Comments on the letter to Weil

Robert Langlands
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Emil Artin and beyond---Class field theory and \(L\)-functions

2015, written 2010
European Mathematical Society

Author's comments: This note Funktorialität in der Theorie der automorphen Formen: Ihre Entdeckung und ihre Ziele was written as commentary to accompany the original letter in a collection of documents on reciprocity laws and algebraic number theory, to appear shortly.

There is a curious ambiguity in the fifth section regarding the location of my office in the old Fine Hall and of a small seminar room. I describe them both as being on the right of the principal entrance, but for my office it is to the right on entering the building, for the seminar room to the right on leaving it. Since I observed this unconscious aspect of my relation to the two rooms only after the article had been published in a book edited by Della Dumbaugh and Joachim Schwermer, I prefer not to make any changes in the article itself.

(July 5, 2015) I add the following letter from James Milne, correcting a careless and incorrect attribution of mine. Unfortunately, it is again too late to correct the article itself, at least as published.

Dear Langlands,

In your article in the Dumbaugh/Schwermer volume, you again credit Borovoi with the proof of Shimura's conjecture. In fact your sentence (p. 205), "Borovois endgültige allgemeine Konstruktion aller Shimuravarietäten war auch von diesem Bericht beeinflusst," makes no sense at all. Borovoi attempted (unsuccessfully) to prove Shimura's conjecture directly. I was certainly the one to prove it via the conjecture in your Bericht.

As I write in an article recently posted on my website (The Riemann Hypothesis....), "Concerning Langlands's conjugacy conjecture itself, this was proved in the following way. For those Shimura varieties with the property that each connected component can be described by the moduli of abelian varieties, Shimura's conjecture was proved in many cases by Shimura and his students and in general by Deligne. To obtain a proof for a general Shimura variety, Piatetski-Shapiro suggested embedding the Shimura variety in a larger Shimura variety that contains many Shimura subvarieties of type $A_{1}$. After Borovoi had unsuccessfully tried to use Piatetski-Shapiro's idea to prove Shimura's conjecture directly, the author used it to prove Langlands's conjugation conjecture, which has Shimura's conjecture as a consequence. No direct proof of Shimura's conjecture is known. Regards,"