# Reference for Deligne's construction of Galois representations attached to modular forms

I was wondering if anyone can suggest some good reference for learning more about Deligne's construction of Galois representations attached to modular forms. I find Deligne's original paper hard to read.

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I know that at one point Brian Conrad was writing a book on such matters, but I think there was some sort of overflow error when the book attained infinite length and now the link on his web page doesn't work (and may even not be there any more). I only hope Brian survived. If you ask him, telling him you want to proof read it, maybe he'll send you something. Be warned through, he never chooses a square root of -1. –  Kevin Buzzard Dec 1 '09 at 22:12
I believe that the book that buzzard was referring to is about to be published by Cambrdige University Press. It is listed on Amazon: amazon.com/Modular-Forms-Ramanujan-Conjecture-Conrad/dp/… though it does not appear to be available quite yet. –  Ben Linowitz Dec 13 '09 at 0:47

I absolutely agree that Deligne is very terse. One thing that I ultimately found very helpful is Carayol's two papers where he proves the analogous theorem for Hilbert modular forms. I say "ultimately" because it took me a long time to read those papers. I would come back to them every few years and learn more, as I matured mathematically. The big problems with using Carayol to understand Deligne will be: (1) Carayol has to work much harder in places than Deligne, because the Shimura curves he uses are not the solution to a moduli problem of abelian varieties plus extra structure, so he has to use extra tricks which Deligne did not have to get into, and this will obfuscate things (I guess perhaps this is only when analysing the bad reduction of the curves, which is perhaps not the paper you'd be wanting to read anyway) and (2) Deligne had to deal with the fact that modular curves need compactifying, so he had to work with parabolic cohomology, which is a technicality he has to deal with and Carayol doesn't. But for the main part, the techniques are the same.

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what are the two papers? I found the one on Ann. Sci. de l'ENS. –  natura Feb 8 '10 at 23:13
From Math Reviews. The first is MR0860139 (88a:11058) Carayol, Henri Sur la mauvaise réduction des courbes de Shimura. (French) [Bad reduction of Shimura curves] Compositio Math. 59 (1986), no. 2, 151--230. (Reviewer: Ernst-Ulrich Gekeler) 11G18 (11G15 14H25 14K22) which computes a lot of explicit stuff about the reduction of the modular curves needed. And the second, the one that constructs the Galois representations and proves local-global for them, is, erm, in the following message. –  Kevin Buzzard Feb 9 '10 at 11:56
MR0870690 (89c:11083) Carayol, Henri Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert. (French) [On $l$-adic representations associated with Hilbert modular forms] Ann. Sci. École Norm. Sup. (4) 19 (1986), no. 3, 409--468. –  Kevin Buzzard Feb 9 '10 at 11:56

Tony Scholl has a paper in which he expands on Deligne's construction in such a way as to explain how to construct Grothendieck motives attached to cuspidal eigenforms. The main focus of Scholl's paper (if I remember correctly) is how to take into account the need for compactification in a precise motivic fashion; still, he gives a useful reprise of Deligne. In any event, depending on your particular difficulties with Deligne, you may find Scholl's presentation helpful.

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can you tell us which paper that is? I couldn't figure it out... –  natura Feb 8 '10 at 23:12
It's a paper of Scholl, in Inventiones, called Motives for modular forms'', or something very similar. –  Emerton Feb 9 '10 at 2:16

Jay Pottharst wrote a short description : "In his famous Bourbaki talk, Deligne described a recipe for attaching -adic Galois representations to elliptic modular forms of integral weight at least 2. As a consequence of the method, one reduces the Ramanujan–Petersson conjecture to the validity of Weil’s Riemann Hypothesis for varieties over finite fields. There seems to exist no brief and precise outline of Deligne’s recipe in circulation, and this note is intended to close this gap in the literature." Conc. Carayol's articles: I found his articles difficult to read, but one learns a lot from them.

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These notes by Takeshi Saito also give a very helpful overview: www.ms.u-tokyo.ac.jp/~t-saito/talk/eepr.pdf

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