All Questions
6 questions
3
votes
0
answers
131
views
Galois cohomology and Levi subgroups
Let $F$ a field and $G$ a smooth connected reductive group with a Levi subgroup $M$. Under what assumptions is $H^1(F, M) \to H^1(F, G)$ injective? In the case $F$ is nonarchimedean local I believe ...
1
vote
1
answer
232
views
Transfer for the group of coinvariants: a reference request
Let $G$ be a group and $M$ be a $G$-module,
that is, an abelian group written additively on which $G$ acts:
$$ (g,m)\mapsto g m.$$
We consider the group of coinvariants
$$ M_G:=G/\langle g m -m\ |\ g\...
18
votes
1
answer
1k
views
Explanation for $\chi(\operatorname{SL}_2(\mathbb{Z})) = -1/12$ with zeta function
$\DeclareMathOperator\SL{SL}\DeclareMathOperator\PSL{PSL}$One can compute the (group cohomological) Euler characteristic of $\SL_2(\mathbb{Z})$ via
$$ \chi(\SL_2(\mathbb{Z})) = \chi(\mathbb{Z}/2) \...
5
votes
0
answers
97
views
Is there a composite-order generalization of the homomorphism on Rep(Z/p) giving total dimension of Tate cohomology?
Let $p$ be a prime, let $\mathbb{Z}_p$ be the ring of $p$-adic integers, and let $G$ be a cyclic group of order $p$. It is rather well-known that finite rank $\mathbb{Z}_p$-free representations of $G$...
13
votes
0
answers
266
views
Group homology $\mathrm{SL}_2$ acting on $\mathrm{Sym}^g$
Let $k$ be a field. We write $\mathrm{Sym}^g(k^2)$ for the $g$-th symmetric power of the (a?) standard representation of $\mathrm{GL}_2(k)$ ($g\geq 0$ an integer). Here I consider $\mathrm{Sym}^g(k^2)$...
2
votes
2
answers
584
views
Lifting projective Galois representation with condition
Let $\bar{\rho}: G_K\to PGL_n(\mathbb{C})$ be projective representation of the absolute Galois group of a number field $K$ and $\varphi\in Aut(G_K)$.
A theorem of Tate tells us that we can always ...