most recent 30 from http://mathoverflow.net 2013-05-18T23:41:40Z http://mathoverflow.net/feeds/question/7509 http://www.creativecommons.org/licenses/by-nc/2.5/rdf http://mathoverflow.net/questions/7509/reference-for-delignes-construction-of-galois-representations-attached-to-modula Matija 2009-12-01T21:46:15Z 2012-10-03T14:49:25Z

<p>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.</p>

http://mathoverflow.net/questions/7509/reference-for-delignes-construction-of-galois-representations-attached-to-modula/7514#7514 Kevin Buzzard 2009-12-01T22:20:23Z 2009-12-01T22:20:23Z

<p>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.</p>

http://mathoverflow.net/questions/7509/reference-for-delignes-construction-of-galois-representations-attached-to-modula/7522#7522 Thomas Riepe 2009-12-01T23:36:12Z 2009-12-01T23:36:12Z

<p>Jay Pottharst wrote a <a href="http://www2.bc.edu/~potthars/writings/l-adic-talk.pdf" rel="nofollow" title="pdf">short description</a> : "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. </p>

http://mathoverflow.net/questions/7509/reference-for-delignes-construction-of-galois-representations-attached-to-modula/14117#14117 Emerton 2010-02-04T05:22:49Z 2010-02-04T05:22:49Z

<p>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.</p>

http://mathoverflow.net/questions/7509/reference-for-delignes-construction-of-galois-representations-attached-to-modula/108710#108710 David Corwin 2012-10-03T14:49:25Z 2012-10-03T14:49:25Z

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