Letter to Serre

Editorial comments: Langlands spent 1967-68 visiting in Ankara, Turkey, and while there wrote this letter to Serre. In it occurs for the first time the question of how to account for `special' representations of the Galois group, such as at primes where an elliptic curve has unstable bad reduction, corresponding to special representations of \(\mathrm{GL}_2\). This correspondence was later expanded to the Deligne-Langlands conjecture, proven eventually by Kazhdan and Lusztig.

Author's comments: This letter is a response to a question of Serre about the gamma-factors appearing in the functional equations of automorphic \(\ell\)-functions. Fortunately Serre's letter to me was accompanied by several reprints, among them apparently the paper Groupes de Lie \(\ell\)-adiqes attachées aux courbes elliptiques that appeared in the volume Les tendances géométriques en algèbre et théorie des nombres.

Although the letter promised in the last line was never written, it is clear what I had in mind. Sometime soon after writing the letter to Weil, perhaps even at the time of writing, I was puzzled by the role of the special representations. The solution of the puzzle was immediately apparent on reading Serre's paper which treated the \(\ell\)-adic representations associated to elliptic curves whose \(j\)-invariant was not integral in the pertinent local field. The special representations of \(\mathrm{GL}(2)\) corresponded to these \(\ell\)-adic representations. The connection between non-semisimple \(\ell\)-adic representations and various kinds of special representations is now generally accepted. The theorem of Kazhdan-Lusztig is a striking example.