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3 votes
0 answers
130 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 ...
C.D.'s user avatar
  • 605
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\...
Mikhail Borovoi's user avatar
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) \...
Yotam Shomroni's user avatar
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 $\mathrm{Sym}^g(k^2)$...
tkr's user avatar
  • 408
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 ...
Bear's user avatar
  • 845
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 $G$...
S. Carnahan's user avatar
  • 45.7k