The Langlands program in one prime

Around Christmas 1640, Fermat told Mersenne that an odd prime is the sum of two squares exactly when it leaves remainder one on division by four. Pull on that thread for three and a half centuries and you arrive at the Langlands program — the most ambitious system of conjectures in mathematics.

Start with the observation itself. 5 is 1 + 4. 13 is 4 + 9. 29 is 4 + 25. 41 is 16 + 25. But 3, 7, 11, 19 and 23 cannot be written as a sum of two squares no matter how long you try, and what separates the two lists is a remainder: the primes that work leave 1 on division by 4, the ones that refuse leave 3. Fermat announced the pattern without proof; the proof took another century and Euler.¹

The right question is why a congruence — the crudest possible information about a prime — should know anything about squares. The modern answer is that p is a sum of two squares precisely when −1 has a square root modulo p, which is to say when the polynomial x² + 1 splits into two factors mod p. Fermat's observation is a splitting law: it describes how one fixed polynomial behaves at every prime, and the answer depends only on the prime's residue. Quadratic #reciprocity, Gauss's golden theorem, extends exactly this kind of rule to every quadratic polynomial.

Class field theory then pushed the idea to its natural limit: whenever a polynomial's symmetry group — its Galois group — is commutative, its splitting behaviour is governed by congruence conditions on p. And there the road ends. For a polynomial like x⁵ − x − 1, whose Galois group is not commutative, no congruence rule exists — provably. The splitting data is real, prime by prime, but it has to be stored somewhere else.²

Langlands' claim is that it is stored in analysis. To each #galois-representations package — the condensed symmetry data of a polynomial — there should correspond an automorphic form: a function of extreme symmetry living on a continuous space, whose Fourier coefficients encode the splitting of every prime at once. Number theory in one column, harmonic analysis in the other, and a dictionary between them in which congruence conditions were only ever the first, commutative page. The #automorphic-forms hold the rest.

A letter and a wastebasket

The conjectures arrived in January 1967 as a seventeen-page handwritten letter from @Langlands, then thirty years old, to @Weil, opened with a covering note of famous diffidence: if Weil would read it as pure speculation, he would be grateful — "if not, I am sure you have a waste basket handy." Weil had it typed instead, and the typescript circulated for decades, quietly reorganising number theory.³

The strongest single piece of evidence is now a theorem. Wiles' proof that elliptic curves are modular is precisely one line of the dictionary — a two-dimensional Galois representation matched with a weight-two modular form — and Fermat's Last Theorem fell out of it as a corollary. When a 350-year-old problem dies as a side effect, the dictionary is carrying real weight.

Held in one prime

Which is why the sum of two squares deserves its billing. On the Galois side sits a question: how does x² + 1 factor modulo p? On the automorphic side sits the answer: the simplest automorphic form there is, a character that sees nothing about p except its remainder mod 4. Fermat's Christmas observation is the one-dimensional, commutative entry of the Langlands correspondence — the single prime pattern in which the whole program is already visible. Everything since is the same sentence with the word commutative deleted.