Fontaine–Mazur geometric Galois-representation conjecture

Fontaine–Mazur geometric Galois-representation conjecture

OpenAI's claim

OpenAI presents the following claims relevant to this question, with manuscript numbers following paper order in Sources. In manuscript 1 (family 010; 2026-09-23), the asserted result is that an odd irreducible two-dimensional two-adic representation over the rationals, finitely ramified and de Rham with unequal Hodge--Tate weights, comes from a classical modular form after a Tate twist.

Mathematical statement of the claimed result

Manuscript 1.

Let $r:G_{\mathbb Q}\to\operatorname{GL}_2(\overline{\mathbb Q}_2)$ be continuous and irreducible. Assume it is odd, unramified outside a finite prime set, and has a de Rham restriction at $2$ whose two Hodge--Tate weights differ. The claimed modularity conclusion is \[ r\simeq r_{f,2}(m) \] for some classical cuspidal eigenform $f$ and integer Tate twist $m$. No residual irreducibility or prescribed local ordinary condition is part of the displayed theorem.

Outline of the argument

For manuscript 1 (family 010; 2026-09-23), local deformation charts and solvable base change create a family with modular support. Integral completed forms and their block action permit patching, after which an ordinary vector and the prescribed local line give classicality and descent.

Formal verification

No Lean formalization is listed at this snapshot for manuscript 1 (family 010; 2026-09-23).

This site has not executed the Lean code. Described statement coverage is not an independent proof check.

Status caveat

The manuscripts were produced by an internal OpenAI model. OpenAI cautions that results without formalization can contain mistakes. This site has not independently checked the proofs; the label records the scope of OpenAI's claim.

Sources

• OpenAI, Fontaine–Mazur modularity at the prime 2, 2026-09-23; citation key OAI:Fontaine-Mazur-modularity-at-the-prime-2-September-23-2026. PDF (GitHub); PDF (direct); Manuscript page and citation; Latest version; Statement source: thm:main.
• OpenAI catalogue entry.

Fontaine–Mazur geometric Galois-representation conjecture

Definitions and mathematical statement

Let $G_K=\operatorname{Gal}(\bar K/K)$ and $\rho:G_K\to\operatorname{GL}(V)$ be an irreducible continuous representation over a finite extension of ${\mathbb{Q}}_\ell$. Being unramified outside a finite set means inertia acts trivially at all remaining finite places. At $v\mid\ell$, being de Rham means the de Rham period module has the full dimension of $V$. The assertion is that \[ V\text{ is a subquotient of }H^i_{\mathrm{\acute et}}(X_{\bar K},{\mathbb{Q}}_\ell)\otimes{\mathbb{Q}}_\ell(m) \] with coefficients extended to those of $V$, for some smooth projective $X/K$, $i\ge0$, and $m\in{\mathbb{Z}}$. Arbitrary Tate twists are allowed; no nonnegative Hodge--Tate-weight condition is imposed.

Formulation and concept references: [S1].

Short English statement

Must every irreducible Galois representation with the specified arithmetic and local geometric properties arise from algebraic geometry?

Research attempt

Current frontier.

Literature and scope check, 27 September 2026. In [S1], printed page 13, Conjecture 3.2 and Remark 3.4 explain exactly how removing the sign restriction on Hodge--Tate weights and allowing Tate twists recovers the Fontaine--Mazur formulation. Those freedoms are retained here. Established potential-automorphy theorems have additional regularity, polarization and residual-image hypotheses; theorems for a compatible system do not automatically apply to an arbitrary single representation [E1].

The August 2026 manuscript [S3], Theorem B, treats two-dimensional representations over totally real fields that are potentially crystalline and ordinary with weights $\{0,1\}$ and totally odd determinant. It proves potential modularity even in cases with reducible residual representation. The text immediately after that theorem separately discusses an additional local condition guaranteeing geometric realization. Theorem D proves modularity over ${\mathbb{Q}}$ under the stated ordinary weight-$\{0,1\}$ hypotheses. These are scoped results, not the all-fields, all-ranks, arbitrary-weight target above. No independent audit of that recent manuscript is claimed.

Attack route.

There are three distinct tasks: find a geometric realization after a finite extension; bring it back to the original number field; and ensure that the realization uses the precise smooth projective category in the question. Potential automorphy addresses part of the first task under special hypotheses, but calling a representation automorphic does not by itself supply the required cohomology. We isolate and completely prove the latter two tasks, including a connectedness safeguard. We also test whether twisting or base extension removes the hypotheses that prevent applying the known potential-automorphy results.

Attempt.

