3
$\begingroup$

I asked this question on Math.StackExchange, to no avail. I try my chance on this one.

Let $M/F$ be an extension of fields. Let $G$ be an algebraic group over $F$, and consider the $F$-group $H$ defined as the restriction of scalars (a la Weil) of $G$ from $M$ down to $F$, i.e. $H=\mathrm{Res}_{M/F} G$. If $T$ is a subgroup of $H$, under what circumstances/hypotheses is it true that there exists a subgroup $\tilde{T}\leq G$ such that $T=\mathrm{Res}_{M/F} \tilde{T}$? Any reference would be greatly appreciated.

Improve this question
$\endgroup$
9
  • $\begingroup$ I don't what went wrong with Latex... $\endgroup$
    – M Turgeon
    Aug 9, 2012 at 18:36
  • 3
    $\begingroup$ Suppose that $M|F$ is a Galois extension with Galois group $A$. Then $H\times_F M\simeq \prod_{a\in A} G^a$, where $G$ is the twist of $G$ by $a$. The natural Galois action of $A$ on $H\times_F M$ is then given by the action by permutation on the right-hand side of the isomorphism. Thus a subgroup $T$ of $H$ corresponds to an $A$-invariant subgroup $T_1$ of $\prod_{a\in A} G^a$. In order for $T$ to be the Weil restriction of a subgroup $\widetilde{T}$, it is necessary and sufficient that $T_1\simeq \prod_{a\in A} G^a_1$, where $G_1$ is an subgroup of $G$ (defined over $M$). $\endgroup$
    – Damian Rössler
    Aug 9, 2012 at 19:11
  • 1
    $\begingroup$ @Damian: Your criterion can be simplified because $T$ is given over $F$: all one really needs to check is that $T_M$ decomposes as a direct product of subgroups of the various $G^a$'s (these factors are then automatically a Galois orbit, since $T$ was defined over $F$). For example, this applies to all maximal $F$-tori and parabolic $F$-subgroups (assuming $G$ is smooth and connected). Also, this works even if $M/F$ is just finite separable (not necessarily Galois), by using Galois descent from the Galois closure. $\endgroup$
    – user22479
    Aug 9, 2012 at 20:51
  • 1
    $\begingroup$ @M Turgeon. I don't have a direct reference for this; the best thing is to study the construction of the Weil restriction given in sec. 7.6 of the book "Néron Models" by Bosch et al. $\endgroup$
    – Damian Rössler
    Aug 10, 2012 at 7:16
  • 2
    $\begingroup$ Weil's original book "Adeles and algebraic groups" should be helpful to you, however it is difficult to get hold of a copy nowadays. $\endgroup$
    – Daniel Loughran
    Aug 10, 2012 at 7:31

0

Reset to default

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.