Modularity at two: geometric representations and completed Hecke families

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Family 010 · Number theory
Importance 5/5 · Provisional editorial rating
Sources consulted 2026-10-10

1. The question

Algebraic numbers have symmetries: an automorphism can permute roots of polynomial equations while preserving addition and multiplication. Modular forms, by contrast, are highly symmetric analytic functions. A major theme of modern number theory is that these two worlds encode the same information in different languages.

The Fontaine–Mazur conjecture predicts that suitably constrained representations of arithmetic symmetries come from geometry. In the odd, regular, two-dimensional setting over $\mathbb Q$, this predicts a more specific source: classical modular forms. Family 010 claims the remaining unrestricted result at the prime two, including representations whose reduction modulo two is reducible or scalar.

The family also contains a broader occurrence theorem without the geometric local hypothesis, and a theorem describing the dimension of the completed Hecke algebra in which those representations occur. Occurrence in that large algebra is called pro-modularity. It does not by itself mean that a representation comes from one classical form. Understanding this distinction is the organizing theme of the three manuscripts.

2. Background and essential definitions

Arithmetic symmetries as matrices

The absolute Galois group

$$G_{\mathbb Q}=\operatorname{Gal}(\overline{\mathbb Q}/\mathbb Q)$$

consists of automorphisms of the algebraic numbers that fix the rationals. A two-dimensional 2-adic representation is a continuous homomorphism

$$r:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(E),$$

where $E$ is a finite extension of the 2-adic number field $\mathbb Q_2$. The 2-adic metric regards numbers as close when their difference is divisible by a high power of two. Continuity makes the matrix action compatible with this arithmetic topology.

The representation is absolutely irreducible if it has no invariant line even after extending its coefficient field. It is odd if complex conjugation $c$ satisfies

$$\det r(c)=-1.$$

Being unramified outside finitely many primes means that away from a finite exceptional set, inertia, the local subgroup measuring ramification, acts trivially. At those good primes $\ell$, the action of a Frobenius element has a well-defined characteristic polynomial.

A stable integral lattice allows reduction of the matrices modulo the maximal ideal of the coefficient ring. This gives the residual representation. An irreducible characteristic-zero representation can have a reducible residual representation; a scalar residual representation has every matrix proportional to the identity. At residue characteristic two, several distinctions visible in characteristic zero collapse, making deformation arguments delicate.

Geometric local conditions and modular forms

The de Rham condition is a local condition at two from 2-adic Hodge theory. It requires a full period comparison with a filtered vector space, of the kind supplied by algebraic geometry. The integers recording the filtration are the Hodge–Tate weights. In dimension two, regular means these two weights are distinct. These hypotheses are stronger than continuity or finite ramification.

A modular form is a holomorphic function on the upper half-plane satisfying a prescribed transformation law under integer matrices and suitable conditions at the boundary. Its weight specifies a power in the transformation law, and its level specifies the congruence subgroup. A cusp form vanishes at the cusps. An eigenform is a simultaneous eigenvector for commuting arithmetic operators called Hecke operators.

Classical eigenforms have associated Galois representations whose Frobenius traces and determinants match Hecke eigenvalues. A Tate twist tensors a representation with an integer power of the cyclotomic character, which records the action on roots of unity; it shifts both Hodge–Tate weights together.

The completed Hecke algebra collects compatible congruences among forms of all weights. A point of this algebra is a compatible eigenvalue system, potentially a 2-adic limit rather than one classical form of fixed weight. Its spectrum is an algebraic parameter space. Krull dimension measures the maximum length of chains of irreducible algebraic subsets, not the vector-space dimension of a space of forms.

3. Development history and previous partial results

Wiles and Taylor–Wiles developed modularity lifting by comparing rings parametrizing Galois deformations with Hecke algebras parametrizing modular eigenvalues. A deformation is a representation lifting a fixed residual representation. This framework made the passage from a modular residual system to a specified characteristic-zero lift a central problem.

The proof of Serre’s conjecture supplied modularity for odd irreducible residual representations, including characteristic two. It did not automatically prove modularity of every lift. Reducible residual representations also require different sources of modular points. Skinner–Wiles developed ordinary-family methods in this setting, while Kisin and Emerton brought local deformation geometry and the 2-adic and more general p-adic Langlands framework into the picture.

