Fontaine–Mazur modularity at the prime 2

Proof. There are countably many finite subextensions of \overline{\mathbb Q}_2/\mathbb Q_2. For each one E, the set of group elements whose representation matrices lie in \GL_n(E) is a closed subgroup. These subgroups cover the profinite group. By Baire’s theorem, one has nonempty interior, hence is open. Enlarge E to contain the finitely many matrix entries of coset representatives. Then every matrix lies in that finite extension.

The image is compact. Its entries, and the entries of its inverses, have bounded valuations. The \OO-submodule generated by the orbit of \OO^n therefore lies between two scalar multiples of \OO^n. It is a stable lattice.□

Proof. Let H be the algebraic monodromy group of an irreducible two-dimensional representation over an algebraic closure. Lie–Kolchin and the structure of connected solvable groups [43, Theorems 17.4 and 16.33] give the following alternatives. If H^\circ is solvable and has a nontrivial unipotent radical, the latter has a unique fixed line; normality makes that line H-stable, contradicting irreducibility. The remaining possibilities are that H^\circ is scalar, or that its two distinct character lines are permuted by H.

In the scalar case the projective image is finite. After a finite extension of the base field, the representation is a scalar character on a two-dimensional space. Its two Hodge–Tate weights are equal, which is excluded. In the second case the action on the two lines gives a quadratic extension L/\mathbb Q and

for a continuous character \theta. The character is de Rham and finitely ramified, hence is the 2-adic avatar of an algebraic Hecke character, by the following rank-one consequence of Tate’s theorem and the construction in [52, Chapter III, Sections 1.2 and 2.3]. The integral algebraic exponents on open local units obey the global unit relation. Principal ideals in a suitable ray subgroup therefore receive the prescribed algebraic monomials. A positive power of each of finitely many ray-class representatives is principal in that subgroup, so its character value is algebraic. Adjoining these values places all ideal-character values in one number field and constructs the algebraic Hecke character. This argument does not assume that the given character was initially rational over a number field. If L is real quadratic, its algebraic exponents at the two real embeddings are equal by the unit theorem: a unit of infinite order forces their difference to vanish in the algebraic character relation. This again contradicts regularity. For imaginary quadratic L, the theta-series construction gives the required cuspidal eigenform, with its algebraic Tate normalization; equivalently, use quadratic automorphic induction [28, Proposition 12.1].

Finally, restriction to finite index preserves H^\circ. If a restriction of a non-virtually-solvable representation were reducible, its identity component would lie in a Borel subgroup and be solvable. This is impossible.□

Proof. For representability under \Phi_2, we use [16, Propositions 3.3 and 3.7, Remark 3.5], with the numbering of the cited arXiv version. Finiteness of the universal Cayley–Hamilton algebra and the algebraization of its compatible matrix representations are [62, Propositions 3.2 and 3.6, Theorems 3.7 and 3.8]. Compatibility here includes the characteristic-polynomial equations; no generalized matrix algebra at a scalar residual point is being assumed. Fixing \chi is a base change of these statements.

For clarity, finite type has a direct algebraic explanation. Choose finitely many algebra generators for the finite algebra \mathscr E, assign a 2\times2 matrix to each, and impose the multiplication and characteristic-polynomial identities. These are polynomial equations over the Noetherian base. Completing at a residual matrix point recovers continuity and the chosen residual representation, including its extension class.

Line invariance and the prescribed action on a line are closed equations in the relative projective line. Over the absolutely irreducible locus the Cayley–Hamilton algebra is an Azumaya algebra of rank four; one may split it faithfully flat locally and then descend the proper incidence image. This proves the closedness assertion.

If g_1,\ldots,g_a topologically generate the relevant abelian character quotient, every recorded character value \lambda(g_j) satisfies

Modulo the maximal ideal of the trace image, the corresponding parameters are nilpotent after the chosen residual roots have been fixed. The closed fiber of the completed character algebra is therefore finite. Complete Nakayama proves finiteness of that algebra over the trace image. Finite torsion generators cause no change.

Finally, the absolutely irreducible determinant theorem identifies the Cayley–Hamilton algebra with a matrix algebra after a splitting extension [16, Theorem 2.22 and Corollary 2.23]. Two compatible representations are conjugate and their stabilizer consists of scalars. This gives the final assertion.□

Proof. Contract P to a prime q of \mathcal O_{X,x}. There is a surjection from the completion of \mathcal O_{X,x}/q to \widehat{\mathcal O}_{X,x}/P, and completion preserves dimension [58, Tags 02IJ and 07NV]. The finite-type dimension formula over the excellent local base therefore bounds the latter dimension by \dim(R/p) plus the generic-fiber dimension of the contraction. Over an algebraic closure of \Frac(R/p) the representation is unique up to a three-dimensional conjugacy orbit. Each unspecified invariant line has dimension at most one in \mathbf P^1; with distinct characters its choices are finite. This gives the inequality. Notice that the proof contracts the formal prime before applying the finite-type formula. It does not assert that an arbitrary formal prime itself algebraizes.□

Proof. Use formal Noether normalization, over \kk in the first case and over \OO in the second; the coefficient and finite normalization facts used here are [58, Tags 032A, 032D, 07QW and 07QV]. Thus A is finite over a power-series subring S. Given 0\ne f\in A, integrality supplies a nonzero element of fA\cap S: use an integral equation for f with nonzero constant term in the fraction field extension, and clear denominators.

In the first case choose a one-parameter specialization of S on which that element is nonzero. Such a specialization exists by choosing positive weights for the variables and, if necessary, coefficients in a finite residue extension so that the lowest weighted part does not vanish. Its image is finite over \kk[[t^a]] for some a>0. Lying-over gives a one-dimensional prime of A avoiding f. Excellence makes its normalization finite; a complete equicharacteristic discrete valuation ring with finite residue field has the asserted form.

In the second case choose sufficiently small integral values for the variables on which the selected nonzero series does not vanish. Existence follows successively from the one-variable preparation theorem. A prime above this specialization has a finite extension of \EE as fraction field. Integrality puts the image of A in its ring of integers; locality and finiteness make the map continuous. Again f does not vanish. Applying this to an element of the ideal of the excluded closed subset proves density.□

Proof. We first prove the integral-order assertion underlying the functorial claim. Let C be one of the Artinian algebras, with maximal ideal \mathfrak n, \mathfrak n^N=0, and \varpi^a C=0. Write

The inverse image of \kk[[t]] under C\to K is A_0+\mathfrak n. A bounded subset of this inverse image lies in a finite A_0-order: subtract coefficientwise lifts of its residual entries, choose a finite A_0-lattice U\subset\mathfrak n containing the resulting bounded differences, and put

Powers here mean finite module spans of products. Nilpotence makes B an algebra finite over A_0. Its quotient by B\cap\mathfrak n is \kk[[t]], with nilpotent kernel. Hence B is local with residue field \kk. The finite local ring B/zB is Artinian, so a power of \mathfrak m_B is contained in zB. Its maximal-ideal topology is therefore its finite-module topology; B is complete and compact.

Apply this to the compact set of matrix entries of a continuous lift of \rho_*. These entries have integral residues because the residual basis was chosen above. Their determinants have residual values in \kk[[t]]^\times, and so are units of B. We obtain a continuous representation in \GL_2(B) reducing to precisely \overline\rho. Negative powers of z in nilpotent entries belong to B\cap\mathfrak n and disappear at that closed residual point. There is no uniform lower valuation bound as C varies, nor is one needed. The same construction simultaneously contains any finite collection of local data, and two such orders admit a common enlargement.

Universality gives a map R\to B. Adjoin z using its coefficient value. Elements outside \mathfrak p have nonzero residue in K, so are units in C, while \mathfrak p^N maps to zero after increasing N. The map factors uniquely through R_*. Conversely, a local map R_*\to C factors through a finite jet. Write the original formal coordinates as integral lifts of their curve values plus nilpotent increments. Expansion in the increments has only finitely many terms at this jet depth; the residual coordinates, after their constants are removed, lie in (t). Substitution in each original formal series consequently converges in the Laurent-series topology. It is continuous, and so gives a continuous lift of the universal representation. The two constructions are inverse. They plainly preserve determinant equations, line incidences, and restriction maps. For determinants use their characteristic coefficients, followed on the absolutely irreducible locus by the matrix-algebra identification in Proposition 3.1.

For dimensions, adjoining z increases a component dimension from e to e+1, while the enhanced curve quotient has dimension one: the map on \kk[[z]] is already onto \kk[[t]]. The dimension formula for complete equidimensional excellent local rings gives height e at the enhanced prime. Localization and completion have dimension e; see also the dimension and catenarity facts in [58, Tags 00KW, 032C, 0ECF and 07NV]. At a horizontal characteristic-zero coefficient point the integral quotient again has dimension one, but no new variable has been adjoined. Its height and completed dimension are therefore e-1. The deformation functor uses the usual 2-adic topology.□

Proof. Choose a stable \kk[[t]]-lattice T\subset V. Continuous cochains with values in T are the inverse limit of the cochain complexes for T/t^nT. Finite-level cohomology is finite, so the derived inverse limits vanish and cohomology commutes with this inverse limit. The resulting compact cohomology modules are finite over \kk[[t]]: their quotients by t inject into the appropriate finite-level cohomology group, and compact Nakayama applies. Every continuous cochain into V has bounded image, since its domain is compact. Thus its cochain complex is obtained by inverting t, and the same is true of cohomology.

Apply finite-coefficient local duality and Poitou–Tate to T/t^nT and the dual torsion lattice, then pass to the inverse limit and invert t. This gives the perfect field-valued pairings, with Tate cohomology at real places, and the exact global sequence. Taking lengths in the finite-level Euler formulas and dividing by n gives the displayed dimensions; bounded torsion contributes zero in the limit. The real-place terms pass through the same calculation and must not be discarded at 2. Intersections with lattices and saturation allow any linear local condition and its annihilator. The finite-coefficient inputs are [40, I, Corollary 2.3, Theorems 2.8, 2.13, 4.10 and 5.1]. The identical lattice argument with an \OO-lattice gives the finite-2-adic-field version.□

Proof. The relative tangent space in the one-frame problem is the kernel of restriction on Z^1(D). We spell out the obstruction bound since it is also used at enhanced curves. Across a small extension, choose continuous matrix lifts with the prescribed determinant. Multiplication errors form a continuous 2-cocycle. Given the specified local lifts, its class belongs to the kernel of restriction on H^2(D). If that class vanishes, correct the matrices to a global representation. Its differences from the chosen local lifts are 1-cocycles, and the second obstruction belongs to the cokernel of restriction on Z^1(D). Continuous set sections of finite-length coefficient modules suffice: coefficientwise digit sections supply them. The determinant map on invertible matrices is smooth, so determinant correction needs no division by two.

Consequently the relation number is at most the sum of these two obstruction dimensions. One can verify that this bounds actual minimal relations as follows. For a minimal relative presentation H\twoheadrightarrow R with kernel I, test the small extension

for j large enough that its kernel is I/\mathfrak m_HI. The two obstruction maps must jointly detect its dual relation directions. Indeed, quotient the kernel onto an undetected one-dimensional direction. Vanishing of both obstructions would give a lift compatible with the local base. Its relative coordinates differ from the tautological ones by elements of the small kernel, which is killed by \mathfrak m_H. Changing these coordinates therefore alters a relation only through its residual linear terms in the relative variables. Those terms vanish by minimality, so the nonzero surviving relation cannot become zero. This contradicts the existence of the lift and proves the relation bound for the complete rings.

Poitou–Tate bounds the first obstruction dimension by

Subtracting it and the Z^1 cokernel from the tangent dimension gives

For the last equality, the real terms in (3) cancel the real local terms; the dyadic Euler characteristic is -3. This proves (4) in both coefficient characteristics. Base change carries the same presentation to the extra local conditions.

