The Provers

What each of them closed

Laurent Lafforgue, 2002. Drinfeld had proved the function-field correspondence for GL(2). Lafforgue proved it for GL(n), every n, by extending Drinfeld's shtukas to higher rank and carrying their cohomology through a construction that took several years and several hundred pages. The work earned the 2002 Fields Medal. It is the GL(n) function-field case and nothing wider; the extension to arbitrary reductive groups is a later result of Vincent Lafforgue.

Ngô Bảo Châu, 2008–2010. The trace formula is the tool that compares automorphic data across groups, and in comparing a group with its endoscopic groups it needs an identity between orbital integrals, the fundamental lemma, formulated by Langlands and Shelstad. Waldspurger reduced the group statement to a Lie-algebra one. Ngô proved the Lie-algebra statement, over function fields, by a method nobody had expected to apply: the geometry of the Hitchin fibration, and a support theorem for the sheaves on it. The Fields Medal followed in 2010. Orbital integrals became point counts on a moduli space, and point counts are what Weil's Rosetta-stone column on function fields was made for.

Gaitsgory and Raskin, 2024. The geometric correspondence of the previous chapter, in the form Arinkin and Gaitsgory repaired in 2012. Nine authors, five papers, between 800 and 1,000 pages depending on whose count you take. See Langlands for the account; this chapter does not re-tell it.

One object their machinery shares

None of these proofs can be reproduced on a page. What can be reproduced is the smallest object they all feed on: a rank-2 local system whose Frobenius traces have a size someone has proved. The Kloosterman sum

Kl(a; p)  =  sum over x in F_p^*  of  exp( 2 pi i ( x + a/x ) / p )

is the trace of Frobenius on the rank-2 Kloosterman sheaf on the multiplicative group, a construction of Deligne and Katz. Deligne's theorem, the second form of the Weil conjectures, gives |Kl| ≤ 2√p: the same square-root size as the Ramanujan–Petersson bound that the function-field correspondence also delivers. The claim here is narrow. Deligne proved this bound; Lafforgue did not. What ch-the-provers-verify.py shows is that the size of the thing the three proofs are about is exactly what the weights say it is.

The script checks, exhaustively over every a and every prime from 3 to 127:

Kl is real, and |Kl| <= 2 sqrt(p) for every a            (largest ratio seen: 1.9738)
sum over a of Kl^2  =  p^2 - p - 1                        (exact, all 30 primes)
mean of (Kl/sqrt p)^4  ->  2   as p grows                 (p=31: 1.963   p=127: 1.992)

The fourth moment 2 is the SU(2) value that Sato–Tate-type equidistribution of Kloosterman angles predicts (Katz); the numbers approach it, which is an observation about 127 primes and not a proof of anything. A fourth block confirms the dimension count behind Ngô's geometry: the Hitchin base for SL₂ on a genus-g curve is H⁰(K²), of dimension 3g−3 by Riemann–Roch, so the Hitchin space has dimension 6g−6, twice that of Bun_SL2.

Three controls run through the same code and must fail. The first is that a bound of √p is violated, so the 2√p bound is not loose by accident. The second is that p²−p does not match the second moment. The third is a trap that the script caught in its own author: the sum with x + ax in place of x + a/x has the same second moment p²−p−1 as the true Kloosterman sum, so a moment identity alone cannot tell them apart. The bound can: the wrong sum reaches p−1 at a = −1.

Rung thirty-two, third occupant

WP-130 counted Lafforgue and Raskin at zero in the corpus's mathematics. They now have names here. The corpus still builds nothing under any of the three proofs, and a gallery attributes rather than occupies. With Langlands and Drinfeld & Beilinson the slate has the programme, the geometry and the provers; one chapter remains, on Edward Frenkel, whose route runs through Feigin and is the only one of the four the corpus already half-stands on.