# English version of Deligne's paper“Les Constantes des Equations Fonctionnelles Des Fonctions L”

I want to learn something about Weil groups, so I want to read Deligne's paper"Les Constantes des Equations Fonctionnelles Des Fonctions L". But I feel not so comfortable to read a so long French paper. It is better if some one had translated it into English.

Does any one has the English translation of this paper?

• I would be very surprised if anyone had translated it into English. – Joël Mar 26 '11 at 18:52
• Reading mathematics in French is really not very difficult. If you want a very compact dictionary with many math terms in it, I'd suggest math.princeton.edu/~klan/documents/french-glossary.pdf, which has helped many a grad student pass their French language exam. – Jack Huizenga Mar 26 '11 at 19:24
• @Jack Huizenga: I may be wrong, but I'm guessing from the name of the OP that they are likely Chinese. While it might be quite easy for a native english (or italian, or german, ...) speaker to read math in french, it's probably an entirely different thing for a native speaker of an asian language who might've only recently become well-acquainted with english. – Rob Harron Mar 26 '11 at 21:26
• I second Matt Emerton suggestion, and add a strong recommendation to look at Chapter 7 of Bushnell & Henniart's book ''The Local Langlands Conjecture for'' GL(2). It presents (with proofs) the material on Weil groups and their (Weil–Deligne) representations as well as proving the existence of the "local constants", thus covering a lot of what is in Deligne's article. It even has a proof of the $\ell$-adic monodromy theorem previously only available in the appendix to Serre–Tate! – Rob Harron Mar 27 '11 at 4:12
• Perhaps Tate's article Local constants, in Algebraic number fields: $L$-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 89–131. Academic Press, London, 1977, might also be of some help. – Chandan Singh Dalawat Mar 27 '11 at 6:14

While it seems unlikely that Deligne's article has been translated, there are several alternative sources in English that may be of use, including:

1. Tate's article Number-theoretic background from volume 2 of the Corvalis proceedings. It provides a lot of introductory material on Weil groups (some of it taken from Deligne's paper).

2. Chapter 7 of Bushnell & Henniart's book The Local Langlands Conjecture for GL(2) (Springer-Verlag, 2006, MR2234120). It presents (with proofs) the material on Weil groups and their (Weil–Deligne) representations as well as proving the existence of the "local constants", thus covering a lot of what is in Deligne's article. It even has a proof of the ℓ-adic monodromy theorem previously only available in the appendix to Serre–Tate!

3. Tate's article Local constants, in Algebraic number fields: L-functions and Galois properties (Proc. Sympos., Univ. Durham, Durham, 1975), pp. 89–131. Academic Press, London, 1977.

[Anyone reading this should feel free to add more bibliographic information, further references, or additional commentary on the various suggestions.]

• previously only available in the appendix to Serre–Tate! See also Illusie (Luc), Autour du théorème de monodromie locale, Périodes $p$-adiques (Bures-sur-Yvette, 1988). Astérisque No. 223 (1994), 9–57. – Chandan Singh Dalawat Mar 29 '11 at 10:53

(I write this as an answer because I can't leave comments yet; if someone can turn it into a comment, please do it!)

As a complement to the other suggestions, it is perhaps worth pointing out that a nice introduction to Weil(-Deligne) groups can also be found in Knapp's survey article "Introduction to the Langlands program", which is available here.