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Sep
10 |
comment |
Restricting representations to lattices
@Misha: Over $\mathbf{C}$ what you write is correct, but over $\mathbf{R}$ you need to impose simply connectedness in additional to semi-simplicity. The usual counterexample is the inverse of the analytic isomorphism ${\rm{SL}}_n(\mathbf{R}) \rightarrow {\rm{PGL}}_n(\mathbf{R})$ for odd $n > 1$. The "problem" is that an $\mathbf{R}$-isogeny between connected linear algebraic $\mathbf{R}$-groups can have finite kernel with no nontrivial $\mathbf{R}$-points; over $\mathbf{C}$ this isogeny issue (invisible to Lie algebras) cannot be missed at the level of $\mathbf{C}$-points. |
Sep
10 |
comment |
Why are $S$-arithmetic groups interesting?
It seems worth reflecting (a bit!) on the case of GL$_1$. Clearly $O_{K,S}^{\times}$ and its finite-index subgroups are the arithmetic subgroups of ${\rm{GL}}_1(K) = K^{\times}$ (relative to the $K$-group ${\rm{GL}}_1$). Chevalley's theorem that this satisfies the congruence subgroup property is crucial for defining Serre tori, which underlie the clean formulation of the important Artin-Weil theorem relating CM fields to general algebraic Hecke characters (the definition of which doesn't explicitly mention CM fields). It doesn't need a general theory of $S$-arithmeticity, but is so classical. |
Sep
10 |
comment |
Why are $S$-arithmetic groups interesting?
@Agol: Yes indeed, and more generally any finitely generated subgroup of $G(\overline{K})$ lies inside an $S$-arithmetic subgroup of $G(F)$ for some finite extension $F$ of $K$ inside $\overline{K}$ and a finite set $S$ of places of $F$. I prefer to think about passage to "$S$-statements" as something we do after first getting everything over a single number field; i.e., once we've gotten ourselves inside $G(F) = G_F(F)$ for a specific number field $F$ then we "spread out" to an $S$-integral statement over $F$, with $S$ living on $F$. |
Sep
9 |
comment |
Why are $S$-arithmetic groups interesting?
For connected reductive $G$ over $K$ it can be useful to consider open subgroups of $G(A_K)$ of the form $G(K_S) \times U$ for a compact open subgroup $U$ of $G(A_K^S)$. This meets $G(K)$ in an $S$-arithmetic subgroup. Note also that a typical finitely generated subgroup of GL$_n(K)$ cannot be conjugated inside GL$_n(O_K)$ but lies in GL$_n(O_{K,S})$ for some $S$. Overall, $S$-arithmetic groups give a robust theory of "integrality away from $S$" inside $G(K)$ that isn't tied to a specific flat affine $O_{K,S}$-model of $G$ (no reductive one may exist!) and it's "functorial" in $G$ over $K$. |
Sep
8 |
revised |
Philosophy behind Mochizuki's work on the ABC conjecture
added 503 characters in body |
Sep
8 |
comment |
When is an orbit spherical?
@Jim Humphreys: I think that what I wrote was accurate: in Theorem A of Richardson's paper "Affine coset spaces of reductive algebraic groups" in Bulletin of the LMS 9 (1977), pp. 38--41, he uses Haboush's theorem (Mumford's conjecture) to prove the result in arbitrary characteristic. He states explicitly early in the paper that his purpose is to record a proof valid in any characteristic. I'm not sure which papers of Richardson and C-P-S you have in mind, but please look at the Richardson paper I have just mentioned. |
Sep
7 |
comment |
Philosophy behind Mochizuki's work on the ABC conjecture
Dear Emerton: As you know, there is a purely algebraic proof of the "classical" (not "Szpiro") formulation for F(t), and that is what I was pretty sure David Speyer was considering to present. That argument (as one finds in elementary expositions) is what I meant isn't relevant, much like the F(t)-version of FLT proved by bare-hands algebra rather than by genus; sorry for being unclear. I agree that the F(t)-case proved via a classifying map from a curve to a moduli space of elliptic curves, thereby highlighting the role of Kodaira-Spencer maps, is a crucial perspective in Mochizuki's work. |
Sep
7 |
comment |
Philosophy behind Mochizuki's work on the ABC conjecture
@James: OK, good to hear the clarification. But one of Mochizuki's survey papers does address exactly what you suggest you'd like to hear about at the end of your comment, though using the sophisticated language of moduli stacks (which, if you pretend are schemes, can be inspiring even if you don't know about stacks): see 1.3.1 of "A survey of the Hodge-Arakelov Theory of Elliptic Curves I". Mochizuki is an extremely good writer!! If you elide unclear technical issues and try to just digest the flavor, you can get a lot of inspiration. I am reminded about "reading the masters"... :) |
Sep
7 |
answered | Philosophy behind Mochizuki's work on the ABC conjecture |
Sep
7 |
comment |
Philosophy behind Mochizuki's work on the ABC conjecture
@David: The link between ABC and Szpiro's Conjecture (which is the content of the application of the Frey-like construction) long predates Mochizuki's work, and the "function field case" of ABC seems to have nothing to do with the ideas relevant in Mochizuki's work in the number field case much as the "function field case" of FLT is totally irrelevant to the actual proof of FLT. So although each aspect is very interesting for someone who has never heard of the ABC Conjecture, neither of them sheds light on anything that has happened since the time Mochizuki began his work on these matters. |
Sep
7 |
comment |
When is an orbit spherical?
