Drinfeld & Beilinson

A curve over a finite field is a number field that holds still

The previous chapter named three strands. The function-field one is the one where proofs came first, and the reason is the second column of Weil's Rosetta stone: a curve C over a finite field F_q has a geometry, and a number field does not. The objects the correspondence talks about — automorphic forms on GL(n) — become functions on the set of rank-n vector bundles on C, and bundles can be moduli-counted, compared, and degenerated.

Drinfeld's route to GL(2) ran through objects he invented for the purpose: elliptic modules (now Drinfeld modules), and then shtukas, which are bundles with a Frobenius-twisted gluing. The Galois side of the correspondence is carved out of their cohomology. That is the content of the function-field proof, and nothing in this corpus reproduces it.

Bun_G, and what Beilinson and Drinfeld asked of it

Replace “functions on bundles over F_q” by “sheaves on the moduli stack Bun_G of G-bundles on a curve”, and the finite field drops out of the statement. A function becomes a sheaf by Grothendieck's dictionary — the function is the trace of Frobenius on the stalks — and the arithmetic content is recovered by specialising back. What remains is a conjecture about complex curves. On one side, D-modules (or sheaves) on Bun_G; on the other, local systems for the dual group ^LG. The correspondence sends a local system to a Hecke eigensheaf, a sheaf that the Hecke operators merely rescale by the local system's own monodromy.

Beilinson and Drinfeld built such eigensheaves for a special class of local systems, the opers, by quantising the Hitchin integrable system on the cotangent bundle of Bun_G. The same opers are what Feigin and Frenkel find at the critical level, which is why ch-feigin and this chapter describe two ends of one construction: the centre of the affine algebra at critical level on a disc, and the differential operators on Bun_G on a whole curve.

The volume of Bun₂ on the projective line

Nothing about the correspondence itself is checkable by hand. One number about the space it lives on is. Over F_q, every rank-2 bundle on P¹ is a sum of two line bundles O(a) ⊕ O(b) (Grothendieck, 1957), so the groupoid is completely enumerable. Count each bundle E with weight 1/|Aut E|, which is how a stack is counted. For a > b the automorphism group has order (q−1)² · q^a−b+1 (two units on the diagonal, and Hom(O(b), O(a)) = H⁰(O(a−b)) of dimension a−b+1); for a = b it is GL₂(F_q).

The Siegel–Harder mass formula says the whole sum, in any fixed degree, is q⁻³/(q−1) · ζ_P¹(2), where ζ_P¹(s) = 1/((1−q^−s)(1−q^1−s)). The script enumerates and compares in exact rational arithmetic:

degree 0:   1/|GL_2(F_q)| + sum_{a>=1} 1/((q-1)^2 q^(2a+1))
degree 1:   sum_{a>=1} 1/((q-1)^2 q^(2a))
both equal  q^-3 / (q-1) * zeta_P1(2)         for every q tested

They match for eighteen field sizes from 2 to 32, in every degree from −3 to 5, with the infinite series compared against both its closed form and an 80-bundle partial sum. Multiplying the mass by q⁴ gives 1.002 at q = 10³ and 1.000000002 at 10⁹: the stack has dimension (g−1)·dim GL₂ = −4, which is the one place the phrase “dimension of the moduli stack” is a number you can watch converge. Four controls run through the same code — a wrong automorphism order, a dropped ζ factor, a wrong Hom dimension, and the rank-3 mass — and each must fail to match. They do.

Rung thirty-two, second occupant

WP-130 found Drinfeld and Beilinson at zero occurrences, and the single mention of Gaitsgory in a list of sources to obtain. This chapter gives them names and one number; it does not build the correspondence. Taken with Langlands and Feigin, the gallery now attributes the arithmetic strand, the geometric strand's formulation, and the vertex-algebra route through the critical level. It still occupies none of them.