What is a Weil-Deligne representation?
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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
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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
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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
2,16411422
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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,
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– Matt E
Sep 2 '11 at 1:25
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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 |
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1 Answer
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1 Answer
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$begingroup$
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
2,16411422
$endgroup$
$begingroup$
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
2,16411422
$endgroup$
$begingroup$
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
2,16411422
$endgroup$
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
2,16411422
edited Dec 30 '18 at 3:34
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,16411422
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,16411422
answered Sep 1 '11 at 22:50
Kevin VentulloKevin Ventullo
2,16411422
2,16411422
$begingroup$
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 |
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