He wrote seventeen pages to a man who already has a chapter in this gallery, apologised for them, and offered him a waste basket. The letter is the reason rung 32 has a name. This chapter is here because the corpus has been using that name without having built under it.
In January 1967 Langlands, then thirty, wrote to André Weil. The letter ran seventeen handwritten pages and set out a correspondence between two subjects that had no business being related: the arithmetic of Galois groups on one side, and harmonic analysis on the other. He ended by telling Weil that if he was willing to read it as pure speculation he would appreciate it — and if not, that he was sure Weil had a waste basket handy.
Weil did not use the basket. He had it typed, and the typescript circulated. What was in it became the Langlands conjectures.
The organising image in that correspondence is older than the letter and also belongs to Weil. Writing to his sister Simone from a prison in Rouen in 1940, Weil described an analogy in three columns — number fields, function fields over a finite field, and Riemann surfaces — three settings in which the same statements keep reappearing in different dialects. He called it a Rosetta stone. The Langlands program has three strands for the same reason that letter has three columns, and Weil's own chapter in this gallery is about the point at which one of those columns stops translating.
Arithmetic. For a global field F and a reductive group G over it, a conjectural correspondence between automorphic representations of G(AF) and parameters valued in the dual group LG — a construction Langlands introduced, and which carries his name. Class field theory is the abelian case, G = GL(1). Modularity of elliptic curves over Q is a case of GL(2).
Function fields. Drinfeld proved GL(2) in 1983; Drinfeld and Gérard Laumon formulated the general GL(n) geometric statement in 1987; Laurent Lafforgue proved GL(n), Fields Medal 2002. The trace-formula machinery the arithmetic side needs rested on a statement Langlands and Shelstad conjectured and could not prove, the fundamental lemma, which Ngô Bảo Châu proved — Fields Medal 2010.
Geometric. A correspondence between sheaves on the moduli of G-bundles over a curve and the moduli of LG local systems; the formulation is Beilinson and Drinfeld's. Proved in 2024 by a nine-person team led by Dennis Gaitsgory and Sam Raskin, with Dima Arinkin, Nick Rozenblyum, Joakim Færgeman, Dario Beraldo, Lin Chen, Justin Campbell and Kevin Lin, across five papers. Gaitsgory took the 2025 Breakthrough Prize in Mathematics for it; Raskin a New Horizons Prize. Sources disagree on the length — Quanta reports more than 800 pages, Wikipedia more than 1,000 — and this chapter does not pick one.
One detail belongs in this corpus more than the rest of it does. Beilinson and Drinfeld's conjecture could not be proved as stated. By Quanta's account their “best hope” had been “a bit too simplistic about how the relationships… should work”, and Gaitsgory and Arinkin made it into a precise conjecture in 2012 — twenty-five years after the formulation, twelve before the proof.
A conjecture whose statement had to be repaired before it could be true is the same object this project has now caught four times in its own Lean: a name that promised more than its statement said. The difference is that theirs was repaired in public, by the people who went on to prove the repaired version.
Almost none of this is checkable without machinery. Two cases are, and
ch-langlands-verify.py checks both exhaustively, in the standard library.
GL(1): reciprocity. Quadratic reciprocity is the abelian case of the correspondence, and it is the oldest piece of the subject — the statement that for distinct odd primes p, q, whether p is a square mod q determines whether q is a square mod p, up to a sign fixed by the residues mod 4. The script verifies it over every ordered pair of odd primes below 200, by exhaustive residue computation rather than by the law.
GL(2): modularity. Take the elliptic curve y² + y = x³ − x², of conductor 11, and for each prime p count its points over Fp by brute force, setting ap = p + 1 − #E(Fp). Separately, expand the eta product q∏(1−qn)²(1−q11n)², the weight-two newform of level 11, by polynomial multiplication. The two sequences have nothing to do with each other: one counts solutions of a cubic over finite fields, the other is a coefficient in a product of power series.
They agree at every prime below 100:
p 2 3 5 7 13 17 19 23 29 31 37 41 a_p -2 -1 1 -2 4 -2 0 -1 0 7 3 -8 p 43 47 53 59 61 67 71 73 79 83 89 97 a_p -6 8 -6 5 12 -7 -3 4 -10 -6 15 -7
That is the Langlands correspondence, in the one place a reader can watch both sides being computed and see them land on the same integers. It is also, in a sense that is not decorative, the Taniyama–Shimura case that Wiles proved on the way to Fermat — and Wiles is the man Langlands shared the Wolf Prize with.
This chapter is in the gallery for the same reason Grothendieck's is: to name a rung the corpus has been standing on without having built.
WP-130 measured it. Nine tracked files carry the word Langlands; three are index pages, three are inventories recording that the material is wanted, one is the paper that measured the absence. Two carry mathematics. Drinfeld, Beilinson, Harish-Chandra and fundamental lemma occurred zero times; the single mention of Gaitsgory was in a list of sources to obtain.
Nothing in this corpus reaches the Langlands program, and this chapter does not claim otherwise. WP-82's correction of 2026-09-11 narrowed rung 32 to the vertex-algebra route — Feigin–Frenkel, z(ĝ) ≅ W(Lg), which ch-feigin carries — precisely because that was the part that could be reached. That judgement stands. A gallery attributes; it does not occupy.
MacTutor History of Mathematics, biography of Robert P. Langlands (birth date and
place, positions, prizes, the January 1967 letter and its closing line).
· Quanta Magazine, “Monumental Proof Settles Geometric Langlands
Conjecture”, 19 July 2024 (the nine-person team, five papers, the 2012 repair).
· Wikipedia, Geometric Langlands correspondence (Drinfeld 1983,
Drinfeld–Laumon 1987). · Frenkel, Recent advances in the Langlands
program (Berkeley).
· Checks: book7/ch-langlands-verify.py.
This chapter makes no claim about Langlands' present circumstances; the sources consulted confirm his biography only to the dates given.