For independent frames, first choose a global cohomology class vanishing at the prescribed local places. Local choices of trivialization contribute the local invariant spaces, and a common global infinitesimal automorphism is quotiented out. This gives (5). If the global representation is absolutely irreducible, the common automorphisms are exactly the scalars; the quotient of the |S| frame spaces therefore has dimension 4|S|-1.□

Proof. Hold the line fixed in an adapted basis. The coefficient Lie algebra is the trace-zero upper triangular algebra of dimension two. The local Euler formula gives \dim Z^1-\dim H^2=4, so the relation bound gives local dimension at least four. Moving the line adds one formal variable. At a real place the odd involution has two distinct eigenvalues in characteristic zero and its conjugacy orbit has dimension two. The final assertion is (4) with d=m=1.□

Proof. There are only finitely many quadratic extensions of F unramified outside S. This follows from Hermite–Minkowski, since fixed local degree bounds the finitely many possible local discriminants. For each such extension F'/F, use the character deformation space of G_{F',S} with the product of the character and its conjugate fixed. Class field theory identifies its free pro-2 parameters with a quotient of the dyadic local units; non-dyadic unit groups and class groups contribute only finite groups. The dyadic unit rank over F' is 2d, and its minus rank for the quadratic involution is d. Fixing the product fixes the plus part up to finite groups: multiplication by two is an isogeny of the relevant free \ZZ_2-modules even though two is not a unit. Thus the reduced special character space has dimension at most d.

Its map to pseudorepresentations has finite image algebra. Indeed the finitely many generating character values satisfy the characteristic polynomials of the induced traces, and complete Nakayama applies as in Proposition 3.1. Its image is consequently closed and has dimension at most d. Taking the finite union gives the dihedral bound. The argument applies to curve points after finite coefficient and curve extensions, and Lemma 3.3 gives the asserted closed-locus formulation.

If the projective image is finite, each scalar in the representation has square fixed by its determinant. The determinant has finite image in the special fiber, so the scalar image, and hence the whole image, is finite. Its traces are algebraic over a finite field. An element of \kk'[[t]] algebraic over a finite field is constant; its constant reduction is already prescribed. Thus these traces give exactly the residual pseudorepresentation.

Finally, a reducible determinant is a pair of characters with fixed product, so one character determines the other. For an abelian totally real F, Brumer’s theorem on Leopoldt’s conjecture [15], in the form proved in [25, Theorem 1 and Proposition 8, pp. 9-08–9-09 and 9-14], gives free pro-2 character rank one. The same finite character-image argument therefore gives a closed locus of dimension at most one.□

Proof. Use the finite universal Cayley–Hamilton algebra \mathscr E for each family. Its reduction modulo the base maximal ideal is finite. After killing that finite residual group image locally, the group acts through the pro-2 group 1+\mathfrak m\mathscr E. At a place of residue characteristic \ell\ne2, wild inertia and the non-2 part of tame inertia therefore act trivially. Write \tau for the remaining tame generator and \phi for arithmetic Frobenius, so that

At any field-valued point the multiset of eigenvalues of \tau is stable under the qth-power map. Each eigenvalue consequently has order dividing q^2-1. A further fixed tame extension kills this semisimple inertia at every point, leaving unipotent inertia. The same finite extension works for the finite list of source families.

In characteristic two a unipotent 2\times2 matrix has square one. After these extensions the inertia image is therefore finite: it is generated by the image of the single pro-2 tame generator. The original inertia image is finite as well, since the inertia subgroup upstairs has finite index in the original one. Now use the Frobenius chosen in the original local group, whose trace is prescribed by the curve constraint. Its characteristic polynomial has coefficients in a finite field, since that trace is the residual constant and the determinant has finite image. Its eigenvalues have finite order, and its possible Jordan part has order at most two. This original Frobenius consequently has finite order. The full local image is an extension of the finite inertia image by the finite quotient generated by Frobenius, and is therefore finite.

Every finite Galois group over a nonarchimedean local field is solvable: wild inertia is an \ell-group, and the successive tame inertia and residue quotients are cyclic. A composition tower therefore realizes each required local extension through cyclic prime-degree steps. At each step the prime-degree case of the Grunwald–Wang theorem [42, Chapter VIII, Theorem 2.4] prescribes the finitely many local extensions while imposing splitting at dyadic and real places. For degree two this is simply weak approximation for squareclasses; no higher-power Grunwald–Wang assertion is needed. Composita and Galois closure preserve solvability, total reality, and complete dyadic splitting. An unramified local extension of even degree makes an odd residue cardinality 1 modulo 4. The determinant’s finite non-dyadic inertia can be killed at the same time.

Finally, inertia in the enlarged field maps into inertia downstairs. A representation unramified at a place outside the original allowed set remains unramified after restriction, even if the field extension itself ramifies there. This proves the last assertion.□

Proof. The unrestricted real and dyadic base. We first establish the assertion about A^{\mathrm{rd}}. An odd real framed ring is given by trace zero and determinant -1. These equations are equivalent to the involution relation with that determinant, by Cayley–Hamilton and invertibility. At the scalar residual point it has the form

Its special fiber a^2+bc=0 is a geometrically normal surface: it is an integral hypersurface whose singular locus has codimension two [58, Tag 031S]. At a non-scalar residual involution one obtains a smooth chart of the same relative dimension two.

For a dyadic factor, the fixed-determinant framed deformation theorem [11, Theorem 1.5] gives a flat complete-intersection normal domain of dimension seven and a normal special-fiber domain of dimension six. The theorem applies at 2, including scalar and reducible residual representations; the prime restriction on a different density theorem is not being used. The same statement holds after each finite coefficient extension.

The completed tensor product of these special-fiber rings over the finite, hence perfect, field is a normal domain. Here we use the completed tensor-product normality theorem [55, Theorem 3.3 and Proposition 3.1]; the residue tensor product is the field itself, so the product is local and connected. Its dimension is 8d. The mixed-characteristic completed product is a flat complete intersection: concatenate presentations, first checking the regular sequences modulo \varpi, and then lift them together with the nonzerodivisor \varpi. Its special fiber is the normal domain just obtained, and the regular-element normality criterion makes the product normal. It is a domain as well: the lowest nonzero \varpi-adic terms of a putative product of two nonzero elements could not multiply to zero in the special fiber.

Adjoining z, localizing at the enhanced curve, and completing preserve normality by excellence [58, Tags 07QW and 0C23]. They preserve the asserted dimensions by Lemma 3.4, and the same applies to the special fiber. The element \varpi remains a nonzerodivisor, so the enhanced ring is flat over the discrete valuation ring \mathcal L. This proves the assertion about A^{\mathrm{rd}} and also shows why no special dyadic hypothesis is needed. The construction can be repeated after any of the finite extensions in the statement.

The integral incidence for the i-type. We prove the tame assertions using proper incidences, keeping their images distinct from the incidences themselves. In the i-case, Cayley–Hamilton gives \tau_j^2=-1, hence \tau_j^{q_j}=\tau_j. Adjoin a \phi_j-stable i-eigenline for each \tau_j. This gives a closed proper incidence Y_i in a product of projective lines over \Spec B_i. In an adapted basis on a slope chart, its matrices are

and their sole remaining equation is

The line slopes are free chart coordinates.

Here is a completion argument which also works for all places at once. On a product of affine slope charts put

This is completion along the entire closed fiber, not at a selected slope origin. Restricted coefficient series embed it into A[[s_1,\ldots,s_u]], so C is a domain. Exactness of completion identifies C/\varpi C with the corresponding restricted-series ring over A/\varpi A, also a domain. Completion of the incidence chart along that fiber is

Reduce each monomial by replacing paired factors x_jn_j by 2ih_j until they never occur together. Assign weights 1,1,2 to x_j,n_j,h_j. Reduction preserves weight, so it converges in the power-series topology and yields a normal form. The difference between a monomial and its reduction is a sum of the relations (7) times series whose weights are lower by two; these series converge as well. The substitution

embeds these normal forms into \Frac(C)[[x_j,n_j,d_j:1\le j\le u]]. Indeed the two output exponents recover the exponent of h_j as their minimum, and each output coefficient receives only finitely many input terms. This proves uniqueness of the normal form and shows that (8) is a domain. This argument needs no uniform bound on denominators. It also shows that the successive relations are nonzerodivisors and gives dimension \dim A+4u=e on the incidence.

We explain why chart completions suffice. At every point of the closed fiber, localization of the chart maps faithfully flatly to a localization of its completed chart. The latter is a domain, so the incidence is locally integral there. Every irreducible component of a proper scheme over a local base meets the closed fiber; so does any nonempty closed intersection of components. Local integrality therefore makes the components disjoint. The closed fiber is (\mathbf P^1_K)^u, which is connected, hence there is only one component. A nonempty nilradical support would also meet the closed fiber, so the incidence is reduced. Thus Y_i is integral.

After inverting two the eigenline is uniquely given by the idempotent (\tau_j+i)/(2i), and the incidence projection is an isomorphism. In characteristic two, commuting matrices have a common eigenline after algebraic extension, so the proper projection covers every geometric point as well. Its image is therefore all of \Spec B_i set-theoretically. The integral incidence is horizontal and generically finite over that image. This proves assertion (1). It does not assert that the raw ring B_i is reduced.

The horizontal unipotent patterns. For \eta=1, write N_j=\tau_j-1. The equations give N_j^2=0 and \phi_jN_j=q_jN_j\phi_j. At a field point either N_j=0, or its kernel and image are the same line. The first alternative is the unramified factor, with \phi_j unrestricted. For the closure of the second alternative use the proper line incidence with adapted matrices

At a nonzero N_j the relation forces exactly this diagonal ratio. Its chart has three formal matrix coordinates and one slope.

Choose either alternative at each place. The resulting proper incidence has free completed slope charts over A, so the same closed-fiber argument proves that it is integral, horizontal, and of dimension e. Its projection is an isomorphism where every chosen ramified N_j is nonzero, because the line is then \im N_j=\ker N_j. Each image is consequently an integral horizontal closed subset of dimension e. These finitely many images cover all geometric points of \Spec B_1.

Specialization and uniqueness of the containing component. The dimension calculation alone is not enough for support transfer: we must know exactly which horizontal component contains each generic special-fiber point. It remains to identify their specializations and distinguish them. The two raw presentations reduce to the same one modulo \varpi, since i\equiv1 and q_j is odd. At one place write

The common special matrix factor is

Its two reduced components are

Both have dimension four and are geometrically integral. For R, prove this first in the homogeneous polynomial quotient. On b\ne0 one eliminates c=a^2/b and f=a^2e'/b^2, obtaining a domain. The locus b=0 has dimension three, whereas every component of the two-equation quotient has dimension at least four. There is thus only one component, generically reduced. The quotient is a complete intersection and has no embedded primes, so it is a domain. Completion is the product of its finite homogeneous pieces; lowest-degree nonzero terms show that it remains a domain. The same proof works over every field extension. This argument is made before completion; it does not substitute into arbitrary formal series after inverting b.

For several places over A/\varpi A, the polynomial products of the chosen U and R factors are flat over that base and have geometrically integral generic fiber. They are domains, and the same homogeneous completion argument applies, completing the base coefficients and then the positive-degree variables. Each pattern therefore gives an integral special-fiber component of dimension e-1. Reduction of its proper incidence is integral and maps onto precisely this component. Proper surjectivity onto the image is preserved by base change, so this is the entire underlying specialization of the corresponding horizontal image.

The patterns are distinct. On the generic ramified special piece \tau_j\ne1, whereas every unramified piece has \tau_j=1. On the generic unramified piece \tr\phi_j\ne0, whereas every ramified special piece has \tr\phi_j=0. Thus no horizontal image contains the generic special point of a different pattern. The images already cover the raw spectrum, proving assertions (2) and (3), including uniqueness of the horizontal component through each special-fiber generic point.□

Proof. Move a marked line in its one-dimensional formal chart and choose the corresponding adapted basis. After the scalar twist, the diagonal entries are characters \lambda,\lambda^{-1} congruent to one. Their reduced character ring is

Indeed \QQ_2^\times has maximal pro-2 quotient \ZZ/2\ZZ\times\ZZ_2^2, and the first factor contributes only a nilpotent character parameter in characteristic two.