@Jesko: I think your affineness condition on the orbit $G/H$ is equivalent to your hypothesis that the underlying reduced scheme of the identity component $H^0$ is reductive. This equivalence is a theorem of Borel & Harish-Chandra in characteristic 0 (proved via topological methods over $\mathbf{C}$), Richardson in any characteristic (proved via Haboush's work on Mumford's conjecture in GIT), and finally proved by Borel in any characteristic via etale cohomology (adapting the argument with H-C). |
Sep
5 |
comment |
On the structure of commutative group schemes
@S. Carnahan: True, but Witt groups are nonetheless "split" (in the sense that they have a composition series over $k$ with successive quotients $k$-isomorphic to $\mathbf{G}_a$), so a more striking kind of example answering Will's question in the negative is one which doesn't even contain $\mathbf{G}_a$ as a $k$-subgroup. Of course there are no such examples when $k$ is perfect, but for imperfect $k$ Rosenlicht gave the classic example $y^p = x - a x^p$ for $a \in k - k^p$ with any $p = {\rm{char}}(k) > 0$. Tits studied this phenomenon in detail (in any dimension) in his Yale lecture notes. |
Sep
5 |
comment |
Rational automorphisms of semisimple algebraic groups
It's well-known; see SGA3, XXIV, 3.9-3.10. Here's the concrete proof. Let $(G_0,B_0,T_0)$ be the split form. Use a pinning to define a section to the map from ${\rm{Aut}}_{G_0/k}$ onto its finite constant component group ${\rm{Out}}_{G_0/k}$, valued in the $k$-subgroup of automorphisms preserving $(B_0, T_0)$. Doing twists by hand, the map ${\rm{H}}^1(k,{\rm{Out}}_{G_0/k}) \rightarrow {\rm{H}}^1(k,{\rm{Aut}}_{G_0/k})$ has image consisting of quasi-split forms whose Aut-sequence is a semi-direct product and meeting every class of inner forms. Now use uniqueness of quasi-split inner forms. |
Sep
4 |
comment |
On the structure of commutative group schemes
@Pooya: Consider two affine finite type $X$-groups $G$ and $H$ (e.g., $H$ could be $\mathbf{G}_a$). By general spreading-out principles, any $\eta$-scheme map $H_{\eta} \rightarrow G_{\eta}$ spreads out to a $U$-scheme morphism $H_U \rightarrow G_U$ for some dense open $U$ in $X$. If the given $\eta$-map is a homomorphism (resp. closed immersion) then we can arrange the same for the $U$-map by shrinking $U$ a bit. Here we use that "homomorphism" amounts to a commutative diagram (i.e., an equality of some maps) and "closed immersion" is one of those properties which can be "spread out". |
Sep
4 |
comment |
Rational automorphisms of semisimple algebraic groups
The phenomenon as in the middle of your 2nd paragraph happens already in type A (for more elementary reasons), as noted in my other comment. |
Sep
4 |
comment |
On the structure of commutative group schemes
@Pooya: No, not over the entirety of $X$. For example the underlying additive group of a line bundle with no nonzero global section is a counterexample to the global statement. But in your question you only asked for the property over some (dense) open subset of the base, and that much follows from standard "spreading out" arguments. |
Sep
4 |
comment |
On the structure of commutative group schemes
The quotient $G/T$ exists as a smooth affine group, and by design it is unipotent on geometric fibers over the integral base in char. 0. The generic fiber is unipotent over the char. 0 function field of the base. Your definition of unipotence is wrong in char. $> 0$ (where there are many smooth connected commutative unipotent groups containing no $\mathbf{G}_a$), but in char. 0 it is equivalent to the "right" definition (the filtration condition on geometric fibers). Such a composition series on the actual (not just geometric) generic fiber spreads out over some dense open of the base. |
Sep
4 |
comment |
Rational automorphisms of semisimple algebraic groups
You say "pinning" but don't assume $T$ is split, so the Question is unclear. The functor $S \mapsto {\rm{Aut}}_{S-{\rm{gp}}}(G_S)$ is represented by a smooth affine $k$-group ${\rm{Aut}}_{G/k}$ with identity component $G^{\rm{ad}}$. The identity component can fail to split off as a semi-direct product since projection from ${\rm{Aut}}_{G/k}$ to the component group need not be surjective on $k$-points when $G$ isn't quasi-split. A counterexample is $G = {\rm{SL}}(D)$ for a central division algebra that isn't 2-torsion in ${\rm{Br}}(k)$. So it fails in anisotropic cases over every local $k$! |
Sep
3 |
comment |
a question on TITS' note “Reductive groups over local fields”
@Igor Rivin: the question is how to define $\phi$. Although it might make Alex Trebek unhappy, on MO questions don't need to be stated in the form of a question. :) |
Sep
3 |
revised |
a question on TITS' note “Reductive groups over local fields”
added 373 characters in body |