Pan’s work treated residually reducible representations and, with earlier results, established the regular two-dimensional conjecture for primes at least five. Zhang’s work at three removed the remaining odd-prime configuration. M3 places its claim at the remaining prime two.

There had already been important dyadic results. Tung’s modularity lifting theorem removed a local restriction while retaining a globally nonsolvable residual-image hypothesis. Covering a scalar restriction at the local prime two is different from covering a globally scalar residual representation. M3 claims all global residual cases.

A second history concerns families. Hida’s ordinary families, the Coleman–Mazur eigencurve, and Gouvêa–Mazur’s infinite fern revealed how modular forms vary and how families meet. Emerton formulated a dimension conjecture for the full varying-weight Hecke algebra and established the lower bound four. Newton and Thorne’s adjoint Selmer vanishing theorem supplies a key ingredient for the upper bound in M2. These precedents explain why the dimension result and modularity claims interact.

4. What the new manuscripts claim

M3: classical Fontaine–Mazur modularity

M3, Theorem 1.1, assumes that a continuous irreducible two-dimensional representation over $\overline{\mathbb Q}_2$ is odd, unramified outside finitely many primes, and de Rham at two with distinct Hodge–Tate weights. It claims that the representation is a Tate twist of one associated with a classical cuspidal eigenform.

There is no hypothesis excluding scalar or reducible semisimple residual representations. This is the odd, regular, two-dimensional Fontaine–Mazur statement over $\mathbb Q$ at two, not the conjecture in arbitrary dimension or over arbitrary number fields.

M1: occurrence without a de Rham condition

M1, Theorem 1.1, starts with a continuous, absolutely irreducible

$$r:G_{\mathbb Q}\longrightarrow\operatorname{GL}_2(E)$$

for finite $E/\mathbb Q_2$, assuming oddness and finite ramification. It imposes no de Rham condition at two.

Its conclusion is a continuous homomorphism

$$\lambda:\mathbb T_2(N)\longrightarrow\mathcal O_E$$

for some odd positive integer $N$, such that at every prime $\ell\nmid2N$,

$$\lambda(T_\ell)=\operatorname{tr}r(\operatorname{Frob}_\ell), \qquad \lambda(\ell S_\ell)=\det r(\operatorname{Frob}_\ell).$$

Here $\mathcal O_E$ is the integral coefficient ring and $T_\ell,\ell S_\ell$ are the Hecke operators normalized to give the characteristic polynomial

$$Z^2-T_\ell Z+\ell S_\ell.$$

The odd level $N$ contains every odd ramified prime but may include auxiliary primes. The theorem does not promise occurrence at the smallest possible level or at a prescribed set of tame primes. All residual representations are allowed.

M2: dimension of the full algebra

For fixed odd $N$, let the finite-stage Hecke algebra act on all holomorphic modular forms of weights one through $k$ and level $\Gamma_1(N)$, including Eisenstein forms. Using the operators away from $2N$, complete at two and pass to the inverse limit over $k$. This is $\mathbb T_2(N)$.

M2, Theorem 1.1, claims that every irreducible component of its spectrum has Krull dimension four. This is the integral algebra over $\mathbb Z_2$, with all weights and all residual systems retained. It is not a claim that each eigencurve or fixed-weight space has dimension four.

5. Why the advance matters

Provisional importance: 5/5. The regular two-dimensional Fontaine–Mazur conjecture is a central bridge between abstract arithmetic representations and classical modular forms. Removing all residual restrictions at the smallest prime would complete that setting over $\mathbb Q$, alongside the stated odd-prime results.

The companion occurrence theorem reaches representations beyond the geometric local class, placing them in a coherent modular parameter space. The dimension theorem describes the size of that full space and supports the mechanism by which modularity propagates through families.

The classical result also has downstream arithmetic uses. The height argument for Hilbert’s tenth problem in family 004 invokes its weight-two modularity consequence without a residual-image hypothesis. That application needs additional arithmetic and logic; it is not an immediate restatement of Fontaine–Mazur. The score concerns the combined significance of the claims if correct.

6. The main idea

Establish classicality through an auxiliary field

M3 first places the target in completed spaces of automorphic forms over a solvable totally real extension $F/\mathbb Q$, chosen to split completely at two. Totally real means every embedding of $F$ into $\mathbb C$ lands in $\mathbb R$; solvability allows descent through a sequence of manageable field extensions.