The maximal pro-2 Galois group of \QQ_2 has three generators and one relation. The odd-degree dyadic case was established by Serre [50, Sections 4.4 and 8]; in Labute’s classification, the q=2, odd-local-degree case gives generators with relation g_1^2g_2^4[g_2,g_3]=1 [37, §5, Theorem 8]. If b(g) is the upper-right entry of the triangular matrix with diagonal entries \lambda(g),\lambda(g)^{-1}, put c(g)=\lambda(g)b(g). Matrix multiplication gives the left-cocycle identity

Its values u_j=c(g_j) satisfy one homogeneous linear equation

All these deformations factor through the maximal pro-2 quotient: their reduction is the identity and their finite coefficient quotients have pro-2 congruence image. Thus this presentation does describe the complete triangular deformation ring after the diagonal reduction, rather than just a subfunctor of it.

Write B=1+x=\lambda(g_2) and C=1+y=\lambda(g_3). The reduced diagonal relation gives \lambda(g_1)=B^{-2}. Using [g_2,g_3]=g_2^{-1}g_3^{-1}g_2g_3 and this left-cocycle identity, direct substitution gives

Multiply (11) by the unit B^2C^2 and set

This triangular formal change of variables transforms the relation into x^2v+y^2u_2=0. Thus the coefficient ideal is exactly (x^2,y^2) and its radical is (x,y); the ideal itself is not radical.

Let A_\infty be the completed real special-fiber product. The real calculation in Proposition 3.10 makes it an excellent Cohen–Macaulay domain of dimension 2d. Adjoin the independent pairs x_j,y_j and the line variables s_j, obtaining

Before completing the upper entries, form the graded algebra

where each f_j is the corresponding relation (11). The coefficients of f_j involve only x_j,y_j and have common-zero locus (x_j,y_j).

On a base stratum where exactly r coefficient rows vanish, the base loses 2r dimensions and the upper-entry fiber gains only r dimensions. Thus the corresponding locus in \Spec P has dimension at most

Every minimal prime of the homogeneous relation ideal is homogeneous and is contained in (\mathfrak m_{A_0},u_{jr}). In that excellent equidimensional local ambient ring, the height theorem for the m equations makes every component have dimension at least \dim A_0+2m. No component can therefore lie over a nonempty rank-drop stratum. Where all coefficient rows are nonzero, the quotient is a vector bundle of rank 2m over an integral base. There is consequently only one component, generically reduced.

The same argument for every initial subset of the equations shows that their heights are the expected ones. They form a regular sequence in the Cohen–Macaulay ambient ring. The quotient has no embedded associated primes, and generic reducedness now proves that P is a domain.

Finally, each homogeneous piece P_n is a finite, torsion-free A_0-module. Over F_0=\Frac(A_0) the m independent linear relations eliminate one upper-entry variable in each triple, so

is an embedding of rings. Since A_0 and its finite modules are complete, the left side is precisely the completed local product in question. It is a domain, of dimension \dim A_0+2m=2d+5m. It is also a complete intersection by the regular sequence already exhibited.

Killing the original diagonal torsion parameters changes no prime ideal: each is nilpotent, and the ideal they generate is nilpotent. The obtained quotient is already reduced, proving the assertion about the original reduction. The real normality and the cocycle and dimension arguments hold after every finite constant extension, proving the final assertion.□

Proof. Finiteness of quaternion ideal classes gives finitely many double cosets at compact open level [61, Main Theorem 17.7.1]; passing to a smaller compact level has finite fibers. Stabilizers modulo the center are finite as well. Their reduced norms have even valuation at every finite place. The group

is finite: the class of the half-valuation ideal maps it to \operatorname{Cl}(F)[2], with kernel given by unit squareclasses, and both groups are finite [41, Theorems 4.4 and 5.1]. The kernel of reduced norm modulo squares has, after scalar rescaling, norm-one representatives in a fixed compact subgroup; these are finite by total definiteness [61, Lemma 17.7.13]. Shrink K until it meets none of these finite projective stabilizers nontrivially. Choosing local sections for the central and tame transformations identifies the function space with continuous functions on a finite disjoint union of copies of K. Its compact dual is a finite direct sum of \OO[[K]]. This also proves admissibility and exact coefficient reduction. Compatibility of the transformation characters on their intersections is part of the definition; the central character is totally even, since both \chi and \eps are odd.

The identification of locally algebraic vectors is the definite double-coset description of algebraic automorphic forms. Jacquet–Langlands transfers the regular noncharacter packets to Hilbert cusp forms [28, Theorems 14.4 and 16.1]; their Galois representations, weights and de Rhamness are given by [4, Theorem 2.1.1]. At the dyadic places we use the potentially semistable comparison, including ramified types: [3, Corollary 1.2]. For the principal types used here, the two distinct inertial characters force monodromy to be zero. Thus the local representation is potentially crystalline and crystabelian, with the weights specified by the algebraic factor.

We spell out the density assertion with fixed type. Let K_0=\prod_{v\mid2}\GL_2(\OO_{F_v}). Matrix coefficients of algebraic representations of central degree zero form a subalgebra of the continuous functions on K_0/Z(K_0). They contain constants and separate points: the adjoint representation is faithful on the projective group. Nonarchimedean polynomial approximation, equivalently Stone–Weierstrass for this compact totally disconnected space, makes this algebra dense.

Choose a nonzero smooth type \tau and an algebraic representation V with the needed central degree. The matrix coefficients a_1,\ldots,a_N of \tau\otimes V have no common zero, since every representation matrix is invertible. For a continuous section f with the prescribed central transformation, choose a finite clopen partition subordinate to sets on which some a_i is bounded away from zero. The functions f/a_i on these clopens, extended by zero, are continuous of central degree zero. Approximate them by the algebra just described and multiply by a_i. Every resulting product is a matrix coefficient of \tau\otimes V\otimes A, where A is algebraic of central degree zero. Complete reducibility in characteristic zero decomposes V\otimes A into algebraic irreducibles of exactly the same central degree. The smooth factor \tau is unchanged. Averaging over the finitely many projective stabilizers gives density in the double-coset function space; its fixed denominators cause no problem over E.

At every dyadic factor we can choose the two inertial characters to be distinct with the required product, using characters of sufficiently large 2-power order. The associated compact principal-series type forces the smooth local representation to be principal series of that type. Here we use Henniart’s compact types [14, Appendix A, Sections A.1.1–A.1.4 and A.2.3–A.2.4], whose scope includes residue cardinality two. In particular it excludes norm characters. The algebraic central degree is common at all embeddings, as required for classical weights. This proves the assertion about a dense span of regular cuspidal packets.

The integral Hecke algebra and its residual factors. Form the Hecke algebra as the closure of the integral Hecke operators in the endomorphisms of the finite Iwasawa module. Reduction modulo its maximal augmentation ideal gives a finite-dimensional commuting algebra. Its topologically nilpotent kernel permits lifting idempotents, giving finitely many residual factors.

Interpolation as an identity of integral operators. We next construct a continuous Galois trace in this integral Hecke algebra. To pass from classical traces to actual operators, we first specify finite tests for the operator topology. Work with the full compact module M\simeq\Lambda_K^r, where \Lambda_K=\OO[[K]] and \mathfrak a=(\varpi,\ker(\Lambda_K\to\OO)). The topology on \End_{\Lambda_K}(M) is its \mathfrak a-adic matrix topology. An operator A is zero at depth j precisely when A(M)\subseteq\mathfrak a^jM, or equivalently when its dual kills the finite \OO-module

Choose finitely many generators of Q_j. Exact coefficient reduction in the function model represents them by \varpi^{-e_i}f_i modulo integral functions, with each f_i integral. Thus the finitely many tests A^\vee f_i\in\varpi^{e_i}\widetilde S_\psi(U^2,\OO) detect depth j.

Approximate these f_i, to the required precisions, by finite sums of classical eigenvectors. Clear the denominators of this finite collection once. Continuity of the finitely many classical Galois traces then makes the corresponding Hecke operators agree to those precisions on the approximations whenever their Frobenius conjugacy classes are sufficiently close. Integral Hecke operators and their differences have norm at most one, so the approximation errors are controlled uniformly. Hence a Chebotarev sequence approaching any Galois conjugacy class is Cauchy at every depth j. Completeness gives a unique integral limit, independent of the sequence; the same finite tests prove continuity of the resulting trace function. Restriction to the integral Hecke-idempotent summands gives the assertion for each residual factor, with its inherited operator topology.

The pseudodeformation quotient and its support. The determinant is the fixed character \chi. The polynomial identities for a two-dimensional determinant hold on all classical packets and hence on their dense span. Boundedness and torsion-freeness make these identities integral. The universal property gives a continuous map from R_F^{\mathrm{ps},\chi} onto the completed trace algebra: the Frobenius traces generate its dense subalgebra, and the source is compact.

An operator vanishing at every classical eigenpacket vanishes on the dense classical span, hence is zero. Classical Hecke actions are semisimple. A nilpotent operator therefore vanishes there, proving reducedness; the action on integral functions proves \OO-torsion-freeness. The same test proves the asserted closure of the classical coefficient points.□

Proof. The one-factor input. We first specify the one-factor results. The integral center map and finiteness theorem of Paškūnas–Tung apply to every block, including the scalar residual block at 2 [48, Theorems 1.1–1.4]. In particular E_v is finite over R_v^{\mathrm{ps}}; an integral-surjectivity claim about the center is unnecessary. More precisely, integral finiteness is Theorem 1.3; the center comparison in Theorem 1.4 can have integral cokernel killed by 2. Their compact projective-generator equivalence is exact. Their generic reducible and absolutely irreducible characteristic-zero fibers are described in [48, Corollary 6.10 and Proposition 6.11]. The all-prime correspondence and locally algebraic comparison are supplied by [18, Theorems 1.1 and 1.3] and [17, Theorem 0.20]. Thus the old residual-scalar restriction at 2 does not enter the present application.

The product category. These statements extend to the required product category by the argument of [26, Lemmas B.7–B.8], adapted from the projective group to a fixed central character using the all-prime inputs just stated. Here are the details. Given a smooth admissible vector, fix compact opens in all but one factor. Its cyclic span under the remaining factor is finitely generated and admissible, so is finite length by [21, Lemma 2.3.6 and Theorem 2.3.8]. The commuting actions give a surjection from the tensor product of the one-factor cyclic spans onto the full cyclic span. Absolutely split simple factors tensor to simple product objects; filtering proves finite length. Hence the smooth product representation is locally finite.

Apply the one-factor compact equivalences successively. On dual finite-length objects this can be checked on simple tensors and extended by exactness. Taking inverse limits gives the equivalence for the compact locally finite category, with endomorphism ring E_{\mathrm{loc}}. Admissibility gives finite multiplicities in the cosocle: a homomorphism from a simple is detected by a nonzero compact-fixed generating vector. There are only finitely many simples in the block. Compact Nakayama therefore makes \Hom(P,M) finite over E_{\mathrm{loc}}.

Identifying the actual local and global actions. It remains to identify the block and its central action with the global Galois restriction. Test one factor at a time on the dense classical principal types of Proposition 4.1. Local-global compatibility gives a regular crystabelian Galois parameter with distinct inertial characters. For an irreducible parameter, the universal unitary completion of its algebraic-smooth principal series is the corresponding Banach representation by [8, Theorem 4.3.1 and Corollaries 5.3.1–5.3.4] and the locally algebraic comparison in [18]. Distinct inertia excludes both equal-root and special degeneracies.

