milnor (page 29, see below) gives an explicit proof that the zeroth $K$-theory of the group ring $\mathbb{Z}[C_p]$, where $C_p$ is the cyclic group of order $p$ with $p$ a prime agrees with $K_0$ of $\...
- 2,303
4
votes
0
answers
339
views
Beilinson regulator: a road map
I'm approaching to the Beilinson Conjecture and after studying some properties of the Deligne-Beilinson cohomology, I want to understand the regulator maps. But I don't know anything about K-theory ...
4
votes
0
answers
2k
views
Online Number Theory Video?
Are there any graduate level number theory course available on line ? The only video series I am aware of are some MSRI videos, and Ted Chinburg's courses http://www.math.upenn.edu/~ted/noframes.html, ...
- 1,663
2
votes
1
answer
84
views
reduction of torsion modules
Let $G$ be a profinite group.
Let $K(G,\mathbb Z_\ell)$ be the Grothendieck group of the derived category of finitely generated $\mathbb Z_\ell$-modules with continuous $G$-action.
Let $K(G,\mathbb ...
- 135
2
votes
0
answers
124
views
Can K$_3$ of finite fields be related to Teichmüller cocycles?
This is sort of a blind shot, but...
For a ring $R$, its third algebraic K-group is given by $\operatorname K_3(R)=H_3(\operatorname{St}(R))$.
To simplify matters, let $R$ be a finite field $\mathbb ...
- 18.4k
2
votes
0
answers
182
views
On the relative class number of a cyclotomic extension
Let $\Bbb Z[\zeta_p]$ denote the cyclotomic integers where $p$ is a prime and let $h_1 = h_1(p)$ denote its relative class number.
Question: Is it known whether there are infinitely many primes $p$ ...
- 18.8k
2
votes
0
answers
209
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
1
vote
0
answers
331
views
Where can I find the article of A. Borel: "Values of zeta-functions at integers, cohomology and polylogarithms"? [closed]
Where on the internet can I find this article?
I know that it is in this book: Current trends in mathematics and physics, Narosa, New Delhi, 1995.
user51486