1. Descent from a genuinely geometric finite extension. Let $E/{\mathbb{Q}}_\ell$ be the coefficient field, $L/K$ a finite extension, and $V$ a finite-dimensional continuous $E$-representation of $G_K$. No irreducibility assumption is needed for the following implication: \begin{equation}\label{wsref-0-65-71-3a-39-34-30-30-30-34-38-3a-70-6f-74-65-6e-74-69-61-6c} \left.V\right|_{G_L}\text{ is a subquotient of } H^i_{\mathrm{\acute et}}(Y_{\bar L},E)(m) \quad\Longrightarrow\quad V\text{ has a smooth projective realization over }K, \end{equation} where $Y/L$ is smooth projective. The cohomological degree and twist on the right may change if a geometrically connected variety is required.

Put $G=G_K$, $H=G_L$, and $d=[G:H]$. Finite-index induction is a direct sum of $d$ copies of the underlying vector space, hence is exact. The tensor identity gives \[ \operatorname{Ind}_H^G\operatorname{Res}_H^G V \simeq V\otimes_E E[G/H]. \] The maps \[ v\longmapsto v\otimes\sum_{gH\in G/H}[gH],\qquad v\otimes\sum_{gH}a_{gH}[gH]\longmapsto \frac{1}{d}\left(\sum_{gH}a_{gH}\right)v \] are $G$-equivariant and their composite is the identity. Thus $V$ is a direct summand of the induced restriction. Division by $d$ is legitimate in $E$ even if $\ell\mid d$; this is not an integral-lattice assertion.

Regard the scheme $Y$ as a $K$-scheme $Z$ by composing its structural map with $\operatorname{Spec}L\to\operatorname{Spec}K$. Smoothness and projectivity are preserved because the latter map is finite etale. After base change to $\bar K$, its components are the conjugates of $Y$, and their cohomology, with the permutation of components and the transported actions, is precisely \begin{equation}\label{wsref-0-65-71-3a-39-34-30-30-30-34-38-3a-69-6e-64-2d-63-6f-68-6f-6d} H^i_{\mathrm{\acute et}}(Z_{\bar K},E)(m) \simeq \operatorname{Ind}_{G_L}^{G_K} H^i_{\mathrm{\acute et}}(Y_{\bar L},E)(m). \end{equation} Finite disjoint-union cohomology and base change justify this identification; no automorphic descent theorem is being invoked. These cohomological functorialities are as in [E2].

If $\operatorname{Res}V=U/U'$ for $U'\subseteq U\subseteq W$, exact induction exhibits $\operatorname{Ind}\operatorname{Res}V$ as $\operatorname{Ind}U/\operatorname{Ind}U'$ inside a subquotient of $\operatorname{Ind}W$. Taking the inverse image of the summand $V$ in this quotient shows that $V$ itself is a subquotient of the right side of (eq:9400048:ind-cohom). This proves (eq:9400048:potential) in the smooth projective scheme convention without any semisimplicity assertion for arbitrary Galois cohomology.

2. A geometrically connected realization without a category loophole. The preceding $Z$ need not be geometrically connected. To avoid resting the argument on a permissive meaning of "variety", embed it into ${\mathbb{P}}^N_K$ with constant codimension $c\ge2$, increasing $N$ if necessary. Here $Y$ may be taken pure-dimensional, as in a realization by a variety. Let \[ X=\operatorname{Bl}_{Z}({\mathbb{P}}^N_K),\quad j:D={\mathbb{P}}(N_{Z/{\mathbb{P}}^N})\hookrightarrow X,\quad q:D\to Z, \quad \xi=c_1(\mathcal O_D(1)). \] The blow-up is smooth and projective; after extending scalars it remains integral, since it is the blow-up of the integral projective space along a proper smooth center. In particular, $X$ is geometrically connected. The normal bundle of $D$ in $X$ is $\mathcal O_D(-1)$.

Using rational etale cohomology, define \begin{align*} A &: H^i(Z_{\bar K},E)(m)\longrightarrow H^{i+2}(X_{\bar K},E)(m+1),& A&=j_*q^*,\\ B &: H^{i+2}(X_{\bar K},E)(m+1)\longrightarrow H^i(Z_{\bar K},E)(m),& B&=-q_*(\xi^{c-2}j^*(-)). \end{align*} The Gysin self-intersection formula and projective-bundle integration give \[ BA=-q_*\left(\xi^{c-2}(-\xi)q^*(-)\right) =q_*(\xi^{c-1}q^*(-))=\operatorname{id}. \] The normalization is $q_*(\xi^{c-1})=1$ on every projective-space fiber. These are the usual Gysin, projection and Chern-class identities [E2]. All maps are defined over $K$ and hence Galois-equivariant. Thus the old cohomology is a direct summand of one cohomology group of a geometrically connected smooth projective $X$, with the displayed changed degree and twist. Both changes are allowed by the original target. This supplies an explicit splitting, rather than assuming that cohomology of a disconnected scheme is automatically an admissible answer.

