Compute all homomorphisms of $mathbb G_a$ to $SL_2$ over $mathbb Z$
4
$begingroup$
How to compute the set of homomorphisms $mathrm{Hom}left(mathbb G_a , SL_2right)$ between those two group schemes over $mathbb Z$?
Over $mathbb C$, this can be done using classical algebraic geometry, up to $SL_2$ conjugacy.
group-theory algebraic-groups group-homomorphism
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
$endgroup$
add a comment |
4
$begingroup$
How to compute the set of homomorphisms $mathrm{Hom}left(mathbb G_a , SL_2right)$ between those two group schemes over $mathbb Z$?
Over $mathbb C$, this can be done using classical algebraic geometry, up to $SL_2$ conjugacy.
group-theory algebraic-groups group-homomorphism
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
$endgroup$
$begingroup$
You can try working $bmod p$ for all primes $p$.
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 7:22
$begingroup$
What do you mean by "compute the set"? you seem to implicitly mean that you want up to conjugacy in some sense, which is the main difficulty.
$endgroup$
– YCor
Dec 21 '18 at 15:02
add a comment |
4
4
4
2
$begingroup$
How to compute the set of homomorphisms $mathrm{Hom}left(mathbb G_a , SL_2right)$ between those two group schemes over $mathbb Z$?
Over $mathbb C$, this can be done using classical algebraic geometry, up to $SL_2$ conjugacy.
group-theory algebraic-groups group-homomorphism
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
$endgroup$
How to compute the set of homomorphisms $mathrm{Hom}left(mathbb G_a , SL_2right)$ between those two group schemes over $mathbb Z$?
Over $mathbb C$, this can be done using classical algebraic geometry, up to $SL_2$ conjugacy.
group-theory algebraic-groups group-homomorphism
group-theory algebraic-groups group-homomorphism
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
edited Dec 21 '18 at 10:58
Johannes Huisman
3,146617
3,146617
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
asked Dec 21 '18 at 3:25
zzyzzy
2,6691420
2,6691420
$begingroup$
You can try working $bmod p$ for all primes $p$.
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 7:22
$begingroup$
What do you mean by "compute the set"? you seem to implicitly mean that you want up to conjugacy in some sense, which is the main difficulty.
$endgroup$
– YCor
Dec 21 '18 at 15:02
add a comment |
$begingroup$
You can try working $bmod p$ for all primes $p$.
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 7:22
$begingroup$
What do you mean by "compute the set"? you seem to implicitly mean that you want up to conjugacy in some sense, which is the main difficulty.
$endgroup$
– YCor
Dec 21 '18 at 15:02
$begingroup$
You can try working $bmod p$ for all primes $p$.
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 7:22
$begingroup$
You can try working $bmod p$ for all primes $p$.
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 7:22
$begingroup$
What do you mean by "compute the set"? you seem to implicitly mean that you want up to conjugacy in some sense, which is the main difficulty.
$endgroup$
– YCor
Dec 21 '18 at 15:02
$begingroup$
What do you mean by "compute the set"? you seem to implicitly mean that you want up to conjugacy in some sense, which is the main difficulty.
$endgroup$
– YCor
Dec 21 '18 at 15:02
add a comment |
0
active
oldest
votes
Your Answer
StackExchange.ifUsing("editor", function () {
return StackExchange.using("mathjaxEditing", function () {
StackExchange.MarkdownEditor.creationCallbacks.add(function (editor, postfix) {
StackExchange.mathjaxEditing.prepareWmdForMathJax(editor, postfix, [["$", "$"], ["\\(","\\)"]]);
});
});
}, "mathjax-editing");
StackExchange.ready(function() {
var channelOptions = {
tags: "".split(" "),
id: "69"
};
initTagRenderer("".split(" "), "".split(" "), channelOptions);
StackExchange.using("externalEditor", function() {
// Have to fire editor after snippets, if snippets enabled
if (StackExchange.settings.snippets.snippetsEnabled) {
StackExchange.using("snippets", function() {
createEditor();
});
}
else {
createEditor();
}
});
function createEditor() {
StackExchange.prepareEditor({
heartbeatType: 'answer',
autoActivateHeartbeat: false,
convertImagesToLinks: true,
noModals: true,
showLowRepImageUploadWarning: true,
reputationToPostImages: 10,
bindNavPrevention: true,
postfix: "",
imageUploader: {
brandingHtml: "Powered by u003ca class="icon-imgur-white" href="https://imgur.com/"u003eu003c/au003e",
contentPolicyHtml: "User contributions licensed under u003ca href="https://creativecommons.org/licenses/by-sa/3.0/"u003ecc by-sa 3.0 with attribution requiredu003c/au003e u003ca href="https://stackoverflow.com/legal/content-policy"u003e(content policy)u003c/au003e",
allowUrls: true
},
noCode: true, onDemand: true,
discardSelector: ".discard-answer"
,immediatelyShowMarkdownHelp:true
});
}
});
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3048144%2fcompute-all-homomorphisms-of-mathbb-g-a-to-sl-2-over-mathbb-z%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
0
active
oldest
votes
0
active
oldest
votes
active
oldest
votes
active
oldest
votes
draft saved
draft discarded
Thanks for contributing an answer to Mathematics Stack Exchange!
- Please be sure to answer the question. Provide details and share your research!
But avoid …
- Asking for help, clarification, or responding to other answers.
- Making statements based on opinion; back them up with references or personal experience.
Use MathJax to format equations. MathJax reference.
To learn more, see our tips on writing great answers.
draft saved
draft discarded
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
StackExchange.ready(
function () {
StackExchange.openid.initPostLogin('.new-post-login', 'https%3a%2f%2fmath.stackexchange.com%2fquestions%2f3048144%2fcompute-all-homomorphisms-of-mathbb-g-a-to-sl-2-over-mathbb-z%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Sign up or log in
StackExchange.ready(function () {
StackExchange.helpers.onClickDraftSave('#login-link');
});
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Sign up using Google
Sign up using Facebook
Sign up using Email and Password
Post as a guest
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
Required, but never shown
$begingroup$
You can try working $bmod p$ for all primes $p$.
$endgroup$
– Qiaochu Yuan
Dec 21 '18 at 7:22
$begingroup$
What do you mean by "compute the set"? you seem to implicitly mean that you want up to conjugacy in some sense, which is the main difficulty.
$endgroup$
– YCor
Dec 21 '18 at 15:02