3
votes
3answers
458 views

Non existence of cyclic infinite linear algebraic groups

Let $G$ be a linear algebraic group defined over some algebraically closed field $\mathbb{K}$ and also over some subfield $k\subset \mathbb{K}$. There is thus a natural group structure on the set of ...
1
vote
1answer
107 views

Compatibility of two definitions of elliptic elements in GLn

For an element $g$ of a connected reductive group $G$ (over a local field), $g$ is called $elliptic$ if it is semisimple and the maximal split subtorus of the center of the centralizer of $g$ is ...
2
votes
0answers
44 views

on degree zero elements in adelic groups

Let $G$ a split connected reductive group and $G(\mathbb{A})$ his points in the ring of adeles. We have a degree map $G(\mathbb{A})\rightarrow X_{*}(Z)$ where $Z$ is the center of $G$. Let ...
5
votes
0answers
133 views

On Langlands Pairing and transfer factors

In the paper "On the definition of transfer factors" Langlands and Shelstad define a certain number of factors $\Delta_{I}$, $\Delta_{II}$,$\Delta_{III,1}$,$\Delta_{III,2}$, which are roots of unity. ...
2
votes
1answer
241 views

finiteness of class number: a bound for semi-simple groups?

Let $F$ be a number field, and $G$ a connected semi-simple linear algebraic $F$-group, which does not contain anisotropic (simple) $F$-factors. Write $\hat{F}$ for the ring of finite adeles ...
1
vote
1answer
279 views

Centralizer of elliptic elements in $GL(2)$

Consider a global field $F$ and the group $\Gamma =GL(2,F)$. An element $\gamma \in \Gamma$ is called elliptic, if its eigenvalues do not lie in $F$. Now consider a completion $F_v$ of $F$ and $G_v = ...
3
votes
0answers
266 views

idelic closures of units of number fields

Let $K$ be a number field, $\mathcal O _K^\times$ its group of integral units and $\mathcal O _{K,+} ^\times$ its group of totally positive units. Denote further by $\widehat{\mathcal O }_K^\times$ ...
7
votes
1answer
721 views

Parametrization of 2-dimensional torus

The units with norm $+1$ in a pure cubic number field $K$ generated by a cube root of $m = ab^2$, where $a$ and $b$ are coprime and squarefree integers, correspond to integral points on the torus $$ ...