For an ordinary parameter with cyclotomic exponents a>b, the algebraic factor is

The ordinary completion theorem [13, Proposition 2.2.1] identifies the relevant locally algebraic induction with a dense subspace of the continuous unitary induction in the order

where \eta_h,\eta_l have exponents a,b. The following character calculation fixes the inducing order on the dense ordinary test types; that order will matter again in Section 8. For the precise normalization, put n=a-b-1\geq0 and \delta=(\eta_l\eps)\eta_h^{-1}. Twisting by the full unitary character \eta_h^{-1}\circ\det reduces this induction to \Ind_B^{\GL_2(\QQ_2)}(\delta\otimes1)^{C^0}. With x the identity character and \eps=x|\cdot|, the source characters in the cited proposition, with k=n+2, are

Since \delta is unitary of algebraic exponent -n, both \chi_2 and (\chi_2/\chi_1)x^{-n} are smooth, and

The two characters in its completion formula are \chi_1|\cdot|^{-1}x^n=1 and \chi_2|\cdot|x^{-n}=\delta, in the reversed inducing order (\delta,1). Twisting back by \eta_h\circ\det gives the displayed order and changes the algebraic factor \det^{-n}\operatorname{Sym}^n into \det^{b+1}\operatorname{Sym}^{a-b-1}. On each lower-coordinate disk its vectors are precisely bounded-degree polynomial sections. The theorem concerns this locally algebraic subspace, not the whole locally analytic induction. When n=0, the excluded equality \chi_1=\chi_2|\cdot|^2 is exactly \delta=1, which is ruled out by the distinct finite inertial characters. The ordinary and nonordinary statements together supply every test type used in the density argument; see also [48, Propositions 5.3–5.6].

A map from such a classical constituent into completed forms extends continuously from its universal unitary completion. After scaling lattices, it is a bounded map in the compact block category. Naturality of the center then identifies the central trace action on its image with the Galois parameter given by classical local-global compatibility. The normalizations give \rho\otimes\eps, as stipulated above. The dense classical span proves equality of the two actions on the whole Banach space. Torsion-freeness of the integral lattice gives integral equality and shows that every other block projection vanishes.

Integral finiteness and coefficient fibers. It follows from integral Paškūnas–Tung finiteness that L is finite over R_{\mathrm{loc}}^{\mathrm{ps}}. Exactness and the equivalence imply that the action of \mathbb T on L is faithful. The same finite generators generate L over \mathbb T. Moreover,

the right side is finite over the noetherian local base. Thus \mathbb T is finite over that base.

At x, the quotient of L[1/2] by the Hecke maximal ideal is nonzero and finite-dimensional. Select a simple quotient after a finite coefficient extension. The commuting images of the factor endomorphism rings generate a finite-dimensional algebra. Their radicals kill the simple quotient; their central idempotents select semisimple factors; after splitting those factors, the quotient is a tensor product of simple modules for matrix algebras. The original integral actions preserve a lattice. All full finite lattices in this quotient are commensurable. Under the inverse compact equivalence, the tensor lattice corresponds to the completed tensor product of the individual Banach lattices. Dualizing the quotient therefore embeds the asserted Banach tensor product in the eigenspace.

For an absolutely irreducible regular de Rham parameter, the local factor has a nonzero locally algebraic vector, so their tensor does as well. For a generic reducible parameter, substituting \gamma_i\eps into the fiber formula of [48, Corollary 6.10] gives exactly the two displayed inducing orders.□

Proof. Write K_n=1+2^nM_2(\mathbb Z_2), n\ge2. Characters have bounded invariant dimension. For a principal series, the double cosets indexed by \mathbb P^1(\mathbb Z/2^n\mathbb Z) give O(2^n) invariants. The Steinberg case follows from the constants–principal-series–Steinberg exact sequence: its possible H^1(K_n,k) correction is bounded, since K_n has four generators.

For a supersingular \pi, let

Thus I_1 is the standard pro-2 Iwahori subgroup. Let M_\pi be the span of the translates of \pi^{I_1} under \left(\begin{smallmatrix}2^n&b\\0&1\end{smallmatrix}\right), where n\ge0 and b\in\mathbb Z_2; this space is Iwahori-stable [47, Definition 4.5 and Lemma 4.6]. Use the exact sequence of Iwahori representations

from [47, Theorem 1.2, proved as Theorem 6.3]. That statement explicitly includes p=2. By [47, Proposition 4.7 and Lemma 4.10], M_\pi is the sum of two injective envelopes for the upper unipotent group \mathbb Z_2. Each has 2^n-dimensional invariants under 2^n\mathbb Z_2. Conjugation by \Pi shifts the congruence level by at most one. Since \dim\pi^{I_1}=2, the invariants sequence gives

The last term bounds H^1(K_n,\pi^{I_1}). The classification of Barthel–Livné and Breuil [5, 6], in the form of [12, Corollary 1.2], shows that the cases above exhaust the irreducible admissible smooth representations with central character.

Let I be the augmentation ideal of a fixed deep uniform compact group. In characteristic two, the operators g^{2^n}-1=(g-1)^{2^n} belong to I^{2^n}. Thus the compact quotient modulo I^{2^n} is bounded in dimension by the dual K_{n+c}-invariants, for a fixed shift c. Choosing 2^n comparable to an arbitrary exponent gives linear augmentation growth. The central character is trivial on a sufficiently deep scalar subgroup, and the product of that subgroup with a deep determinant-one subgroup has finite index. The same upper bound therefore holds on the latter. Finite extensions preserve upper growth bounds, and tensor products add them. This proves the product assertion.□

Proof. Reducedness and prime avoidance give an element of \mathbb T that vanishes on every component except C and is nonzero on C. Its image on M is a nonzero submodule M' supported on C. The full M is finite free over a sufficiently deep Iwasawa algebra A=\OO[[K]]. The residual summand and M' embed in this free module.

The maximal augmentation filtration, including \varpi, has polynomial associated graded ring of dimension 1+3d, after taking K sufficiently deep. A nonzero submodule of a finite free A-module has that full growth degree: it contains a cyclic copy of A, and Artin–Rees compares its induced filtration with its intrinsic filtration. Consequently M' has growth degree 1+3d.

Let r=\dim C, and choose r parameters of the supporting Hecke quotient. Write J for their ideal. Proposition 4.2 and exactness of the block functor show that M'/JM' is dual finite length. Lemma 4.3 bounds its growth degree by d. Each Hecke parameter is topologically nilpotent; on the finite quotient M'/\mathfrak m_A M', some fixed power is zero. Hence there is c>0 such that

Filter M'/\mathfrak m_A^nM' by J. The degree-j layer is a quotient of at most \binom{j+r-1}{r-1} copies of

Only cn layers occur. Their total length is therefore O(n^{r+d}). Comparing with the degree 1+3d gives 1+3d\le r+d, which is the claim.□

Proof. The finite disjoint-union function model used in Proposition 4.1 is free under the deck action, since the ratio quotient kills scalars and the level was chosen to remove projective stabilizers. Its compact dual is therefore flat over \OO[\Delta], with augmentation equal to the Iwahori-level module. Apply the exact projective functor to the finite-presentation ideal tests for flatness. This proves flatness of the block module and commutation of reduction with that functor.

At one auxiliary prime, let \alpha:M_{\mathrm{old}}^{\oplus2}\to M_{\mathrm{Iw}} be the two degeneracy maps and \beta:M_{\mathrm{Iw}}\to M_{\mathrm{old}}^{\oplus2} their trace maps. Their composite C=\beta\alpha has diagonal entries q_v+1 and off-diagonal entries given by the unramified Hecke operator, up to central units. Write D=\det C. On the Iwahori module, D denotes the same scalar expression in the interpolated Frobenius trace and central operators. In particular this is one compatible scalar operator on both levels, not a determinant defined only on the old module. The standard degeneracy calculation expresses it, up to a central unit, as

where \phi_v is a Frobenius lift. At the chosen characteristic-two point, its value is a unit times the nonzero square of the Frobenius trace. Thus D becomes invertible in this localization.

Before inverting D, we have the integral identity

The local alternatives here are the Iwahori component of Henniart’s description [14, Appendix A, Sections A.1.3 and A.2.3]. Indeed an Iwahori-spherical classical cuspidal local representation is unramified principal series or special. In the former case its two Iwahori vectors are spanned by the degeneracies, and the identity follows from the matrix adjugate identity. In the latter case the old space is zero and the Frobenius eigenvalue ratio is q_v^{\pm1}, so the displayed scalar expression for D is zero as well. Classical density and torsion-freeness prove the identity on integral completed forms and their compact duals. Consequently D^{-1}\operatorname{adj}(C)\beta is both a left and a right inverse to \alpha after the selected localization. Iterating gives 2^h old copies.

The maps are Hecke equivariant. Interpolation identifies their Galois trace actions; at the absolutely irreducible enhanced curve, the framed representation action is consequently the actual old unramified action. Thus the augmentation factors through the old deformation quotient, with its original action, rather than only having the same multiplicity. The flatness and Iwahori augmentation established before localization remain valid at the selected point on both sides. The exact block functor and the finite-ideal tests used in Section 5 preserve them under the indicated flat coefficient extensions, localizations, completions, and smooth framings.

On a classical point with a diamond eigencharacter, local Langlands gives semisimple inertia characters equal to that character and its inverse, because \chi is unramified. Their sum is T(\tau_j). The displayed monic quadratic identity follows at every such point, and hence on the integral module by density. Fixed tame character transformations congruent modulo \varpi identify their residue function spaces. The associated trace comparisons show that this identification also preserves the residue deformation actions.

For the last assertion, multiply a form by the quadratic character composed with reduced norm. The central character is unchanged because its value on a scalar norm is a square. On the ratio-defined auxiliary subgroup, the two residue diagonal entries have the same image in the character quotient; their product is therefore killed by a quadratic character. Thus the level is preserved. Triviality at 2 and infinity preserves the local actions there, and the Galois parameter twists by the same character. Its residual reduction is one, so it fixes the characteristic-two point and its residual frames. This gives the asserted semilinear intertwiner.

Finally, adding a coefficient variable preserves flatness by the complete-module local criterion. Localization, completion of finite noetherian modules, and smooth framing extension preserve the stated flatness and module identities. The intertwiners are local and preserve maximal-ideal filtrations. Hence every assertion survives the operations used at the finite patching stages.□

Proof. First suppose that F'/F is Galois. Transfer the dense classical points by solvable base change, iterated through prime-degree cyclic extensions [2, Chapter III, Theorems 4.2 and 5.1]; for the original \GL_2 construction see [38, Chapter 11, Lemma 11.3 and Proposition 11.4]. Apply the unitary formulation after a norm twist when necessary, then undo the twist. Use Jacquet–Langlands [28, Theorems 14.4 and 16.1]. A regular cuspidal packet stays cuspidal along each prime-degree cyclic step: a loss of cuspidality would make it induced from a character of a totally real quadratic field, whose parallel algebraic exponents contradict regularity. Local-global compatibility and the uniform inertia control of Lemma 3.9 bound the conductors at a fixed finite set of places. The wild kernel is common and the rank is two, so the possible tame monodromy contribution is uniformly bounded. Thus a single tame level contains every transferred packet.

Their trace closure contains the restricted locus by density and continuity. Thus this locus is pro-modular over the specified field F', at the common tame level just chosen. After the inertial types are killed, only unramified principal or special local factors remain at a bad non-dyadic prime, both with Iwahori invariants. The stated level follows.

For a general solvable extension, pass to its solvable Galois closure. It is still totally real and completely split at 2. The Galois case then proves potential pro-modularity of the restricted locus. We use only this forward base-change assertion here; classical descent is applied later to an actual cuspidal packet.□

Proof. Put M=\ad\rho_*. The pairing (A,B)\mapsto\tr(AB) on M_2(K) is perfect also in characteristic two. Thus the dual coefficient module is again M: the mod-two cyclotomic twist is trivial. The relative Selmer condition is zero at S and full at Q_n. The dual condition therefore consists of classes unramified outside S whose localizations at Q_n vanish.

