Breuil-Mezard conjecture: equality of the Galois and automorphic sides

Breuil-Mezard conjecture: equality of the Galois and automorphic sides

Definitions and mathematical statement

Source correction. The catalogue writes $\sigma_{\mathrm{Gal}}=\sigma_{\mathrm{Aut}}$ as an equality of representations, but its actual source, Volpato's workshop notes, Section 1.1, Conjecture 1, writes a numerical equality $\mu_{\mathrm{Gal}}=\mu_{\mathrm{Aut}}$. For a residual two-dimensional representation $\bar\rho$ of $G_{{\mathbb{Q}}_p}$, a Hodge type $k$, and inertial type $\tau$, the Galois quantity is a Hilbert--Samuel multiplicity of a potentially semistable deformation ring modulo a uniformizer. The automorphic expression is \[ \mu_{\mathrm{Aut}}(k,\tau,\bar\rho)=\sum_{n,m}a(n,m)\mu_{n,m}(\bar\rho), \] where $a(n,m)$ are the multiplicities of weights $\operatorname{Sym}^n{\mathbb{F}}^2\otimes\det^m$ in the semisimplified reduction of the prescribed type lattice. Hilbert--Samuel multiplicity is the normalized leading coefficient of the local length-growth polynomial. The source's determinant and deformation conditions must be retained. The printed representation-equality target lacks definitions supporting that literal interpretation; its wording is preserved below rather than silently declared equivalent to this numerical conjecture.

Formulation and concept references: [S1].

Short English statement

The cited Breuil–Mézard formulation compares a Galois deformation-ring multiplicity with a weighted automorphic multiplicity, not the undefined representation equality printed in the catalogue.

Research attempt

Current frontier and scope audit.

The source defect already recorded above is substantive. On p. 2 of the workshop notes, Conjecture 1 is the numerical identity $\mu_{\mathrm{Gal}}=\mu_{\mathrm{Aut}}$; Conjecture 2 is its framed Hilbert--Samuel version. Neither defines the catalogue's two supposed representations $\sigma_{\mathrm{Gal}},\sigma_{\mathrm{Aut}}$. [S1], Section 1.1. The literal representation equality therefore has no specified objects on which to attempt a proof. The original text is retained, and the deductions below are explicitly about the source-defined numerical problem, not a silently repaired literal representation theorem.

The numerical problem has substantial established solutions. Paškūnas proves it, with a cycle refinement, for two-dimensional representations of $G_{{\mathbb{Q}}_p}$ with scalar endomorphisms when $p\ge5$. [E1], Theorem 1.1. A later approach of Feng--Le Hung constructs cycles in higher-rank generic settings and verifies the predicted identities for sufficiently generic tame types and small Hodge--Tate weights. Its hypotheses cannot be discarded. [E2], Abstract and Section 1.5. The old workshop statement is not being used as a current assertion that every numerical case remains open. The August 2026 date generated inside the HTML rendering of the 2014 Paškūnas version is not its scientific revision date.

Attack route.

Patching aims to identify geometric cycles and then obtain their multiplicities; it requires actual compatible deformation modules. A numerical route instead tries to determine a fixed vector of weight multiplicities from many Hodge and inertial types. We investigate the second route: can finitely many exact tests force all others, and what would a failed test really show? This produces a finite-obstruction lemma, but does not manufacture the arithmetic compatibility needed to apply it. We also test the false implication from numerical equality back to isomorphism of representations or local rings.

Attempt.

1. Separate fixed data, type data, and the proposed unknowns. Fix $p$, a residual representation $\bar\rho$, and the determinant and deformation conventions of the source. We do not change the determinant while comparing types. Let $\mathcal W$ be the finite set of Serre weights used there, represented by \[ \operatorname{Sym}^n{\mathbb{F}}^2\otimes\det^m, \quad 0\le n\le p-1,\quad0\le m\le p-2. \] For an admissible type $t=(k,\tau)$, let $a_{t,w}\in{\mathbb{Z}}_{\ge0}$ be the semisimplified type-lattice multiplicities, and let $e_t\in{\mathbb{Z}}_{\ge0}$ be the appropriate Galois Hilbert--Samuel multiplicity. These are the source's numerical inputs. For the algebraic diagnostic, temporarily treat $\mu_w$ as unknown nonnegative integers and examine \begin{equation}\label{wsref-0-33-30-30-30-36-39-38-31-3a-73-79-73-74-65-6d} e_t=\sum_{w\in\mathcal W}a_{t,w}\mu_w \qquad\text{for every admissible }t. \end{equation} In the actual conjecture, the right-hand multiplicities are prescribed by the representation-theoretic construction; finding some solution of (30006981:system) would not by itself identify that prescribed vector. This distinction is essential. The weight formula and its determinant dependence come from [S1], Section 1.1.

