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) ?
What is a Weil-Deligne representation?
1 Answers
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 $p\neq\ell$, and let $W_K$ be the Weil group of $K$ (this is $\phi^{-1}(\mathbb{Z})$ where $\phi:G_K\rightarrow\hat{\mathbb{Z}}$ 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 $N:V\rightarrow V(-1)$ 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.
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0Dear 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. – 2011-09-02