Set

The tower group is solvable. Nonvirtual solvability of \rho_* therefore implies that the geometric reduced Zariski closure of \rho_*(H) contains \SL_2; the same is true for every open subgroup of H. Indeed, take the closure over an algebraic closure of K. Its smooth connected derived group lies in \SL_2. If proper, it has dimension at most two and is solvable, forcing the identity component itself to be solvable, a contradiction [43, Corollaries 6.19 and 17.27]. Consequently regular semisimple elements are dense in every open coset of H.

We give the detection argument explicitly. Let c\in Z^1(G_{F,S},M) have zero diagonal evaluation at every regular semisimple \sigma\in H. Then the first-order representation

has the same trace as \rho_* on those elements. Indeed, the quotient

is the diagonal subspace in an eigenbasis, and zero evaluation kills both diagonal entries. Continuity and the density just observed give equality of traces on all of H.

Choose h_1,\ldots,h_4\in H such that A_j=\rho_*(h_j) form a basis of M_2(K). The lifts \rho_c(h_j) form a basis over K[\epsilon]. The Gram matrix \tr(A_jA_k) is invertible, and equality of traces on products of two and three group elements shows that this Gram matrix and all multiplication structure constants are unchanged after lifting. The linear map taking A_j to \rho_c(h_j) is therefore an algebra automorphism of M_2(K[\epsilon]), reducing to the identity. Such an automorphism is conjugation by a matrix congruent to the identity: this follows, for example, by carrying a system of matrix units to its image and choosing the corresponding basis. Pairing with the four basis matrices then shows that this conjugation carries \rho_*|_H to \rho_c|_H on every element of H. Hence [c]|_H=0.

The invariants M^H are the scalars. Inflation–restriction bounds the kernel of all these evaluations by

This space has dimension at most two. To see the last assertion, the corresponding tower over \QQ is pro-two and unramified away from 2. Its quadratic subfields are \QQ(i), \QQ(\sqrt2), and \QQ(\sqrt{-2}). Its elementary two-character rank is therefore two. The field F is disjoint from that tower: a nontrivial intersection would have a quadratic subfield, whereas none of these three fields has 2 split completely. Thus the tower group over F has the same two-character rank.

The ambient cohomology space is finite dimensional. A finite list of regular elements consequently detects its quotient by a subspace of dimension at most two. Extend K once to split the eigenvalues of this list. Both regular split eigenvalues and the nonvanishing minors of the evaluation matrix persist on sufficiently small neighborhoods. For each n, Chebotarev approximation supplies Frobenius elements in these neighborhoods which are trivial at the required finite stage of the tower [42, V, Theorem 3.23]. All conditions concern finite quotients after the neighborhoods have been chosen. Extra such primes can be added, so the cardinality can be any fixed sufficiently large h. The dual Selmer dimension is then at most two.

The auxiliary evaluations have reduced the undetected dual Selmer space to at most two scalar tower directions. The next count turns this into the uniform presentation bound g=2h+1. For completeness, write h^j_v=\dim_K H^j(G_{F_v},M). Local duality and Euler characteristics give

For trivial complex conjugation, the dimensions are (h^0_v,h^1_v)=(4,4). For nontrivial unipotent complex conjugation they are (2,0). Thus the dyadic contribution 4d cancels the real contribution -4d. The localization image in the direct sum of the local H^1 spaces is its own annihilator under Poitou–Tate. For a subspace of local conditions L, elementary linear algebra therefore gives the Selmer difference as \dim L-\tfrac12\sum_v h^1_v. Here L is zero at S and full at Q_n. At each auxiliary prime regularity gives h^0_v=2 and h^1_v=4. Consequently the identity is

The finite-coefficient formulas are [40, I, Corollary 2.3, Theorems 2.8, 2.13, 4.10, and 5.1]; their passage to K is the lattice-and-limit argument of Lemma 3.5. With independent S-frames the relative tangent dimension is

Generators with a fixed old composite. To obtain the last assertion as well as the variable bound, write A_\eta=R_{\mathrm{loc},\eta} and I_{\eta,n}=\ker(R_{\eta,n}\to R_\eta). The relative cotangent spaces are

The first maps onto the second. Its kernel is the image of the actual ideal I_{\eta,n}: if a representative maps into \mathfrak m_\eta^2+\mathfrak m_{A_\eta}R_\eta, lift a decomposition of its image into \mathfrak m_{\eta,n}^2+\mathfrak m_{A_\eta}R_{\eta,n} and subtract it. This leaves a representative in I_{\eta,n} of the same relative cotangent class.

Choose a basis of the old space Q_\eta with fixed representatives r_{\eta,1},\ldots,r_{\eta,q} in R_\eta. Choose their reductions first in the common special-fiber diagram and then lift to both values of \eta. Their relative cotangent spaces agree modulo \varpi, so the number q is the same. For each n, lift these fixed old representatives to R_{\eta,n} and complete their independent cotangent classes to a basis using representatives in I_{\eta,n}. The tangent bound just proved shows that at most g=2h+1 elements are needed; pad the list with zeros to length g.

These choices can have identical reductions for \eta=1,i. First choose the lifts and the kernel-complement elements in the common special-fiber diagram. Lift an element arbitrarily to R_{\eta,n}. If its image in R_\eta differs from the required fixed old value by \varpi a, subtract \varpi times any preimage of a in R_{\eta,n}. This preserves the chosen special-fiber reduction and makes the old image exact. For a kernel-complement element the required old image is zero. No flatness of the old global ring over the coefficient DVR is used.

Complete relative Nakayama now gives (13). Every composite to R_\eta sends the first q variables to the fixed r_{\eta,j} and all remaining variables to zero, independently of n. The two presentations still agree modulo \varpi. The variable count, and hence every later ambient dimension, is unchanged.□

Proof. Let S_2 be the dyadic places. Take the ordinary ray class group of modulus \prod_{q\in Q_n}q, and quotient by the dyadic ideal classes. Omitting real places from the modulus requires them to split; at the dyadic places, both the units and uniformizers are trivial in this quotient. The ray class exact sequence is

This is the usual ideal-theoretic ray sequence, with the primes in S_2 inverted [42, V, Theorem 1.7]. The group of dyadic S-units has free rank

Here we use the S-unit theorem [41, Theorem 5.11]. The direction generated by 2 is primitive in its free part: its valuation at each dyadic prime is one, since 2 splits completely in F. Since F is totally real, generators for these units may therefore be chosen as -1,2 and 2d-2 further units. For n\geq2, put a_n=n-1 and take the cyclic residue-unit quotient of order 2^{a_n} at each auxiliary prime. Both -1 and 2 vanish in these quotients: the first is a 2^{a_n}th power because \mu_{2^{a_n+1}} is present, and the second is such a power by the imposed Kummer splitting. The elementary divisor theorem shows that quotienting (\ZZ/2^{a_n}\ZZ)^h by the images of the remaining 2d-2 generators leaves a quotient (\ZZ/2^{a_n}\ZZ)^{h-2d+2}. The image of the local units in \Gamma_n thus has such a quotient. A finite abelian group containing a subgroup with this quotient also has this quotient: in invariant-factor form, the subgroup forces at least this many factors of order at least 2^{a_n} in the containing group. Apply this to \Gamma_n. Class field theory turns its quotient into the required characters [42, V, Theorems 5.3 and 5.5]. Discarding the first finitely many indices is harmless.□

Proof. At each depth, taking an ultrapower extends the scalars in each finite-dimensional associated graded piece from K to K^*. Thus the graded ring of B^* is generated by the images of any fixed finite set of generators of \mathfrak m. Successive expansion in these generators, followed by completeness, identifies the jet filtration with powers of the ideal they generate. The complete filtered-ring argument then proves noetherianity: initial ideals are finitely generated in the noetherian associated graded ring, and successive approximation lifts those generators.

For an exact sequence of fixed finite modules, Artin–Rees makes the induced and intrinsic filtrations on its submodule cofinal. Use the induced filtration to obtain exact sequences of jets; cofinality identifies their inverse limits with the construction using intrinsic filtrations. The resulting limit is exact. A fixed finite presentation identifies M^* with B^*\otimes_B M. This gives flatness by the finite-ideal criterion, and the nonzero residue extension gives faithful flatness. Lengths of the graded pieces are unchanged after residue-field extension, so their Hilbert growth, and hence the support dimension, is unchanged.

Suppose next that B is a domain and is not a field. Choose a discrete valuation v of its fraction field centered at \mathfrak m, for example one supplied by the normalized blowup of \mathfrak m. It is positive on \mathfrak m. The ideals

have zero intersection. Chevalley’s lemma for a complete local ring implies that for every q there is an a(q) with I_{a(q)}\subseteq\mathfrak m^q. Thus being nonzero modulo a fixed \mathfrak m^q bounds v(x) from above, uniformly in x. If c_0>0 is the minimum valuation of fixed generators of \mathfrak m, then v(\mathfrak m^r)\geq rc_0. For two fixed nonzero jets, these inequalities give a depth at which the product of any representatives of the two jets is nonzero. Applied on a set belonging to \mathfrak U, they show that the product of two nonzero elements of B^* cannot vanish. The assertion is immediate when B is a field.

For the varying construction, choose a uniform number of generators at each level. Each jet ultraproduct is then a quotient of a fixed finite free module over the corresponding jet of B^* and has finite length. Images of kernels at any fixed depth form a descending chain in a finite-length module, hence stabilize. The Mittag–Leffler argument gives surjectivity on inverse limits and a finite generating set over B^*. The same argument applied to a right-exact sequence proves the cokernel assertion. This does not assert exactness for arbitrary varying submodules.

For the uniform assertion, choose one finite free presentation F\twoheadrightarrow C, with kernel L, and lift it to f_n:F\to H_n. Then H_n=P_n+f_n(F). If x\in P_n\cap\mathfrak a^jH_n, write

Since x maps to zero in C, we have y\in L. Artin–Rees for the one fixed inclusion L\subset F gives y\in\mathfrak a^{j-c}L, with c independent of n. As f_n(L)\subset P_n, this proves (19). Applying this successively to the fixed quotients of a finite filtration gives uniform strictness for each inclusion. Uniform cofinality transfers these bounds to the ambient jets, proving the final statement.□

Proof. Polynomial layers with a fixed old action. Fix \eta, and write \mathfrak m for the maximal ideal of B_\eta. The presentations of Lemma 5.2 give surjections B_\eta\to R_{\eta,n} whose composites to R_\eta are independent of n. Regard

as a fixed finite B_\eta-module through this common composite. Let J=(\varpi,V_1,\ldots,V_h) denote the ideal of commuting operators on the auxiliary modules and on their patched limit.

Fix a truncation length r. For all sufficiently large n, the relations (1+V_j)^{2^n}-1 have J-order greater than r. Flatness in (15), together with the actual old-action identification (16), gives multiplication isomorphisms

where T_0,T_j represent \varpi,V_j. These are isomorphisms of B_\eta-modules. Indeed the Galois action commutes with the J-operators, so its action on each layer is obtained from its action on N_{\eta,n}/JN_{\eta,n}=C. The fixed old composite therefore makes every degree a fixed finite direct sum of C. Only the extensions joining these layers vary with n.

Exactness at each fixed truncation length. Put H_n=N_{\eta,n}/J^rN_{\eta,n} and F_j=J^jH_n for 0\leq j\leq r. All these modules are finite over B_\eta. Their successive quotients are the fixed modules in (22), so their numbers of generators are bounded independently of n for this fixed r. If \mathfrak q_n is the maximal ideal of R_{\eta,n}, the surjection from B_\eta gives

Thus the \mathfrak m-adic filtrations are precisely the maximal-ideal filtrations used for patching these modules.