2. A finite-obstruction lemma for the numerical consistency problem. Let $\mathcal W$ be any finite set and $a_{t,w},e_t$ any nonnegative integers, indexed by an arbitrary family of rows $t$. Then the following are equivalent: every finite subsystem of (30006981:system) has a solution in ${\mathbb{Z}}_{\ge0}^{\mathcal W}$, and the entire system has such a solution.

Here is a proof which supplies the needed boundedness instead of invoking compactness of an unbounded integer lattice. Call a column active if it appears positively in some row. For each active $w$, choose a row $t(w)$ with $a_{t(w),w}\ge1$, and let $\mathcal T_0$ be this finite collection of rows. Any nonnegative solution of those rows satisfies \begin{equation}\label{wsref-0-33-30-30-30-36-39-38-31-3a-62-6f-78} 0\le\mu_w\le \left\lfloor\frac{e_{t(w)}}{a_{t(w),w}}\right\rfloor. \end{equation} Inactive columns can be set to zero. Thus the candidates satisfying $\mathcal T_0$ form a finite set $S$. Under finite consistency $S$ is nonempty. If no member of $S$ satisfied all rows, choose for each member one row that it violates. Those finitely many rows, together with $\mathcal T_0$, would be inconsistent, a contradiction. This proves the lemma. In the empty-active-set case finite consistency simply says that every $e_t$ is zero, and the zero vector works.

The proof gives a precise negative-certificate principle: failure of nonnegative-integral compatibility can always be witnessed by finitely many types. It does not provide an a priori bound on the complexity of the witnessing types, nor a method for computing arbitrary $e_t$. It also does not prove that the Galois values really are compatible. If a known representation-theoretic vector $\mu^{\mathrm{Aut}}$ is already supplied, the actual assertion is the stronger direct test $e_t=a_t\cdot\mu^{\mathrm{Aut}}$ for every type.

3. Why a few consistent rows do not extrapolate. Integer row relations give necessary identities. If finitely many integers $c_t$ satisfy $\sum_t c_t a_t=0$, then any solution requires \begin{equation}\label{wsref-0-33-30-30-30-36-39-38-31-3a-72-6f-77-72-65-6c-61-74-69-6f-6e} \sum_t c_t e_t=0. \end{equation} This can be checked by exact integer linear algebra, not floating-point least squares. But neither separate positivity nor solvability one row at a time enforces it. For example, the rows \[ a_1=(1,1),\quad a_2=(1,0),\quad a_3=(0,1), \qquad e_1=e_2=e_3=1 \] are individually soluble over nonnegative integers, while their combination requires simultaneously $\mu_1+\mu_2=1$ and $\mu_1=\mu_2=1$. Even a nonnegative rational solution is not enough: the one-row equation $2\mu=1$ has no integer solution. These examples are tests of the proposed inference, not fabricated deformation-ring multiplicities for an actual $\bar\rho$.

If a finite collection of rows has full column rank over ${\mathbb{Q}}$, it fixes at most one rational vector. Checking that vector on those rows does not imply an identity for a new row. A missing structural theorem would have to show that every new Galois multiplicity obeys all the same relations as the automorphic type multiplicities, and that the resulting vector is the prescribed one. The cycle identities in [E1], Theorem 1.1, provide exactly such substantive geometry in their proven scope; they are not consequences of fitting a finite linear system.

