# Deligne's letter to Jean-Pierre Serre

I'm looking for another letter of Pierre Deligne, this time to Jean-Pierre Serre (from around 1974 I think), in which he proves that the Galois representation associated to a certain Hecke eigenform is of specific form at p. This letter is mentioned, for example, in the remark 1.4 in here, or in the paper of B. Gross "A tameness criterion for Galois representations associated to modular forms (mod p)".

Additional question: Is it the same letter that is mentioned in Buzzard, Gee "The conjectural connections between automorphic representations and Galois representations"?

I will be grateful for your help.

edit: I'm pushing the question up. Does anyone have the mentioned letter?

• The letter mentioned in Buzzard-Gee is a different letter, written only a few years ago. – Kevin Buzzard Nov 23 '11 at 19:40
• In retrospect, your comment shows me that we should have said something like "a 2007 letter from Deligne to Serre" or something. Thanks. I'll change this before it appears in print. – Kevin Buzzard Nov 23 '11 at 19:41
• I wonder if we'll ever start getting these kind of questions about e-mails. – Gunnar Þór Magnússon Nov 23 '11 at 19:44
• @Kevin: I sometimes use the "poor-man's method" of editing comments: I copy the text of the old comment into a new comment, edit it, post the revised version, and delete the old comment. – Timothy Chow Nov 23 '11 at 20:04
• Dear Przemyslaw, I have a copy of the letter (which I got from Dick Gross many years ago). I can send one to you. (I should have time in the next week or so.) Regards, Matthew P.S. Its arguments are different to those in the literature, e.g. in Mazur and/or Wiles. It more directly uses $p$-adic vanishing cycles. – Emerton Dec 3 '11 at 7:22