Apply Lemma 5.4(3) over the one fixed ring B_\eta to

Its fixed-quotient Artin–Rees bound makes each inclusion uniformly strict. Composing the finitely many bounds compares the induced filtration from H_n with the intrinsic filtration on every F_j. The constants may depend on r, but are independent of n. Consequently this finite filtration patches exactly. No exactness for arbitrary varying submodules is required.

The ideal J^r is generated by a fixed finite number of monomials in its commuting generators. The cokernel assertion of Lemma 5.4(2) therefore identifies the patched H_n with N_{\eta,\infty}/J^rN_{\eta,\infty}; the same argument identifies its quotients by J^j. These identifications respect the multiplication maps and are compatible as r varies. We obtain

Here C^* is the scalar extension of the fixed B_\eta-module C from Lemma 5.4(1).

Regularity, dimension, and augmentation. The finite joint action algebra generated by the U_j is complete local: modulo the deformation-ring maximal ideal, every V_j is nilpotent. Its module topology agrees with the one used for patching. Since J lies in its maximal ideal, the finite module N_{\eta,\infty} is J-adically separated and complete. A nonzero element killed by \varpi would therefore have a first nonzero J-initial form, contradicting injectivity of T_0 in (23). The same initial-form argument gives

Hence the associated graded module after quotienting by \varpi is obtained by setting T_0=0. Repeating the argument for V_1,\ldots,V_h proves that the stated sequence is regular.

The final quotient by this regular sequence is C^*. Its dimension is at least 2d+4s-1 by (17) and Lemma 5.4(1). Each member of the regular sequence raises support dimension by one on passing back to the original module. Dimensions over the finite joint action algebra and over R_{\eta,\infty} agree, giving (20). Finally, apply cokernel patching to the diamond augmentation (16), without reducing by \varpi. The old B_\eta-action is fixed by the same common composite, so this gives (21) with that actual old deformation action.□

Proof. Finite twisting and its dimension contribution. The finite characters of Lemma 5.3 give a local ring map after patching,

Here one takes bounded total jets in the ring and character variables. At each such depth the equations (1+l_j)^{2^{a_n}}-1 disappear for large n. Coefficientwise patching of the finitely many monomials at each depth identifies the target with the displayed power-series ring. This explains both existence of the map and its target.

The map (24) is finite. Since A_n is a character-group quotient, choose, for each n and each j, a Galois element detecting exactly its jth coordinate. Its determinant divided by \chi maps under twisting to (1+l_j)^2. Although these Galois elements may vary with n, their determinant functions determine elements in every patched jet, and hence in R_{\eta,\infty}. The resulting elements have residue one. On the closed fiber of (24) they impose l_j^2=0. At l_1=\cdots=l_b=0 the map is the fixed-determinant surjection. Its closed fiber is therefore an Artinian local algebra, of length at most 2^b, with the same residue field. Complete Nakayama proves finiteness.

Let T_\eta denote N_{\eta,\infty}[[l_1,\ldots,l_b]], restricted along (24) and then regarded as a finite B_\eta^*-module. Its support is the closed image of the power-series-extended support. Finiteness of the map preserves its dimension, and (20) gives

Furthermore \varpi is regular on T_\eta, since it is regular before power-series extension and restriction of scalars. No minimal support prime of T_\eta is vertical.

Full support on the i-model and transfer to the 1-model. The dimension estimate now reaches the entire ambient dimension after twisting. The spectrum of B_i^* is irreducible of dimension e. It follows that T_i has full support. On reducing modulo \varpi, the finite-stage modules, actions, and twisting maps for \eta=1,i are identical. Reduction is a cokernel, and power-series extension commutes with it and with the jets. Thus T_1/\varpi T_1 and T_i/\varpi T_i are the same module over the identified special fibers. For any finite module T, local Nakayama gives

It follows that \Supp T_1 contains the generic point of every top-dimensional special-fiber component of B_1^*.

We give the component argument explicitly. Let \mathfrak q be one of these generic points, and let \mathfrak p be the unique minimal prime of B_1^* below it. The local geometry gives \dim B_1^*/\mathfrak p=e and \dim B_1^*/\mathfrak q=e-1; equivalently \mathfrak q/\mathfrak p has height one. The latter also follows from the principal ideal theorem, since it is a minimal prime of the divisor \varpi on this component. Choose a minimal support prime \mathfrak a\subseteq\mathfrak q of T_1. The uniqueness assertion forces \mathfrak p\subseteq\mathfrak a. The prime \mathfrak a is horizontal, so it is not \mathfrak q; there is no prime strictly between \mathfrak p and \mathfrak q. Thus \mathfrak a=\mathfrak p. Every ambient component is obtained this way, and T_1 has full support.

Removing twists by actual sign intertwiners. It remains to remove the formal twists. For each fixed sign tuple \sigma\in\{1,-1\}^b, the finite-level quadratic twists described above give local ring automorphisms and semilinear module isomorphisms. They preserve maximal-ideal filtrations, and therefore patch, together with their inverses. In particular the support of N_{1,\infty} is stable under the resulting sign automorphisms of R_{1,\infty}^f. Evaluation of (24) at l_j=\sigma_j-1 is the fixed-determinant quotient followed by this patched sign twist, by its finite-level construction. The substitution is well-defined on formal series, since its values are 0 or -2.

Let x\in\Spec R_{1,\infty}^f. Full support of T_1 supplies a point y in the support of N_{1,\infty}[[l_1,\ldots,l_b]] mapping to x. The determinant calculation, since x has fixed determinant, gives in the residue field \kappa(y)

For every j, primality therefore chooses l_j=0 or -2. Choose the corresponding sign tuple. Quotienting by these sign equations leaves a nonzero localized module, by Nakayama, and identifies the quotient with N_{1,\infty} at a point x' of its support. Its image x is the sign transform of x'. Sign stability puts x in the untwisted support. In characteristic two the two signs coincide, with the same conclusion. This proves the first assertion.

Diamond augmentation and the old action. Finally consider the old quotient (R_1^f)^*. Let \mathfrak p be any of its primes, viewed in \Spec R_{1,\infty}. It belongs to the support just proved. In the finite joint action algebra generated by the U_j, there is a support prime \mathfrak P above \mathfrak p. On the old quotient each \tau_{j,n} is trivial, so (18) becomes

Hence V_j\in\mathfrak P for all j. The nonzero finite module localized at \mathfrak P remains nonzero after quotienting by all the V_j, by Nakayama. Equation (21) shows that (N_1^{\oplus2^h})^* has support at every prime of (R_1^f)^*. Faithful flatness in Lemma 5.4 now gives \Supp_{R_1^f}N_1=\Spec R_1^f.□

Proof of Theorem 5.1. We have proved full support on the enhanced completed global deformation germ with independent S-frames, global determinant \chi, and unipotent bad inertia. After the initial base changes, the source locus has exactly these local conditions. At its absolutely irreducible curve, the representation and pseudo-deformation problems agree after a smooth choice of frames. Adding z, passing to a finite curve-field extension, and taking this completed localization are faithfully flat on the indicated germ, by Lemma 3.4. Consequently the closed Hecke support contains that source germ.

To pass from the germ to the asserted irreducible closed locus, work on its reduced irreducible coordinate ring. The image of any equation of the Hecke support is nilpotent in the completed source germ, by the set-theoretic containment just proved. Faithful flatness descends this nilpotence to the localization at the enhanced curve. That localization is reduced, so the image is zero there. It therefore vanishes on the entire irreducible locus: localization of a domain is injective. Thus the locus is pro-modular over the field reached by the allowed solvable base changes. This is precisely potential pro-modularity and proves the theorem.□

Proof. If the narrow class number of K were even, the nonzero \FF_2-space \operatorname{Cl}^+(K)/2 would have a nonzero invariant linear functional for \Gal(K/E). Here we use the elementary fact that a nonzero module of a finite 2-group in characteristic two has nonzero invariants, applied to the dual module. Class field theory would give a quadratic extension H/K, unramified at every finite prime, which is Galois over E. The group G=\Gal(H/E) has order four. Above the unique finite prime ramified in K/E, its inertia group I has order two and maps isomorphically to \Gal(K/E). It thus has trivial intersection with \Gal(H/K). This excludes G\simeq\ZZ/4\ZZ, whose only subgroup of order two is \Gal(H/K). Hence G is the Klein four group. All inertia groups above that prime equal I, since G is abelian, and all other finite inertia groups are trivial. The quadratic extension H^I/E is consequently unramified at all finite primes, contrary to oddness of the narrow class number of E. Ramification at infinity is allowed by the narrow class field and causes no exception to this argument.

For the last assertion, the signature cokernel of the units is a 2-group embedded in the narrow class group, so it is zero. For a totally real field of degree e, Dirichlet’s unit theorem [41, Theorem 5.1] gives |\mathcal O_E^\times/(\mathcal O_E^\times)^2|=2^e. The signature map from this group onto \{\pm1\}^e is therefore an isomorphism. Its kernel is the group of totally positive units modulo squares.□

Proof. Let X^{\mathrm{rel}} have full condition at T and the old conditions elsewhere. The relevant part of the Poitou–Tate comparison sequence is

The last arrow is the transpose of localization on Y under the perfect pairings H^1_v/L_{\mathrm{ur},v}\times L_{\mathrm{ur},v}\to k_0. It is injective by the hypothesis, so X^{\mathrm{rel}}=X. The new Selmer group is a subspace of this unchanged relaxed group. Its localizations must belong both to the old unramified line and the new transverse line, hence must be zero. This gives the asserted kernel exactly.□

Proof. Set N=G_{B K_\alpha E_2}. The finite quotient

has C=\Gal(K_\alpha/\QQ) of odd order and P a 2-group. On k_0(\beta) the group C acts nontrivially. Averaging over C and using k_0(\beta)^C=0 gives H^i(C\times P,k_0(\beta))=0 for all i\ge0. Inflation–restriction thus makes restriction to N injective. On N the coefficients are trivial, so cocycles are additive homomorphisms and coboundaries are zero.

For independent classes c_1,\ldots,c_t, their joint evaluation map N\to k_0^t has additive image stable under multiplication by \beta(C). These scalars generate k_0 over \FF_2, so its image is a k_0-subspace. If it were proper, a nonzero linear functional annihilating it would give a nontrivial linear combination of the c_i restricting to zero on N, a contradiction. Its image is therefore all of k_0^t. The paired assertion follows by applying this to W+\sigma W; trivial intersection makes its two summands independent. The conjugation formula for cocycles identifies these with the evaluations at the two conjugate places.

For the inert assertion choose g\in G_{\QQ} acting nontrivially on B and trivially on K_\alpha E_2. Then g^2\in N, and g represents \sigma on cohomology. Since \beta(g)=1, for h\in N the cocycle identity gives

The first term is a fixed affine offset. The proven evaluation surjectivity on (1+\sigma)W shows that this expression is uniform in W^\vee if 1+\sigma is injective on W. Also c(g^2)=(\sigma c)(g^2) and (1+\sigma)\sigma c=(1+\sigma)c in cohomology. Evaluations on N do not see coboundaries, so (30) gives equality for c and \sigma c, including the affine offset.

For different Frobenius orbits, regard these evaluation images as \FF_2[C]-modules. They have different simple isotypes; semisimplicity of \FF_2[C] makes their joint image the product of the individual images. In the remaining case the automorphism x\mapsto x^{2^j} of k_0 carries cocycles with coefficient \alpha to those with coefficient \alpha^{-1}. It commutes with restriction, conjugation, and evaluation. Every ramified line below is generated by an \FF_2-valued quadratic character, so these lines, as well as the zero, full, and unramified conditions, are preserved. All evaluation matrices are entrywise Frobenius transforms and have the same ranks.

