What is a Weil-Deligne representation?
$begingroup$
Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
reference-request representation-theory galois-theory
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
$endgroup$
add a comment |
$begingroup$
Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
reference-request representation-theory galois-theory
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
$endgroup$
add a comment |
$begingroup$
Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
reference-request representation-theory galois-theory
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
$endgroup$
Can anyone explain (or give reference) Weil-Deligne representations, and how they are linked with Galois representations of p-adic field (on l-adic vector spaces or else) ?
reference-request representation-theory galois-theory
reference-request representation-theory galois-theory
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
asked Sep 1 '11 at 21:48
user10676user10676
6,29021737
6,29021737
add a comment |
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The canonical reference is Tate's "Number Theoretic Background" in the Corvallis volume. He discusses the Weil-Deligne group in Section 4. Here is a brief description:
Fix a local field $K$ with perfect residue field of characteristic $pneqell$, and let $W_K$ be the Weil group of $K$ (this is $phi^{-1}(mathbb{Z})$ where $phi:G_Krightarrowhat{mathbb{Z}}$ is the unramified quotient). A Weil-Deligne representation is a pair
$(V,N)$
where $V$ is a char. $0$ representation of $W_K$ on a discrete vector space and $N:Vrightarrow V(-1)$ is a morphism of representations. One important fact is that the category of $ell$-adic representations of $G_K$ embeds fully faithfully into the category of Weil-Deligne representations.
Anyway, to get the full story, you should probably read Tate's article.
EDIT: Another nice reference is Chapter 1 of (the current draft of) a book by Fontaine and Ouyang.
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
$endgroup$
$begingroup$
Dear Kevin, Note that if $ell neq p$, then every $ell$-adic representation of $G_K$ is "potentially semi-stable", and so corresponds to a Weil--Deligne representation. Regards,
$endgroup$
– Matt E
Sep 2 '11 at 1:25
$begingroup$
Dear Matt, Thank you for your comment. I started out trying to write about an arbitrary complete DVF but changed my mind and ended up with some garbage. I have edited to reflect your remark.
$endgroup$
– Kevin Ventullo
Sep 2 '11 at 2:44
add a comment |
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
});
}
});
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%2f61278%2fwhat-is-a-weil-deligne-representation%23new-answer', 'question_page');
}
);
Post as a guest
Required, but never shown
1 Answer
1
active
oldest
votes
1 Answer
1
active
oldest
votes
active
oldest
votes
active
oldest
votes
$begingroup$
The canonical reference is Tate's "Number Theoretic Background" in the Corvallis volume. He discusses the Weil-Deligne group in Section 4. Here is a brief description:
Fix a local field $K$ with perfect residue field of characteristic $pneqell$, and let $W_K$ be the Weil group of $K$ (this is $phi^{-1}(mathbb{Z})$ where $phi:G_Krightarrowhat{mathbb{Z}}$ is the unramified quotient). A Weil-Deligne representation is a pair
$(V,N)$
where $V$ is a char. $0$ representation of $W_K$ on a discrete vector space and $N:Vrightarrow V(-1)$ is a morphism of representations. One important fact is that the category of $ell$-adic representations of $G_K$ embeds fully faithfully into the category of Weil-Deligne representations.
Anyway, to get the full story, you should probably read Tate's article.
EDIT: Another nice reference is Chapter 1 of (the current draft of) a book by Fontaine and Ouyang.
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
$endgroup$
$begingroup$
Dear Kevin, Note that if $ell neq p$, then every $ell$-adic representation of $G_K$ is "potentially semi-stable", and so corresponds to a Weil--Deligne representation. Regards,
$endgroup$
– Matt E
Sep 2 '11 at 1:25
$begingroup$
Dear Matt, Thank you for your comment. I started out trying to write about an arbitrary complete DVF but changed my mind and ended up with some garbage. I have edited to reflect your remark.
$endgroup$
– Kevin Ventullo
Sep 2 '11 at 2:44
add a comment |
$begingroup$
The canonical reference is Tate's "Number Theoretic Background" in the Corvallis volume. He discusses the Weil-Deligne group in Section 4. Here is a brief description:
Fix a local field $K$ with perfect residue field of characteristic $pneqell$, and let $W_K$ be the Weil group of $K$ (this is $phi^{-1}(mathbb{Z})$ where $phi:G_Krightarrowhat{mathbb{Z}}$ is the unramified quotient). A Weil-Deligne representation is a pair
$(V,N)$
where $V$ is a char. $0$ representation of $W_K$ on a discrete vector space and $N:Vrightarrow V(-1)$ is a morphism of representations. One important fact is that the category of $ell$-adic representations of $G_K$ embeds fully faithfully into the category of Weil-Deligne representations.
Anyway, to get the full story, you should probably read Tate's article.
