Episode 010, October 2026

Even the prime 2

Unrestricted pro-modularity and Fontaine–Mazur at the prime 2

Covers result 010 in OpenAI’s catalog: all 3 papers. See its connections to other results and the work it cites.

Play Episode 010: Even the prime 2? Fontaine–Mazur modularity at the last remaining prime. Green circles around the odd primes 3 to 17, and a large gold circle around 2.

5:19. English captions available. The video loads from YouTube when you press play.

Chapters

The claim, in plain words

For the elliptic curve y² + y = x³ − x² − 10x − 20, take each prime p, count the solutions modulo p, and subtract the count from p. The resulting list, −2, −1, 1, −2, 1, 4, …, is the list of coefficients of the power series q∏(1 − qⁿ)²(1 − q¹¹ⁿ)², a modular form. Behind the first list sits a Galois representation: the symmetries of all algebraic numbers acting by 2 × 2 matrices with 2-adic entries, whose traces at Frobenius elements reproduce the list. The two-dimensional Fontaine–Mazur conjecture over ℚ predicts that every irreducible, odd representation of this kind that is unramified at all but finitely many primes and de Rham at p with distinct Hodge–Tate weights comes from a modular form, up to a twist.

That was known at every odd prime. Result 010 claims the prime 2 with no restrictions on the representation’s reduction mod 2, in three papers:

Pro-modularity (the catalog’s main paper, 29 pages). Every continuous, odd, absolutely irreducible ρ : G_ℚ → GL₂(E), for E a finite extension of ℚ₂, unramified outside finitely many primes, occurs in the full 2-adic Hecke algebra 𝕋₂(N) for some odd N, with no condition at 2. This is a statement about the big Hecke algebra that glues together modular forms of all weights 2-adically, not yet about a single classical form.
The Hecke algebra (12 pages). Every irreducible component of the full 2-adic Hecke algebra at odd level has Krull dimension 4, as Emerton predicted.
Fontaine–Mazur at 2 (72 pages, posted first). If an irreducible, odd ρ unramified outside finitely many primes is also de Rham at 2 with distinct Hodge–Tate weights, it is a Tate twist of the representation of a classical cuspidal eigenform, for every residual representation, scalar and reducible ones included.

The pro-modularity paper uses the other two as inputs. With the odd-prime results, the last paper would complete the odd, regular, two-dimensional Fontaine–Mazur conjecture over ℚ at every prime.

What it doesn’t claim

Pro-modularity alone doesn’t produce a classical modular form; that needs the de Rham condition with distinct Hodge–Tate weights. Nothing here is about representations of dimension higher than two or over number fields other than ℚ. The elliptic curves in the video were already known to be modular (Wiles; Breuil, Conrad, Diamond and Taylor); they illustrate the statement and are not new cases.

Where it stands

Posted as preprints: OpenAI, September 23 to October 6, 2026
Peer reviewed: Not yet
Checked in Lean: No: OpenAI published no Lean proof for this result

Last checked October 19, 2026. The claim has not been withdrawn; the October 6 editions only update citations. It relies on recent preprints: Jack Thorne’s August 2026 preprint (used by the Fontaine–Mazur paper, and so by the main paper) and, for the odd primes, Pan (2022) and a 2025 preprint by Zhang. We found no public critique. Episode 4’s claimed proof about Hilbert’s tenth problem over ℚ uses the Fontaine–Mazur paper for its weight-two modularity input.

What was known

Eichler (1954) found the match between the curve’s list and the power series above; Wiles and Taylor–Wiles (1995) proved modularity for semistable elliptic curves, the route to Fermat’s Last Theorem.
Kisin (2009) and Emerton proved many cases of Fontaine–Mazur through the p-adic Langlands correspondence. At odd primes, Pan (2022) settled all but one local case at 3, and Zhang (2025 preprint) removed it.
At 2, earlier results restricted the reduction mod 2: Allen (2014) treated nearly ordinary, residually dihedral cases, and Tung (2021) needed nonsolvable residual image. Thorne’s August 2026 preprint handles ordinary cases of weights 0 and 1 at every prime, with no condition on the reduction. Scalar and reducible reductions remained open in general.

How the proof goes

The three stages shown in the video are the Fontaine–Mazur paper’s:

A family. Over a solvable totally real field, split completely at 2, the target is connected through characteristic-2 curves of deformations to a known modular seed, and potential pro-modularity propagates along them. In the scalar and reducible cases, a small proportion of the places above 2 are marked with invariant lines, which keeps information that traces alone would lose; a Selmer calculation keeps the reducible locus from disconnecting the family.
A classical vector. Locally at 2, the p-adic Langlands correspondence (Colmez; Colmez, Dospinescu and Paškūnas) and Paškūnas–Tung block finiteness supply the input. When the target is reducible at 2, a dense set of special weights falls under Thorne’s Theorem D, and a finiteness and polynomial-orbit argument produces a nonzero classical vector at the target.
Descent. The resulting classical form over the totally real field is brought back to ℚ along cyclic steps of prime degree, twisting by Hecke characters.

The pro-modularity paper instead builds a family of trace data, finds a regular de Rham point on it, applies the Fontaine–Mazur theorem there, and returns to ℚ using the Hecke-algebra paper and a Selmer vanishing result of Newton and Thorne.