The finite collection of cocycles factors through a finite quotient after adjoining their coefficient fields. Chebotarev realizes the conjugacy class of the chosen element of this quotient. Choosing a place in the corresponding finite Galois extension above the rational prime then chooses a Frobenius representative; one may choose the prescribed representative in that class. Conjugacy scales coefficient evaluations and may interchange the two split places, which preserves every rank and nonvanishing condition in use. Removing finitely many primes leaves infinitely many choices.□

Proof. The intersection is stable under \sigma. If nonzero it has a nonzero invariant class, which descends to \QQ because k_0(\beta)^{G_B}=0. Its dyadic localization is zero: intersecting either pair of conjugate dyadic conditions imposes zero at both w_1 and w_2. At infinity it satisfies at least the full condition defining X_0^\beta; this observation also covers W=Y_B^\beta. At every conductor, splitting in B makes descent of the rational line k_0\delta immediate. At other primes unramified in B/\QQ, the unramified condition descends by restriction to inertia. There is no condition to check at q_0: the coefficient is unramified with nontrivial Frobenius there, so local H^0 and H^2 vanish and the local Euler formula gives H^1=0. Thus the nonzero descended class lies in X_0^\beta, a contradiction.□

Proof. For absolutely irreducible residue this is Serre’s modularity theorem, including characteristic two [33, Theorem 0.2]; the results are due to Khare–Wintenberger and Kisin [31, 32, 35]. The characteristic-two parity condition is automatic.

For reducible residue, the existence of a cuspidal lift, including in characteristic two, is already proved by Billerey–Menares [9, Theorem 2.1]. The geometric argument below uses the same Eisenstein difference as their [9, Theorem 2.3]; we retain the details of integrality, cuspidality, and lifting needed here.

Write the reducible residue as \bar\chi_1\oplus\bar\chi_2 and put \alpha=\bar\chi_1/\bar\chi_2. After a finite-order twist it suffices to lift the eigensystem of 1\oplus\alpha. The character \alpha has odd order, is unramified at 2, and has an even Dirichlet lift of odd conductor. Choose an even weight k'\ge4 and a normalized Eisenstein series f with this character pair and a_1(f)=1. Take an odd level N\ge5 divisible by the conductor. Let \ell be a prime congruent to one modulo N2^a, where a will be sufficiently large. Put

Regard this as a form of level \Gamma_1(N\ell). Its positive Fourier coefficients at infinity are 2-integral, and its constant term there is zero. The tame modular curve has good reduction, with geometrically integral special fiber [30, Corollary 10.9.5(2) and Table 10.9.6]. The q-expansion principle [29, Theorem 1.6.1 and Corollary 1.6.2] therefore extends g integrally: a vertical pole would be detected at that component’s cusp expansion.

We verify cuspidality modulo \varpi at every cusp. For a cusp matrix \gamma\in\SL_2(\mathbb Z), the primitive-column decomposition is

Since \ell\equiv1\pmod N, one can choose \gamma'\equiv\gamma\pmod N. The two constant terms in g|_{k'}\gamma therefore differ by the factor

There are only finitely many constant terms of f at level N. Choose a larger than their denominator valuations, with an additional power sufficient for reduction modulo \varpi. The displayed factor makes every constant term of g vanish in the residue field. Its coefficient of q is one, so the reduction is nonzero. Away from N\ell, it is an eigenvector with the Eisenstein residual eigensystem 1\oplus\alpha.

This mod-\varpi cusp form lifts to the characteristic-zero cuspidal space in the same weight. Indeed on the tame modular curve, Kodaira–Spencer gives \Omega^1\simeq\omega^2(-\mathrm{cusps}). For this formula and positivity of \omega, use [29, Section 1.5 and Appendix A1.5.4]: after adjoining the required roots of unity, a symplectic full-level component maps to the \Gamma_1-curve by a finite cover of odd degree. The logarithmic Kodaira–Spencer isomorphism descends along this tame cover, and positivity descends along the finite map. On the special fiber, Serre duality [58, Lemma 53.4.1 and Remark 53.4.3, Tags 0BS2 and 0BS4] identifies the obstruction H^1(\omega^{k'}(-\mathrm{cusps})) with the dual of H^0(\omega^{2-k'}), which is zero for k'\ge4. The corresponding integral H^1-module is finite by proper coherent cohomology [58, Lemma 30.19.2, Tag 02O6]. The multiplication-by-\varpi exact sequence and the special-fiber vanishing make multiplication by \varpi surjective on this module. Nakayama’s lemma makes it zero, so reduction of cuspidal sections is surjective. The finite integral cuspidal Hecke module has a characteristic-zero eigenpacket specializing to this residual eigensystem: its Hecke algebra is finite and torsion-free over \OO, and the residual maximal ideal lies above a horizontal minimal prime. Finally undo the finite-order twist. The construction also covers \alpha=1.□

Proof. Retain the real local problems and the m marked dyadic line problems as the local base. Reducing this base changes no spectrum. By Lemma 3.12, it is a complete local domain of dimension

The relative framed presentation of Proposition 3.6 has generators minus relations at least

The last inequality uses the three-dimensionality of (\ad^0)^\vee(1). A quotient of a complete local domain with this presentation has all components of dimension at least 2d-m. Grothendieck’s connectedness theorem gives connectedness in dimension at least 2d-m-1: in the form used here, cutting a complete local domain of dimension N, after adding a formal variables, by b equations gives connectedness in dimension at least N+a-b-1. This is the component-chain formulation of the connectedness theorem [27, Exposé XIII, Théorème 2.1].

We bound the reducible locus separately. It is empty for absolutely irreducible residue. Otherwise add a global invariant line by proper incidence. At a closed point of that incidence, choose an adapted basis and first hold this line fixed. Its movement costs one parameter. The diagonal global characters with fixed product vary in dimension at most one, by abelian Leopoldt over F; see Lemma 3.8. Freeze those character parameters.

At a marked place the closed local matrices are scalar and the marked line is transverse to the chosen global line. An infinitesimal upper extension, with diagonal characters frozen, must therefore vanish locally: a nonzero upper nilpotent matrix cannot preserve a second transverse line. Thus its cohomology class is strict at the marks. Moving a marked line costs at most one further tangent parameter, giving m in total. There is at most one upper coboundary parameter. The strict Selmer bounds in (26) therefore give the following dimensions:

Here the three remaining parameters account for the diagonal base, the global line, and the possible coboundary. The relative tangent estimate bounds the number of generators over the diagonal base, so it is an upper bound for the whole formal reducible chart, not only for its tangent space at a fixed diagonal character. Properness of the global-line incidence transfers the bound to its image. Finite residue extensions do not affect these dimensions.

The first two inequalities in (39) make both bounds smaller than 2d-m-1. This proves the lemma.□

Proof. Work first in the completed chart. By Lemma 7.3, two successive components in a connecting chain intersect in dimension at least 2d-m-1. Choose an irreducible component of maximal dimension in that intersection. Its generic global representation is absolutely irreducible, because the reducible locus has smaller dimension. Pass first to the completed trace-and-character image, retaining the characters on the marked lines. Forgetting the global frame and the line choices loses at most 3+m dimensions, by Lemma 3.2. Proposition 3.1 makes this image finite over its trace image, so forgetting the recorded characters causes no further dimension loss. Thus the pseudo-image of the chosen component has dimension at least

At each allowed non-dyadic prime impose that the trace of a chosen Frobenius lift equal its residual value. These u_F equations leave a closed locus of dimension at least 2d-2m-4-u_F>d. The bad pseudo-locus bounds of Lemma 3.8 and curve avoidance therefore produce a non-virtually-solvable curve in it. Lemma 3.9 gives finite local images at the non-dyadic allowed places.

If the preceding component is potentially pro-modular, this curve is potentially pro-modular. Theorem 5.1 makes the next component potentially pro-modular. A starting curve as in the statement first makes its containing component potentially pro-modular by that same theorem. Iterating crosses the entire chain. Only finitely many extensions and levels occur. Take the compositum of their Galois closures and a common deeper level, using Lemma 4.6.

Completion is faithfully flat. Every component of Y through the closed point is the contraction of a minimal prime of the completed chart. The component pseudo-images are the corresponding trace contractions. The conclusion therefore descends from the chart to Y.□

Proof. Work over an algebraic closure of the residue field. For scalar residue with marks, a transverse residual representation is a nonzero additive extension after a scalar twist. It has a unique global invariant line. Its cocycle is zero at the marks. With the global line fixed, these cocycles form a vector space with the origin removed. The marked complementary lines form products of affine line charts. Conjugating the basis gives a connected parameter space whose image is the entire transverse fiber.

For distinct residual characters, use both upper triangular orientations. In either orientation, with its global line fixed, the extension cocycles form a vector space. In the marked case they vanish locally at the marks. The allowed choices of marked lines form a nonempty open subset of the product with projective lines, hence an irreducible parameter space. Varying the basis preserves connectedness. The two orientations meet at the split representation; when there are marks, choose lines avoiding both global invariant lines. Without marks there is no transversality condition, and the same two families meet at that split representation. For absolutely irreducible residue, the residual fiber is a conjugacy orbit. These parameter spaces are finite-dimensional algebraic spaces of cocycles: this follows either from the residual representation scheme or from the finite-dimensional residual H^1. This proves geometric connectedness.

There are finitely many components of the noetherian scheme Y. Their nonempty intersections with the transverse residual fiber form a finite closed cover. Make a graph with these components as vertices, joining two vertices when they meet in that fiber. The graph is connected because the fiber is connected. Choose a spanning tree and, for each edge, a closed point of the corresponding intersection in the transverse locus, after a finite coefficient extension. At that point Lemma 7.4 gives the same potential pro-modularity status to the two components.

Only finitely many such points and completed-chart component chains are needed. Their field extensions have a common Galois compositum, and forward base change gives one common deeper tame level. Thus the conclusion holds simultaneously for every component meeting the transverse fiber, and hence for every transverse chart. Every component through a transverse point is accounted for by completion, as in the preceding lemma. The finite coefficient extensions are harmless because the integral forms and their Hecke support commute with them.□

Proof of Theorem 7.1. The seed and target loci over A. For the seed and fields already chosen, solvable base change and Jacquet–Langlands give a cuspidal residual packet over A. It remains cuspidal by regularity, as in Lemma 4.6.

Its central character need agree with \psi=\chi\eps only modulo \varpi. At a sufficiently deep tame level their transformations coincide on reduction. The exact integral function model of Proposition 4.1 therefore gives residual occurrence also for the specified central character \psi. Proposition 4.4 supplies a pro-modular component C_{\mathrm{seed}} in the fixed-determinant pseudo-base over A with

Independently choose an unrestricted global fixed-determinant framed component through r|_{G_A}, or containing the restricted framed support of the prescribed family. Such a component exists by taking a minimal prime below that support. Its generic representation is absolutely irreducible because it contains the absolutely irreducible target point. The relative presentation over the real local rings gives dimension at least

before forgetting the frame. The loss under trace contraction is at most three. Its pseudo-image C_{\mathrm{target}} therefore has dimension at least 2d_A-2. This large unrestricted component contains the trace image of the initially prescribed family; no lower dimension bound on that smaller family is required.

Choosing curves in the two loci. In each of C_{\mathrm{seed}} and C_{\mathrm{target}}, pass to the special fiber and prescribe residual Frobenius trace at every bad non-dyadic place. If there are marks, prescribe in addition that the entire local pseudorepresentation at v_0 is its constant residual one. This last condition costs at most the number of topological generators of one fixed local pseudo-ring for \mathbb Q_2, a constant independent of d_A. All the other costs are bounded by 1+u_A, with a fixed additional constant for the target dimension estimate. Taking d_A sufficiently large leaves dimension greater than d_A. By Lemma 3.8, each locus contains a non-virtually-solvable curve. The Frobenius constraints make its bad local images finite.