4. Equal leading coefficients do not identify the objects. Let $\kappa$ be a field, and consider the one-dimensional complete local rings \[ R=\kappa[[x,y]]/(xy),\qquad S=\kappa[[x,y]]/(y^2). \] For every integer $N\ge0$, both have \begin{equation}\label{wsref-0-33-30-30-30-36-39-38-31-3a-6c-65-6e-67-74-68} \operatorname{length}(R/\mathfrak m_R^{N+1}) =\operatorname{length}(S/\mathfrak m_S^{N+1})=2N+1. \end{equation} For $R$ the monomial basis is $1,x,\ldots,x^N,y,\ldots,y^N$. For $S$ it is $1,x,\ldots,x^N,y,xy,\ldots,x^{N-1}y$, with the second list empty when $N=0$. Thus both Hilbert--Samuel multiplicities are two, and even their entire displayed length functions agree. But $R$ is reduced, since $(xy)=(x)\cap(y)$, whereas the nonzero element $y\in S$ is nilpotent. The rings are not isomorphic.

There is an equally direct representation-theoretic test. Over a field of characteristic $p$, let a generator of $C_p$ act on $\kappa^2$ either by $I_2$ or by \[ J=\begin{pmatrix}1&1\\0&1\end{pmatrix}. \] Both define representations because $(I+N)^p=I+N^p=I$ for $N^2=0$. Their semisimplifications have two trivial constituents. Their spaces of invariant vectors, however, have dimensions two and one respectively, so they are not isomorphic. Consequently neither equality of Jordan--Hölder multiplicities nor equality of local-ring multiplicities can justify the undefined literal representation equality in the catalogue. A specified functor and an extension-class theorem would be additional data, not notation that can be silently inferred.

Stress test.

The finite-obstruction proof used positivity to obtain (30006981:box); without it an unbounded integer system need not have this property. It is not a proof that a finite selection of the existing BM tests is universally sufficient, because the excluded candidates may require types of unbounded complexity to detect. More importantly, consistency does not identify the multiplicities already fixed on the automorphic side. The elementary examples do not disprove the genuine numerical conjecture; they disprove shortcuts that would bypass its arithmetic content.

No global modularity hypothesis from Theorem 1 of the workshop notes has been imported into the local conjecture without justification. No scalar-endomorphism or $p\ge5$ hypothesis from the proven theorem has been erased. The rings in (30006981:length) are illustrative local rings, not claimed to be the specified potentially semistable deformation rings. The source's warning that its notes were made during a workshop is retained as a reason to check later precise theorems, not as license to invent an alternative formulation.

Outcome.

Outcome: Exploratory approach with unresolved gap.

The source-defined numerical formulation has been separated from the undefined literal representation statement. A finite-obstruction lemma for nonnegative-integral multiplicity systems is proved, together with exact counterexamples to extrapolation from isolated tests and to recovering isomorphism from numerical invariants. These are limited structural deductions, not a new Breuil--Mézard theorem.

The literal target still lacks definitions of its asserted representations. For the numerical source formulation, the missing work is a deformation/representation-theoretic compatibility theorem in any scope not already covered by established results. Nothing in this attempt supplies that theorem or establishes a universal representation isomorphism.

Sources

• S1 AIM workshop problem list, p-adic Representations, Modularity, and Beyond. https://aimath.org/WWN/padicmodularity/padicmodularity.pdf. Catalogue formulation source. Consulted 24 September 2026: Section 1.1, Conjecture 1 is an equality of numerical multiplicities, not the representation equality printed in the catalogue.
• E1 V. Paškūnas, On the Breuil--Mézard conjecture, Duke Mathematical Journal 164 $2015$, no. 2, 297--359. Consulted arXiv:1209.5205v3, 20 March 2014, Introduction and Theorem 1.1. https://arxiv.org/html/1209.5205v3. The theorem assumes $p\ge5$ and scalar residual endomorphisms; the HTML-generated 2026 internal date is not a revision date.
• E2 T. Feng and B. Le Hung, Mirror symmetry and the Breuil--Mézard Conjecture, arXiv:2310.07006, consulted version 4, Abstract and the revision's Section 1.5 summary. https://arxiv.org/abs/2310.07006. The generic-parameter, tame-type and Hodge-weight qualifications are retained. Both research sources and the workshop PDF were consulted 27 September 2026.