The construction works with whole deformation families. A trace records less information than an actual matrix representation, especially when the residual representation is scalar or reducible. M3 temporarily records invariant lines at selected dyadic places. Their positions impose additional constraints that make globally reducible loci small enough for a connectedness argument to pass modular occurrence from a known seed to the target family.

Occurrence is then upgraded to a classical vector. If the local representation at two is irreducible, local Langlands theory supplies locally algebraic vectors. In the reducible local case, M3 constructs a family carrying both trace and a chosen character. A dense set of specially selected weights becomes classical through Thorne’s ordinary modularity theorem. The manuscript transfers the character information to the target and uses a polynomial-orbit argument to produce a genuine locally algebraic vector there.

Only after obtaining a classical automorphic representation over $F$ does the argument descend to $\mathbb Q$. Finite-character discrepancies are corrected by twists along the solvable tower. This order matters: the manuscript does not assume a descent theorem for arbitrary pro-modular loci.

Determine the dimension at classical points

M2 combines a global lower bound with a local upper bound. It first treats the completed algebra as a finite product of complete Noetherian local rings, so its algebraic geometry is controlled by finitely generated local ideals. At two it uses determinant laws retaining both trace and determinant, avoiding formulas that would divide by two.

Classical points are dense, and the closure of Eisenstein points has dimension at most two. Since every component has dimension at least four by Emerton’s theorem, each component contains a classical cuspidal point of weight at least three.

At such a point, first-order Galois deformations are measured by adjoint cohomology. The adjoint representation describes the action on endomorphisms of the two-dimensional representation. Newton–Thorne’s vanishing theorem eliminates the global finite Selmer contribution, leaving at most three local tangent directions. A dimension comparison adds the one-dimensional arithmetic coefficient direction, giving the upper bound four and matching the lower bound. The argument allows a reducible residual system because the chosen classical point is in characteristic zero.

Spread Hecke occurrence to a nongeometric target

M1 places its unrestricted target in a sufficiently large fixed-determinant family and obtains potential Hecke occurrence after base change. It then constructs an auxiliary regular de Rham point on that family. M3 makes this auxiliary point classical; the original target need not become de Rham.

At that classical point, M2 and the adjoint Selmer bound force the local Hecke and deformation dimensions to agree. The manuscript uses this to show that the kernel cutting out Hecke occurrence vanishes on the relevant family. Specializing back gives the desired eigenvalue system for the original target.

Thus M1 uses M3 and M2. Its broader hypothesis is balanced by its weaker conclusion: a point in the completed algebra, with auxiliary tame primes permitted, rather than a classical eigenform.

7. Scope, limitations, and open questions

M3 requires oddness, dimension two, finite ramification, and regular de Rham behavior at two. Equal Hodge–Tate weights, even representations, higher dimensions, and other base fields are outside its theorem.

M1 drops the de Rham hypothesis but concludes only pro-modularity. The level may grow by auxiliary odd primes. It does not settle the sharper formulation prescribing exactly which tame primes are allowed.

M2 concerns the particular full all-weight algebra, including Eisenstein systems, at odd level. Its dimension statement should not be transferred to a fixed determinant, fixed weight, ordinary locus, or local deformation ring without a separate argument.

The manuscripts distinguish residual restrictions from conditions on the characteristic-zero target throughout. A local theorem that allows scalar reduction is not by itself a global unrestricted modularity theorem. These distinctions also delimit future extensions of the claimed results.

8. Sources and further reading

M1’s Theorem 1.1 states pro-modularity and its level convention; M2’s Theorem 1.1 gives the integral dimension; M3’s Theorem 1.1 gives classical regular modularity. Their introductory strategy sections explain how the family, local representation theory, and dimension arguments fit together.

The primary works linked in Sections 3 and 6 trace the odd-prime, dyadic, and adjoint-Selmer inputs. For optional introductory terminology, see Galois representations, modular forms, and p-adic numbers.

Source manuscripts

• M1. OpenAI, Unrestricted pro-modularity at the prime two (October 6, 2026).
• M2. OpenAI, The Dimension of the Two-Adic Hecke Algebra at Odd Level (October 5, 2026).
• M3. OpenAI, Fontaine–Mazur modularity at the prime 2 (October 6, 2026).

Sources consulted on 10 October 2026 at repository revision fd4aeeb. Download the manuscript citations.