0
votes
0answers
155 views

on the Galois cohomology of reductive groups

Let $G$ a simply connected group over an algebraically closed field. $F=k((t))$ and $\mathcal{O}=k[[t]]$. Let $\gamma\in G(\mathcal{O})\cap G(F)^{rs}$. Let $E=k((t^{1/n}))$ with $n$ prime to the ...
6
votes
2answers
233 views

Taking invariants under pro-p-group is exact?

Let $l$, $p$ be primes. Is it true that the functor of taking invariants under pro-$p$-group $P$ of finite-dimensional $\mathbb Q_l$-vector spaces ($l\neq p$) is an exact functor? Thanks! NOTE 1: I ...
3
votes
0answers
134 views

Modified radical group of a Kummer extension

If $K/k$ is a degree $p$ Kummer extension of number fields (so $k$ contains the $p^r$th roots of unity for some $r \geq 1$ --- let's also assume $K/k$ is not generated by $p$-power roots of unity), I ...
4
votes
1answer
418 views

Tamagawa numbers of abelian varieties and torsion.

Let $A$ be an abelian variety defined over a number field $K$. Fix a prime $v \subset \mathcal{O}_K$, with underlying rational prime $p$. What relationship, known or conjectural (if any), should there ...
15
votes
1answer
997 views

Reference request: The first cohomology of SL(2,Z) with coefficients in homogeneous polynomials

Let $H_k$ be the vector space of degree $k$ homogeneous polynomials in two variables.I'm looking for a reference for the fact that $H^1(SL(2,\mathbb Z);H_k)=M^0(k+2)\oplus\overline{M^0(k+2)}\oplus ...
5
votes
4answers
1k views

Galois cohomology of linear groups over local fields

Let $F$ be a local field of characteristic zero (for simplicity), $\overline{F}$ an algebraic closure of $F$ and $L/F$ a fixed finite Galois extension. If $G$ is a linear algebraic group defined over ...