3. A complete, but already familiar, finite-image sector. If an irreducible representation $\tau$ has finite image, let $L/K$ be the finite Galois extension cut out by its kernel, with group $\Gamma$. The module $H^0((\operatorname{Spec}L)_{\bar K},E)$ is $E[\Gamma]$. Every irreducible $E[\Gamma]$-module occurs in the regular representation: it is a quotient, and averaging a linear splitting over the finite group makes it a direct summand. Consequently, $\tau(m)$ is realized in $H^0(\operatorname{Spec}L)(m)$ for every integer $m$. Construction 2 can replace this finite etale scheme by a geometrically connected variety. This proves the required conclusion whenever $V$ is a Tate twist of a finite-image representation, without an oddness requirement. It is a known special case, not a claim of new arithmetic progress.

4. Why the known hypotheses cannot simply be normalized away. A Tate twist translates all Hodge--Tate numbers by the same integer (with the sign determined by the convention). Differences and multiplicities are unchanged. Thus repeated weights cannot become distinct by twisting. After extending the local field, labeled weights at an embedding are repeated at the embeddings above it, not arbitrarily replaced by new ones. An appeal to regular potential automorphy still needs a regularity hypothesis.

There is a similar elementary obstruction to forcing self-duality by a uniform twist. An essentially self-dual representation with a de Rham multiplier must have its local weight multiset invariant under $h\mapsto a-h$, where $a$ is the multiplier's weight. The multiset $\{0,1,3\}$ has no such symmetry: pairing its extremes requires $a=3$, while its middle weight requires $a=2$. Translation by an integer $t$ changes these requirements to $3+2t$ and $2+2t$, still inconsistent. This is a test of a weight-based normalization lemma, not a construction of a global irreducible representation violating Fontaine--Mazur. The self-duality and distinct-weight conditions in [E1] really are additional restrictions.

Stress test.

Restriction to $G_L$ need not preserve irreducibility. The proof of (eq:9400048:potential) deliberately assumes a realization of the whole restriction and never silently applies an irreducible-case theorem to it. Induction is exact because the index is finite; no averaging over an infinite profinite group has been used. Nor have we inferred that a general subquotient splits. Only the explicitly constructed finite-permutation summand and the explicitly split blow-up summand are split.

The remaining arithmetic hypothesis is substantial: for every $V$ in the original statement, find some finite $L/K$ and an actual smooth projective $Y/L$ with the left side of (eq:9400048:potential). Local de Rham comparison is a necessary property of cohomology, not a construction of such a $Y$. Potential modularity must also be connected to an appropriate cohomological realization; the qualification following Theorem B of [S3] makes this distinction particularly relevant. No unavailable universal polarization, regularity, compatible-system, or residual-image theorem is inserted into that gap.

Outcome.

Outcome: Conditional reduction. Potential geometric realization suffices for the exact base-field problem. The descent, coefficient-field division, subquotient bookkeeping, and geometric-connectedness repair are fully supplied above. Finite-image Tate twists are handled directly. These structural arguments are not asserted to be historically new.

What remains is potential geometric realization for every irreducible, finitely ramified, de Rham representation in the question. At the level of universal existence this potential assertion is equivalent to the target, since a realization over $K$ is already a potential realization. The reduction isolates where an arithmetic advance is needed; it does not make that advance or prove the conjecture.

Sources

• S1 Sug Woo Shin and Nicolas Templier, On fields of rationality for automorphic representations, author version dated 2 July 2014. https://math.berkeley.edu/~swshin/FieldRat.pdf. Catalogue formulation source.
• S2 James Newton, Modularity of Galois Representations and Langlands Functoriality. https://link.springer.com/article/10.1007/s41745-022-00305-0. Catalogue research/status reference; not a proof endorsement.
• S3 Jack A. Thorne, Towards the Fontaine–Mazur conjecture for GL$2$. https://arxiv.org/html/2608.07186v1. Catalogue research/status reference; not a proof endorsement.
• E1 Thomas Barnet-Lamb, Toby Gee, David Geraghty and Richard Taylor, Potential automorphy and change of weight, Annals of Mathematics $2$ 179 $2014$, 501--609; arXiv:1010.2561, introduction, Theorems A and C, and Section 4.5. https://arxiv.org/abs/1010.2561. Established potential-automorphy results with explicit additional hypotheses; consulted 27 September 2026.
• E2 James S. Milne, Lectures on Etale Cohomology, author-hosted course notes, Sections 16 (purity and Gysin), 17 (proper base change), 22 (projection formula) and 23 (Chern classes), especially Theorem 23.2. https://www.jmilne.org/math/CourseNotes/LEC.pdf. Cohomological conventions and functorialities used in the induction and blow-up calculations; consulted 27 September 2026.