The examples in the video

For y² + y = x³ − x² − 10x − 20 (the curve 11a1), p minus the number of solutions mod p equals the coefficient of q^p in q∏(1 − qⁿ)²(1 − q¹¹ⁿ)² for every prime below 200, including the bad prime 11. The Frobenius traces of the curve’s 2-adic representation are these numbers for primes other than 2 and 11.
For y² = x³ − x, the corresponding numbers at the odd primes 3 to 41 are 0, −2, 0, 0, 6, 2, 0, 0, −10, 0, −2, 10, and every one at an odd prime below 200 is even: the curve has all its 2-torsion points over ℚ, so its representation mod 2 is trivial.
Both checked with a short Python script.

Read the source

The main paper (PDF), “Unrestricted pro-modularity at the prime two,” October 6, 2026 edition (29 pages).
Fontaine–Mazur modularity at the prime 2 (PDF), October 6, 2026 edition; first posted September 23 (72 pages).
The Dimension of the Two-Adic Hecke Algebra at Odd Level (PDF), October 5, 2026 (12 pages).
The collection’s catalog, where this is result 010.

The 3 papers in this result

OpenAI groups related papers, such as alternative proofs, supporting results and consequences, into one result; we label the first paper it lists the main paper and the others companions.

Unrestricted pro-modularity at the prime two main paper
The Dimension of the Two-Adic Hecke Algebra at Odd Level companion
Fontaine–Mazur modularity at the prime 2 companion

Transcript

Read the full script

0:00Two lists that match

Here are two lists of numbers. The first comes from an elliptic curve: for each prime p, count its solutions modulo p, and subtract that count from p. −2, −1, 1, −2, 1, 4, and so on. The second comes from a power series: q ∏ (1 − qⁿ)² (1 − q¹¹ⁿ)². Read off the coefficient of qᵖ. −2, −1, 1, −2, 1, 4. We checked every prime below 200: the lists agree. This is modularity, first seen by Martin Eichler in 1954, and Andrew Wiles’ route to Fermat’s Last Theorem. The Fontaine–Mazur conjecture says when this should happen. Its two-dimensional case over the rationals is now known at every odd prime. Result 010 in OpenAI’s math repository claims the last remaining prime, 2, with no extra restrictions.

0:58Galois representations

Behind the first list sits a Galois representation. The Galois group collects the symmetries of all algebraic numbers, and the curve turns each symmetry into a 2 × 2 matrix with 2-adic entries. For each prime p other than 2 and 11, a special symmetry, called Frobenius, gets a matrix whose trace is exactly the number on our first list. The conjecture covers irreducible representations like this one that are odd, meaning complex conjugation has determinant −1, that are unramified at all but finitely many primes, and that look geometric at the prime 2, a condition called de Rham, with distinct Hodge–Tate weights. It predicts every such representation comes from a modular form, up to a twist.

1:44Why 2 is hard

Why is 2 hard? The standard method starts from the representation’s reduction modulo the prime, and needs that reduction to be interesting enough. Modulo 2, it can collapse completely. Take the curve y² = x³ − x. Its numbers at odd primes are all even. 0, −2, 0, 0, 6, 2, every one of them. Modulo 2, every trace looks like the trace of the identity matrix. The reduction carries almost no information. Results at 2 came with restrictions on that reduction. In 2021, Shen-Ning Tung needed its image to be nonsolvable. Earlier work handled special dihedral cases. Scalar and reducible reductions stayed open in general, though this particular curve has long been known to be modular by other means.

2:36The claims

Result 010 has three papers. The main one proves pro-modularity: every odd, absolutely irreducible two-dimensional 2-adic representation, unramified outside finitely many primes, occurs in the full 2-adic Hecke algebra, which glues together modular forms of all weights, two-adically. No condition at 2 is needed. A second paper shows that every component of this Hecke algebra has dimension four, as Matthew Emerton predicted. The third, posted first, assumes the de Rham condition and gets the real prize: the representation comes from a classical modular form, up to a twist. That is the two-dimensional Fontaine–Mazur conjecture at 2, for every reduction, including the scalar ones. The main paper uses both of the others.

3:26How the proof goes

The Fontaine–Mazur paper’s proof has three stages. First, put the target in a family. Over a carefully built totally real field, that paper connects it, through curves of deformations, to a representation already known to be modular, and propagates pro-modularity along the way. In the scalar and reducible cases, it marks a small fraction of the places above two with invariant lines, keeping information that traces alone would lose. Second, find a classical form. The 2-adic Langlands correspondence of Pierre Colmez and his coauthors, with block finiteness from Paškūnas and Tung, supplies the local input. When the local picture at 2 splits, a dense set of special weights falls under a theorem posted by Jack Thorne this August, and from them the paper extracts a genuine classical vector at the target. Third, descend: the modular form lives over the bigger field, and the paper brings it back to the rationals, one cyclic step at a time.

4:26Why it matters

If correct, this completes the odd, regular, two-dimensional Fontaine–Mazur conjecture over the rationals at every prime. And it’s load-bearing. Episode 4’s claimed proof that Hilbert’s tenth problem over the rationals is undecidable uses this theorem for its weight-two modularity input.

4:46The caveats

The caveats. These are three preprints, more than a hundred pages, not peer reviewed, and with no Lean proof. The classical result, and through it the main paper, rely on Thorne’s August preprint, and the odd-prime story on Pan’s 2022 paper plus a 2025 preprint by Zhang. Pro-modularity alone doesn’t give a classical form: that needs the de Rham condition, with distinct weights. Two lists that match, now at the prime 2, for every reduction. If the proof holds.

Spot a mistake?

This episode simplifies on purpose, but it shouldn’t be wrong. If something is, open an issue on GitHub or leave a comment on YouTube. Corrections are noted here with a date.