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Editorial comments: In January of 1967, while he was at Princeton University, Langlands wrote a letter of 17 hand-written pages to Andre Weil outlining what quickly became known as `the Langlands conjectures'. This letter even today is worth reading carefully, although its notation is by present standards somewhat clumsy. It was in this letter that what later became known as the `\(L\)-group' first made its appearance, like Gargantua, surprisingly mature. Because of its historic importance, we give here two versions of this letter, as well as a pair of supplementary notes accompanying it. A typed copy of this letter, made at Weil's request for easier reading, circulated widely among specialists in the late sixties and seventies. The covering note from Harish-Chandra has been helpful in establishing a date for the letter, which is itself undated.
In reply to a question asked by many: there was no written reply from Weil.
Author's comments: The letter to Weil is undated. However, thanks to David Lieberman, I was able to discover that Chern's talk in the IDA Mathematics Colloquium was held on January 6, 1967. Thus the letter was written between then and the date January 16 that appears in the note of Harish-Chandra.
In order to make it easier for Weil to read, the handwritten note was typed some days later. The four footnotes were then added and one or two phrases were modified for the sake of clarity. These modifications are incorporated into the present version. Otherwise the letter has been allowed to stand as it was. Even unfortunate grammatical errors have not been corrected.
The emphasis on explicit, concrete reciprocity laws may surprise the reader. The note A little bit of number theory will clarify what I had in mind.