33
votes
2answers
2k 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 ...
6
votes
1answer
132 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 $n\geq 2$. ...
14
votes
2answers
764 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 $(\ ,\ ...
2
votes
0answers
656 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, ...
11
votes
1answer
707 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 ...
6
votes
1answer
544 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 $$ ...
6
votes
0answers
277 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 ...
21
votes
1answer
1k views

Morava on Shafarevich

Jack Morava has some interesting ideas stemming from stable homotopy theory and geometric topology on the Shafarevich Conjecture. The Shafarevich Conjecture states: $Gal(\bar Q / Q_{cycl})$ is free. ...
15
votes
1answer
1k 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 ...
12
votes
3answers
935 views

Stable Homology of arithmetic groups.

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 ...