Entering the transverse fiber over F. We explain why both curves enter transverse charts over F. Without marks, any stable lattice suffices. With marks, the local pseudo at v_0 is constant and scalar after untwisting, so the curve-field representation has an invariant line there after a finite coefficient extension. Choose a nonzero vector v on this line and let

Compactness bounds this module inside a fixed lattice. Global irreducibility makes it full rank, and it is stable by construction. The vector v is primitive in \mathcal V: if it belonged to t\mathcal V, all its translates would, and Nakayama would force \mathcal V=0. Its reduction lies in no global invariant line, since its orbit spans \mathcal V/t\mathcal V.

Restrict the lattice to G_F. No new global invariant line appears in the residue. For distinct residual characters, a nonsplit extension remains nonsplit by the restriction injectivity in Proposition 6.1; a split representation retains its two distinct character lines. For scalar semisimplification, the spanning property above rules out a split scalar residue, and its nonzero additive extension remains nonzero by the same injectivity. These descriptions hold after extending the residue field. Since F/A is Galois, G_A permutes the globally invariant lines of the restricted residue. Transporting the chosen local line to every prime over v_0 therefore preserves its avoidance of all these lines. Hence every transported marked line is transverse. Both curves pass through transverse points of the same residual scheme Y.

Returning from the target curve to its containing locus. The seed curve remains potentially pro-modular after restriction to F. Its finite bad local images allow Theorem 5.1 to start the propagation in its transverse chart. Lemma 7.5 then gives potentially pro-modular images at every transverse chart. In particular the restricted target curve is potentially pro-modular. By definition, the original target curve over A is therefore potentially pro-modular as well. Apply Theorem 5.1 over A to this original curve and its containing locus C_{\mathrm{target}}. The whole locus, and therefore the target and the prescribed family trace image, becomes pro-modular after a common allowed extension.

The finite component graph in Lemma 7.5 and the finite chains inside its selected charts require only finitely many propagation steps. Replace their fields by one Galois compositum and their levels by one deeper tame level, using forward base change. Total reality, complete splitting at 2, and solvability over \mathbb Q are preserved. This proves both assertions.□

Proof. Characters occurring as subquotients of a de Rham representation are de Rham. Suppose a chosen subcharacter has smaller exponent than the quotient, and let \theta be their ratio. Its exponent is strictly negative, so D_{\mathrm{dR}}(\theta)/\Fil^0=0 and H^0(\QQ_2,\theta)=0. The character \theta^{-1}\eps has strictly positive exponent. If its finite inertial part is nontrivial, its crystalline module is zero; otherwise its crystalline Frobenius eigenvalue has nonzero valuation and cannot be 1. Equation (41) therefore gives H^1_g=0. The extension splits, and its other summand is the required line.□

Proof. Use a framed global deformation chart with fixed determinant and an invariant line at 2, in an integral basis for the target and its line. Consider its completed local ring at the characteristic-zero target point. Corollary 3.7 gives dimension at least five for the local line chart and dimension two for the odd real factor, over the characteristic-zero coefficient field. The relative presentation of Proposition 3.6 has generators minus relations at least -3: its dual invariant term vanishes because the global representation is not virtually solvable. The global completed local ring therefore has dimension at least 5+2-3=4. The characteristic-zero dimension comparison in Lemma 3.4 then gives a horizontal integral component through the target in the original chart of dimension at least five.

Contract that component to its global trace and local subcharacter values. On the absolutely irreducible locus the only positive dimensional fibers are conjugacy fibers, of dimension three. The local characters are distinct on a dense open set, so recording the character does not leave an extra moving-line fiber there. The dimension comparison of Lemma 3.2 gives a domain \mathscr D of dimension at least two. Properness of the invariant-line incidence preserves the recorded subcharacter condition on the irreducible locus. Theorem 7.1, applied to an unrestricted framed component containing this family, gives one common F and one tame level with the asserted trace support.

We give the finiteness argument, including the recorded character. Write T_{\mathscr D} for the trace of the family. Let A\subset\mathscr D be the closed restricted trace image, and let q=[G_{\QQ}:G_F]. For every g\in G_{\QQ}, g^q\in G_F after replacing q by the order of the finite quotient if necessary. The identity

has P_q(X,Y) monic of degree q in X. Hence every downstairs trace is integral over A. A local character value satisfies

and is integral as well. The global determinant algebra has finitely many topological trace generators, and a continuous local character is determined topologically by its values on the uniformizer, -1, and 5. Adjoining these finitely many integral elements to A gives a finite complete algebra, hence a closed subalgebra of \mathscr D; it contains its topological generators and is all of \mathscr D. Finally, Proposition 4.2 makes the Hecke algebra finite over the product of the local central pseudodeformation images. Transitivity of finiteness proves the last assertion.□

Proof. In one factor the identity

proves independence of representatives and preservation of lower invariance. Conjugation by t=\operatorname{diag}(u_1,u_2)\in T_1 replaces a by (u_2/u_1)a, which permutes the residue classes. Thus tU=Ut on the lower-invariant space. The tame Hecke actions commute with these local actions. A common multiple of two periods is a period of the sum of the corresponding vectors. Finally U is bijective on W: on a vector of period N, its inverse is U^{N-1}, independently of the chosen period.

We show that one compact open subgroup fixes W[\varpi]^{T_1}. For j\ge2, x\in2^j\ZZ_2, and a\in\ZZ_2, direct multiplication gives

If f is T_1-fixed and upper-2^{j+1}-invariant, the final two factors fix f. The map a\mapsto a/(1+xa) permutes the lower residue classes, and (44) absorbs the changes of lift. Thus Uf is upper-2^j-invariant. Once upper-4 invariance is reached, it is preserved by the same identity.

Each smooth vector of W[\varpi]^{T_1} starts at some finite upper level. Iterate the preceding improvement far enough, taking an iteration number divisible by its period. The resulting vector is the original vector, so it is fixed by the single open compact subgroup generated by N_0^-, T_1, and \prod_{v\mid2}n^+(4\ZZ_2). Proposition 4.1 gives smooth admissibility; therefore W[\varpi]^{T_1} is finite-dimensional over \kk. Compact Nakayama now gives finite generation of M over \Omega.

Since \Omega is noetherian and M is finite, \End_\Omega(M) is finite over \Omega. The closed algebra in the statement is an \Omega-submodule of this finite module, and hence is finite too.□

Proof. For each s, form S_s=S\widehat\otimes_{\Omega,s}\Lambda, and let T_s be the image of R\widehat\otimes_{\OO}\Lambda in S_s. Both are finite \Lambda-algebras, and their images are closed. Let I_s be the kernel of the latter map. Each point in the dense set kills some I_s after mapping to \mathscr D. Hence

The spectrum is irreducible, so one of these finitely many closed sets is the whole spectrum. Equivalently, the prime kernel of R\widehat\otimes\Lambda\to\mathscr D contains one I_s. The closed trace-plus-weight image B in \mathscr D is therefore a quotient of T_s and is finite over \Lambda.

By Lemma 8.2, \mathscr D is finite over the restricted trace image A, and A\subset B. The same finite set of A-module generators generates \mathscr D as a B-module. Thus \mathscr D is finite over \Lambda. Its dimension is at least two, whereas \dim\Lambda=2; a nonzero prime kernel in the domain \Lambda would lower that dimension. The map is therefore injective, and the dimension is exactly two.□

Proof. The two diagonal characters in (50) are de Rham. Their ratio \theta=(u_h/u_l)\xi\eps has H^0(\theta)=H^0(\theta^{-1}\eps)=0, the latter because the finite inertial character is nontrivial. Thus (40) gives \dim H^1(\QQ_2,\theta)=1. In (41) the de Rham tangent term has dimension one, while the other two terms vanish. Hence every extension class belongs to H^1_g and r' is de Rham.

The p-adic monodromy theorem makes it potentially semistable [7, Theorems 0.6 and 5.19]. Its two finite inertial characters are distinct. Monodromy commutes with inertia, so it preserves their two eigenspaces; it vanishes on each by nilpotence. Thus it is potentially crystalline.

Twisting preserves absolute irreducibility and finite ramification. For a complex conjugation c_\infty, \lambda(c_\infty)^2=1, so the determinant remains odd. Finally, in the convention \HT(\eps)=-1, the dual has weights \{0,1\} and ordinary shape

All the hypotheses of [59, Theorem D] are now satisfied. That theorem makes the dual modular, and hence makes r' modular.□

Proof. The algebra \mathcal A containing U^{\pm1} is finite over S. Since it is the faithful action algebra of M, and M is finite over it, its spectrum is the support of M. Lying over the given point and passing to a finite coefficient field E' therefore gives a joint scalar quotient of M[1/2]. Write \OO' for its ring of integers, with uniformizer \varpi'.

The corresponding nonzero map

is continuous. Indeed, M is finite over weight space and its weight values lie in the formal open unit disk. Its image is a compact \OO'-submodule spanning E', hence a lattice; rescale it to \OO'. The scalar actions preserve this lattice. In particular the U-eigenvalue is a unit because U^{-1} preserves it as well.

Applying \OO'-linear Pontryagin duality, with values in E'/\OO', produces an equivariant injection

The image f_n of \varpi'^{-n} has exact order \varpi'^n and satisfies \varpi'f_{n+1}=f_n. Taking the Tate module gives a nonzero continuous integral completed-forms vector. Explicitly, a compatible sequence of continuous functions modulo \varpi'^n determines a continuous \OO'-valued function, because each of its reductions is continuous; conversely every such integral function gives that sequence. Exact orders ensure that the resulting vector is nonzero. All Hecke and torus eigenrelations, lower invariance, and the U-eigenrelation pass to this inverse limit. This construction finds a divisible copy inside W; it does not require W itself to be divisible.□

Proof. Choose m\ge2 so the stated torus character holds on \prod_v(1+2^m\ZZ_2)^2, and put X=(2^m\ZZ_2)^{[F:\QQ]}. Lower invariance gives

independently of the integral lifts of b. For x\in X the map a\mapsto a/(1-xa) permutes these residue classes. With c=1-xa, direct matrix multiplication gives

Define the V-valued polynomial

Its multidegree is bounded by \delta. Use a G_2-invariant nonarchimedean norm on V. The torus eigenrelation, (53), and the ultrametric inequality give

There is no factor for the number of residue classes: the norm of a sum is bounded by the largest norm of a summand. The unit eigenvalue and unitarity also introduce no growing factor. Continuity at the identity proves the final convergence.

For each v, choose \delta_v+1 distinct points in 2^m\ZZ_2. Evaluation on their product grid and inversion of the resulting Vandermonde matrices recover all coefficients of a polynomial of multidegree at most \delta, continuously in its values. This polynomial space is therefore closed in C(X,V). Equation (54) implies

For completeness, we identify an algebraic representation containing the resulting orbit. Put H=\prod_v(1+2^mM_2(\ZZ_2)) and write h_v=\left(\begin{smallmatrix}A_v&B_v\\C_v&D_v\end{smallmatrix}\right). Its Gauss decomposition is

Every factor lies in the neighborhoods already considered. Hence

These are regular algebraic functions on G_2, also when some e_{1,v} is negative, since powers of the determinant are invertible in its coordinate ring.

Let z_0=\bigotimes_ve_2^{\delta_v}\in A, with the determinant factors understood. In the binomially scaled monomial basis E_\nu of A, one has n^+(x)z_0=\sum_\nu x^\nu E_\nu. Define \Theta:A\to V by \Theta(E_\nu)=v_\nu. Formula (56) says \Theta(hz_0)=hf for h\in H. The vectors kz_0, k\in H, span A, by the same upper-unipotent interpolation. Thus

proves that \Theta is H-equivariant. Since H is Zariski dense, its invariant subspace \ker\Theta\subset A is an algebraic subrepresentation. The representation A is irreducible and \Theta(z_0)=f\ne0, so \Theta is injective. Its image contains f and has algebraic type A. This proves local algebraicity without any assertion that algebraic vectors commute with specialization.□