Unrestricted pro-modularity at the prime two

The explanation

Take the elliptic curve y² + y = x³ − x² − 10x − 20. Count its points modulo each prime p, including one point at infinity. Write a_p for p + 1 minus that count. Then a_2 = −2, a_3 = −1, a_5 = 1, a_7 = −2 and a_13 = 4. Now multiply out q(1 − q)²(1 − q¹¹)²(1 − q²)²(1 − q²²)² and so on forever. The coefficients of q², q³, q⁵, q⁷ and q¹³ are −2, −1, 1, −2 and 4. They match the point counts.

That infinite product is a modular form, a function with a strong symmetry under a group of transformations. The match between counts and coefficients is what "modular" means here. Wiles, Taylor and others proved such matches for every elliptic curve over the rationals by 2001.

The counts can also be read through symmetry. The points of order 2, 4, 8 and so on carry an action of the Galois group. That group collects all symmetries of the algebraic numbers. The action can be written with 2-by-2 matrices of 2-adic numbers. In 2-adic numbers, two numbers are close when their difference is divisible by a high power of 2. For each prime p other than 2 and 11, a special symmetry called Frobenius has trace a_p. This matrix-valued action is a 2-adic Galois representation.

Many representations do not obviously come from a curve. In 1995, Fontaine and Mazur predicted which ones come from geometry. For odd two-dimensional ones over the rationals, the prediction is modularity, up to a standard shift called a Tate twist. Odd means complex conjugation acts with determinant −1. The other condition, regular de Rham at p, is a technical local test at p. It says the representation looks geometric there, with two distinct Hodge–Tate weights.

Earlier work covered most cases. Khare, Wintenberger and Kisin proved Serre's conjecture, which handles the reduction mod p. Pan in 2022 treated many cases with reducible reduction at odd primes. Zhang then finished every odd prime. At 2, earlier theorems needed conditions on the reduction mod 2, such as a large image.

The third paper claims the prime 2 case. Every continuous, irreducible, odd two-dimensional 2-adic representation of this kind is modular up to Tate twistSource 1. It must be unramified outside finitely many primes and regular de Rham at 2. Scalar and reducible reductions mod 2 are includedSource 2.

The first paper claims a statement with no local condition at 2Source 3. Modular forms of different weights can be 2-adically close. For example, E₄ = 1 + 240q + 2160q² + … has every later coefficient divisible by 16. So E₄ is close to the constant 1, a form of weight 0. Hecke operators, written T_ℓ, act on all forms of all weights at once. Completing their algebra 2-adically gives the full Hecke algebra. The paper claims every odd, absolutely irreducible representation of this kind matches a point of this algebra, at some odd levelSource 4. It calls this pro-modularity. It does not claim the representation comes from one classical formSource 3.

The second paper claims a size result. At every odd level, each irreducible component of the full 2-adic Hecke algebra has Krull dimension exactly 4Source 5. Krull dimension counts independent directions of variation. This is the prime 2 case of a conjecture of EmertonSource 6.

The main obstacle is a representation with no geometric condition. The first paper places it in a family of representations of dimension at least 3. It then finds one regular de Rham point in that familySource 7. The third paper's theorem makes that point modular. A dimension count, using the second paper, spreads the match from that point to the whole familySource 8.

The papers depend on each other and on recent outside work, such as a 2026 preprint by Thorne on ordinary representationsSource 9. The Hilbert's tenth problem paper in the same release uses the third paper's theoremSource 10.

How sure is this? No Lean formalization is listed for these papers in the openai/math release. So no computer proof checker has checked them. No specialist review has been published.

What it doesn't show. This does not cover even representations, higher dimensions, or base fields other than the rationals.