Langlands's Notes on Artin \(L\)-functions

Editorial comments:

• Although a part of these notes have circulated as a rather bulky preprint, they remained, for reasons to be described, incomplete, and even the parts completed were never all typed.
• (2024-01-02): Dwork's thesis referred to below is available here: http://alpha.math.uga.edu/~lorenz/articles.html

Author's comments: One project that was formulated after writing the letter to Weil and that was suggested by his 1967 paper on the Hecke theory was to establish a representation-theoretic form of it and to acquire thereby a clearer notion of the implications of the conjectures. In particular, I suppose although I have no clear memories, it was only after writing the letter that the possibility of local forms of the conjectures, over the reals, the complexes, and nonarchimedean fields, presented themselves. As the theory for \(\mathrm{GL}(2)\) worked itself out, with precise product formulas for the factor appearing in the functional equation, it became clear that, as a consequence of the conjectures in the form they were taking, there would have to be a similar product formula for the analogous factor in the theory of Artin \(L\)-functions.

My office in Ankara was next to that of Cahit Arf, and when I mentioned the question to him, he drew my attention to a paper of Hasse that had appeared in a journal not widely read, the Acta Salmanticensia of 1954. He fortunately had a reprint. So I could begin to think seriously about the matter. The critical idea came in April 1968 in a hotel room in Izmir, where I had gone to deliver a lecture. It was the understanding that all identities needed were consequences of four basic ones, formulated in the notes as the four main lemmas. Once this is understood and basic facts about Gauss sums are understood, as in the papers of Lamprecht and Davenport-Hasse, three of these four identities are not so difficult to establish. The second main lemma turned out, on the other hand, to be a major obstacle. Fortunately while leafing idly through journals in the library, either in Ankara or later in New Haven (I no longer remember), I came across Dwork's paper in which the first and the second main lemmas were proved. Dwork had indeed tried to establish a product formula for what has come to be called the \(\epsilon\)-factor but, without the insight that came from the adelic form of the Hecke theory and the conjectured relations of that to Artin \(L\)-functions, did not appreciate the need to introduce the factor \(\lambda(E/F,\psi_F)\) in condition (iii) of Theorem A. So he fell short of the goal, but fortunately not before he had established these two lemmas, which are indeed far more than lemmas, the proof of the second being a magnificent tour de force of \(p\)-adic analysis. Unfortunately he did not publish a proof, and the only material I had available when writing these notes was the thesis of K. Lakkis which reproduced Dwork's arguments, but only up to sign, and this is of course not enough. Nonetheless although many of the calculations are there, I was never able to work my way through them or put them in a form that was at all publishable. What I put down on paper from my attempts to understand the arguments of Dwork as reproduced by Lakkis is included here as fragmentary Chapters 12 and 13. They are included for what they are worth. Chapter 10, in which the proof of the first main lemma is completed, is also missing. Either it was never written or was misplaced. In any case, the material of Chapters 7, 8, and 9 at hand, the proof of the first main lemma is neither long nor difficult. With the exception of Chapters 10, 11, and 12, and perhaps some easy material that was to have been included in Chapters 8 and 9, the notes are complete. The proof is complete if one accepts the two lemmas of Dwork. Whether the complete proofs, which certainly existed, appeared in his thesis, I do not know, nor do I know whether his notes are still extant.

I abandoned my attempt to prepare a complete manuscript when Deligne observed that it is an easy matter to reverse the arguments and to proceed from the existence of the global \(\epsilon\)-factor, known to exist since Artin introduced the \(L\)-functions, to the existence of the local factors. It suffices to be clearly aware of their defining properties. Since these had escaped a mathematician of Dwork's quality, they cannot be regarded as manifest, or in the words of an eminent French mathematician "peu de chose"! Perhaps he was misled once again by partisan sentiments.

What of any possible use remains of the arguments here? First of all a general lemma about the structure of relations between induced representations of nilpotent groups that is conceivably of interest beyond the purposes of these notes, but that has never, so far as I know, found application elsewhere. Perhaps of more importance: although the local proof, which could be reconstructed from Dwork's notes and the material here, is far too long, a global proof of a local lemma is also not satisfactory. So the problem of finding a satisfactory local proof remains open.

The local \(\epsilon\)-factor is often incorporated into characterizations of the local correspondence for \(\mathrm{GL}(n)\). This is also unsatisfactory. The only real criterion for deciding whether a local correspondence is correct is that it be compatible firstly with the global correspondence and secondly with localization for representations of the Galois groups on one hand and automorphic representations on the other. Such a local correspondence established, the existence of the \(\epsilon\)-factor is immediate. At present, however, all aspects of the theory are rudimentary and inchoate. What may ultimately happen---I am not inclined to predictions in the matter---is that the existence of the local correspondence and of the \(\epsilon\)-factor will be established simultaneously, and that some of the arguments of these notes will reappear, but supplemented with information about the representations of \(\mathrm{GL}(n)\) over nonarchimedean fields.

I stress that these notes were written about 1970. I have not examined them in the intervening years with any care. There may be slips of the pen and even small mathematical errors.