# Deligne's letter to Piatetskii-Shapiro from 1973

Could anyone point me to a place where I could find Deligne's letter to Piatetskii-Shapiro from 1973? It is cited for example in Berkovich's "Vanishing cycles for formal schemes II".

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In my office :-/ – Kevin Buzzard Feb 5 '11 at 14:45
And any place on the Internet I could find it? :) – Przemyslaw Chojecki Feb 5 '11 at 15:33
Is Piatetskii-Shapiro one guy? Is this a Mittag-Leffler situation? – Gunnar Þór Magnússon Feb 6 '11 at 19:16
– Did Feb 7 '11 at 6:31
@Gunnar: that's a good band name right there. "The Mittag-Leffler situation." – Qiaochu Yuan Feb 7 '11 at 23:22

I have typeset Deligne's letter, and placed the result here:

http://www.math.ias.edu/~jaredw/DeligneLetterToPiatetskiShapiro.pdf

I have made some minor edits so that the text reads more naturally to a native speaker of English. Also I made a few annotations where I truly believe there is an error in the original. Any other errors are mine.

The letter struck me with how much it accomplishes in such a short space. Actually, I would say that the meat of the argument is confined to the final three pages, with the rest there only to establish notation. This letter ought to be required reading for anyone studying automorphic forms in arithmetic!

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Thanks! could other letters of Deligne be also made available soon? – SGP Feb 18 '11 at 23:56

I have a scan that I made of the copy in Kevin Buzzard's office some years ago, but I probably shouldn't post it without permission. It's also not the most legible copy - I suspect it is many generations of photocopies old. If someone can get Deligne's permission I'd be happy to make the PDF available - but I bet there are other people here with better copies, in any case.

EDIT: I now have permission, so here is my scan.

[removed]

It's possible that a clearer scan will be forthcoming; I will update this post if so.

EDIT: here is an improved scan, from Deligne's own copy.

http://dl.dropbox.com/u/66812/deligne-pish-improvedscan.pdf

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Based on my experience with the "fundamental group of P^1 - 0,1,infty" paper, Deligne is quite happy to give permission for this kind of thing. – JSE Feb 6 '11 at 16:49
@TG Do you think you really need Deligne's permission? This letter has been published, see my answer and that of KConrad! – Mikhail Borovoi Feb 6 '11 at 18:27
For what it's worth, Deligne says that "I view any mathematical letter I write as being an open letter", so presumably the same permission applies for any other letters that people ask about in future. – TSG Feb 7 '11 at 23:19
I suspect this new copy is as good as it gets. – TSG Feb 8 '11 at 18:37
I would like to typeset Deligne's letter if no one else has begun doing so. I agree we should seek his permission again, but this will be easy because he is probably having tea downstairs at the moment. – Jared Weinstein Feb 9 '11 at 20:45

I think it was published obscurely and in a Russian transcription (the original is in English) .

In endnote 25 of the review of Harris-Taylor on his website here, Milne writes:

[Deligne proved it] in an eleven-page handwritten letter to Piatetskii-Shapiro dated March 25, 1973, with a one-page typed covering letter dated April, 1973: “In it I claim (except at 2) to prove for the supercuspidal representations what in your notes [Antwerp Conference LNM 349] you prove for the principal (unramified) series. The idea is that

1. room is left for it in your notes only thanks to the supersingular elliptic curves;
2. supersingular elliptic curves correspond to ideal classes in the quaternion algebra ramified at p and 1;
3. this, by a global argument using Jacquet Langlands, forces the outcome.”

Apparently (see MR 50 7095), the letter was published: Matematika — Period. Sb. Perevodov Inostrannykh Statei, 18 (1974), 110–122. It would be useful if someone would put it on the web, since it was the starting point for Carayol and H&T. In fact, Deligne’s proof of (b) was completed by J-L. Brylinski (appendix to Carayol 1986).

Below is the complete typed covering letter.

Bures-sur-Yvette, April 2, 1973.

Dear Piatetski-Shapiro [the name is handwritten in Russian]

I am ashamed the enclosed letter is so badly written, and written more for my sake than for yours. In it, I claim (except at 2) to prove for the supersingular cuspidal representations what in your notes you prove for the principal (unramified serie [sic]. The idea is that

a) room is left for it in your notes only thanks to the supersingular elliptic curves;

b) supersingular elliptic curves correspond to ideal classes in the quaternion algebra ramified at $p$ and $\infty$;

c) this, by a global argument using Jacquet Langlands \S 14, forces the outcome

Corollary 1: Let $E$ be an elliptic curve $/\mathbb{Q}$, which is a direct factor (up to isogeny) of a jacobian of a modular curve, corresponding to a new form $\omega_E$. Then, except perhaps at $2$, the conductors of $E$ and $\omega_E$ are the same (At 2, I still can prove $2|f_E\iff 2|f_{\omega_E}$)

Corollary 2: (Casselman-Morita): Let $H$ be a quaternion algebra over $\mathbb{Q}$ split at $\infty$, $p$ be a prime at which $H$ ramifies, $\mathcal{O}$ be an order of $H$ maximal at $p$ and $M=X/\mathcal{O}^*$ [$X=$ Poincar\'e upper half plane]. Then, the irreducible components of the reduction of $M$ mod $p$ are rational. [$M$ is defined over a cyclotomic field unramified at $p$].

Mumford has nice ideas about the compactification of $K\backslash G/\Gamma$ ($K\backslash G$ hermitian symmetric). For the moduli of abelian varieties, I hope it will eventually give a nice compactification over $\mathbb{Z}[\frac{1}{\textrm{obvious primes}}]$.

I am looking forward to meeting you again.

Yours most sincerely,

P. DELIGNE

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Incidentally the copy I have doesn't have the typed covering letter, but has 19 handwritten pages. – TSG Feb 6 '11 at 19:43

Find it here. (Edit: Link removed). I hope you can read Russian. Enjoy!

Edit: Find here another (better?) scan, also in Russian.

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I have wanted to see this letter also, so thank you. Now it just remains for KConrad to post a translation of the text... – Jared Weinstein Feb 6 '11 at 20:55
Jared's referring to the MO post mathoverflow.net/questions/29741/…. Let's wait to see if Toby hears from Deligne soon, since otherwise translating back into English something that was first in English could quickly turn out be a wasted effort. – KConrad Feb 6 '11 at 23:58
Why the question mark on "better?"? It is clearly better! Now all the subscripts can be read. – KConrad Feb 7 '11 at 13:53
I had translated up to the start of section B yesterday. Since Toby has now posted a copy of the original letter in English, further translation of the Russian version is not necessary. It's unfortunate that the scan of the original letter is in such poor condition while the Russian version is so clean. – KConrad Feb 8 '11 at 7:51