EDIT: Another nice reference is Chapter 1 of (the current draft of) a book by Fontaine and Ouyang.
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
$endgroup$
$begingroup$
Dear Kevin, Note that if $ell neq p$, then every $ell$-adic representation of $G_K$ is "potentially semi-stable", and so corresponds to a Weil--Deligne representation. Regards,
$endgroup$
– Matt E
Sep 2 '11 at 1:25
$begingroup$
Dear Matt, Thank you for your comment. I started out trying to write about an arbitrary complete DVF but changed my mind and ended up with some garbage. I have edited to reflect your remark.
$endgroup$
– Kevin Ventullo
Sep 2 '11 at 2:44
add a comment |
$begingroup$
The canonical reference is Tate's "Number Theoretic Background" in the Corvallis volume. He discusses the Weil-Deligne group in Section 4. Here is a brief description:
Fix a local field $K$ with perfect residue field of characteristic $pneqell$, and let $W_K$ be the Weil group of $K$ (this is $phi^{-1}(mathbb{Z})$ where $phi:G_Krightarrowhat{mathbb{Z}}$ is the unramified quotient). A Weil-Deligne representation is a pair
$(V,N)$
where $V$ is a char. $0$ representation of $W_K$ on a discrete vector space and $N:Vrightarrow V(-1)$ is a morphism of representations. One important fact is that the category of $ell$-adic representations of $G_K$ embeds fully faithfully into the category of Weil-Deligne representations.
Anyway, to get the full story, you should probably read Tate's article.
EDIT: Another nice reference is Chapter 1 of (the current draft of) a book by Fontaine and Ouyang.
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
$endgroup$
The canonical reference is Tate's "Number Theoretic Background" in the Corvallis volume. He discusses the Weil-Deligne group in Section 4. Here is a brief description:
Fix a local field $K$ with perfect residue field of characteristic $pneqell$, and let $W_K$ be the Weil group of $K$ (this is $phi^{-1}(mathbb{Z})$ where $phi:G_Krightarrowhat{mathbb{Z}}$ is the unramified quotient). A Weil-Deligne representation is a pair
$(V,N)$
where $V$ is a char. $0$ representation of $W_K$ on a discrete vector space and $N:Vrightarrow V(-1)$ is a morphism of representations. One important fact is that the category of $ell$-adic representations of $G_K$ embeds fully faithfully into the category of Weil-Deligne representations.
Anyway, to get the full story, you should probably read Tate's article.
EDIT: Another nice reference is Chapter 1 of (the current draft of) a book by Fontaine and Ouyang.
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
edited Dec 30 '18 at 3:34
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,13911422
2,13911422
$begingroup$
Dear Kevin, Note that if $ell neq p$, then every $ell$-adic representation of $G_K$ is "potentially semi-stable", and so corresponds to a Weil--Deligne representation. Regards,
$endgroup$
– Matt E
Sep 2 '11 at 1:25
$begingroup$
Dear Matt, Thank you for your comment. I started out trying to write about an arbitrary complete DVF but changed my mind and ended up with some garbage. I have edited to reflect your remark.
$endgroup$
– Kevin Ventullo
Sep 2 '11 at 2:44
add a comment |
$begingroup$
Dear Kevin, Note that if $ell neq p$, then every $ell$-adic representation of $G_K$ is "potentially semi-stable", and so corresponds to a Weil--Deligne representation. Regards,
$endgroup$
– Matt E
Sep 2 '11 at 1:25
$begingroup$
Dear Matt, Thank you for your comment. I started out trying to write about an arbitrary complete DVF but changed my mind and ended up with some garbage. I have edited to reflect your remark.
$endgroup$
– Kevin Ventullo
Sep 2 '11 at 2:44
$begingroup$
Dear Kevin, Note that if $ell neq p$, then every $ell$-adic representation of $G_K$ is "potentially semi-stable", and so corresponds to a Weil--Deligne representation. Regards,
$endgroup$
– Matt E
Sep 2 '11 at 1:25
$begingroup$
Dear Kevin, Note that if $ell neq p$, then every $ell$-adic representation of $G_K$ is "potentially semi-stable", and so corresponds to a Weil--Deligne representation. Regards,
$endgroup$
– Matt E
Sep 2 '11 at 1:25
$begingroup$
Dear Matt, Thank you for your comment. I started out trying to write about an arbitrary complete DVF but changed my mind and ended up with some garbage. I have edited to reflect your remark.
$endgroup$
– Kevin Ventullo
Sep 2 '11 at 2:44
$begingroup$
Dear Matt, Thank you for your comment. I started out trying to write about an arbitrary complete DVF but changed my mind and ended up with some garbage. I have edited to reflect your remark.
$endgroup$
– Kevin Ventullo
Sep 2 '11 at 2:44
add a comment |
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.
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%2f61278%2fwhat-is-a-weil-deligne-representation%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
