Global Functions and Nonextendable Spectral Germs on Ordinary GSp₄ Components

1. Introduction and Statements

For ${\mathrm{GL}}_n$, global functions on sufficiently generic Emerton–Gee components admit a spectral Satake description [1,2]. The corresponding question for a symplectic group has two parts. One can ask for an intrinsic comparison with the weight Hecke algebra, or for extension of particular formal spectral functions obtained from a deformation problem. These questions involve different maps, even when the source rings have the same abstract presentation.

The second comparison deserves a preparatory reading, expanded in Remark 1. In isolation, the statement that a nonconstant formal series in the local spectral coordinates is not a global function is close to tautological: global functions are constant along Rees degenerations and therefore see only the semisimplification, whereas the spectral coordinates vary with the extension classes. The content of Theorem 2 lies in three points. First, the divided Frobenius coordinates $u,t$ are well-defined formal functions on the completed germ at all, invariant under changes of Kisin eigenbasis and frame. Second, they complete the two character directions to part of a regular parameter system on the framed residual ring, which is what supports the joint power-series embedding and the fraction-field statement. Third, the two formal fraction fields separate. The theorem therefore identifies the exact formal images of the two comparisons, and the regular parameter system describing them, rather than merely asserting that two displayed functions differ.

The global-function method has a precise antecedent in Hjelle, Jaburi, Knak, Lee and Wang [2] (§§4–6, Appendix A). For sufficiently deep ${\mathrm{GL}}_n$ weights over unramified fields, they use torus restriction, auxiliary charts imposing the prefix-monomial cone, and extension of prefix minors to compute the global ring. Their main theorem assumes $(3n-1)$-deepness, hence 11-deepness for $n=4$. Their Section 7 gives an informal ordinary Hecke interpretation and refers to Lee for its detailed realization for ${\mathrm{GL}}_n$; that section does not treat ${\mathrm{GSp}}_4$. We use this framework and Lee’s integral functions. The symplectic upper bound below uses Siegel and Klingen component morphisms; the lower generators use a separately matched standard-representation morphism to ${\mathrm{GL}}_4$. We verify these maps and their normalization in the stated range.

The local comparison concerns two specified maps into one ordinary germ. Deformations with prescribed unramified characters give a left inverse for the intrinsic global completion. Under the suitable patched hypotheses, combining these directions with the divided Frobenius directions gives four regular parameters on a framed deformation ring. The field-separation corollary is the resulting elementary algebraic consequence. Neither the ${\mathrm{GL}}_n$ global-ring presentation nor the coefficient-extraction argument is claimed as a new general method.

We work with $G={\mathrm{GSp}}_4\left({\mathbf{Q}}_p\right)$ and the reduced symplectic Emerton–Gee stack constructed in [3]. Fix integers

$$a_0<a_1<a_2<a_3,a_0+a_3=a_1+a_2,$$

(1)

and set

$$r=a_1-a_0=a_3-a_2,s=a_2-a_1,S=2r+s=a_3-a_0.$$

The Serre weight in Lee’s coordinates is

$$\sigma =F(r+s-2,r-1;a_0).$$

(2)

Its component $X={\mathcal{C}}_{\sigma }$ contains a dense locus of maximally nonsplit ordinary representations with ordered inertia

$$({\omega }^{a_3},{\omega }^{a_2},{\omega }^{a_1},{\omega }^{a_0}).$$

The similitude on this component has inertial part ${\omega }^{a_0+a_3}$. Write $c:X\to {\mathbf{G}}_m$ for its unramified coordinate and $X_1=c^{-1}\left(1\right)$. This is a coordinate fiber, with its stack automorphisms retained.

Weight coordinates

Two notations for weights of ${\mathrm{GSp}}_4$ occur in the sources we use, and we record both and their conversion. In [3] (§2) the triple $(x,y;c)$ denotes the character of the diagonal torus

$$(x,y;c):diag(t_1,t_2,\nu t_2^{-1},\nu t_1^{-1})⟼t_1^x{t}_2^y{\nu }^c,$$

so the third slot of (2) is the exponent of the similitude character $\nu$. In [4] (§2) the triple $(x,y;z)$, subject to $z\equiv x+y(mod2)$, denotes the character $t_1^x{t}_2^y{\nu }^{(z-x-y)/2}$, so the third slot is the central exponent. The conversion is

$$(x,y;c)⟷(x,y;x+y+2c).$$

(3)

In particular the weight (2) has [4] coordinates $(r+s-2,r-1;a_0+a_3-3)$; its central exponent $z=a_0+a_3-3$ is the third coordinate displayed in the occurrence lemma of §7, where it also matches the weight ${\mu }_{\overline{\rho }}=(a_2-a_0-2,a_1-a_0-1;a_0+a_3-3)$ of axiom (PM6). Replacing z by $z+p-1$ multiplies the central character by ${\nu }^{(p-1)/2}$, the quadratic character; this changes the Weyl character and is not a cosmetic relabelling. All weight displays below follow (3). Within this paragraph and at (3) we retain the sources’ own symbols $(x,y;·)$; elsewhere in this paper generic weight coordinates are written $({\lambda }_1,{\lambda }_2;·)$, and the extension classes of the Fontaine–Laffaille seed below are written $(\alpha ,\beta )$, so that $(x,y)$ occurs only in this quoted convention.

The hypotheses used at different interfaces are summarized here. All interval tests concern the displayed weights or types, not an interchangeable notion of genericity.

Interface	Hypothesis used	Source / application
Global comparison	$r,s\ge 7$, $S\le p-4$	Theorem 1
${\mathrm{GL}}_2$ wall bound	Non-Steinberg; highest-weight differences in $(0,p-1)$ here	Lee, Prop. 5.3.3, §5.4
Ordinary lift on ${\mathrm{GL}}_4$	$\kappa \in X_1\left(T\right)$, 3-deep, ordered inertia $\kappa +\eta$	LLLM, Lemma 6.2.8
Integral prefixes	Type 5-generic ($e_{\mathrm{K}}=1$, $h_{\eta }=3$)	Lee, Prop. 5.1.1, Cor. 5.1.2
Patched pair	Axioms (PM1)–(PM6), congruent at every patching place	§7.1, Lemma 12
Local separation	$r,s\ge 8$, $S\le p-8$, $\beta NQ\ne 0$, congruent patched pair at every place	Theorem 2, §7
Ordinary crystalline ring	Adjacent jumps $\ge 2$, spread $\le p-2$	Enns–Lee, Prop. 2.4.2
Restriction of cocycles	Triangular adjoint; no cyclotomic constituent	Shapes, Def. 3.8, Prop. 3.12

The displayed ranges are sufficient conditions, not necessary ones, and we make no claim outside them. Since $S=2r+s$, the global range (4) forces $S\ge 21$ and $p\ge S+4\ge 25$, hence $p\ge 29$ because p is prime. The local range (6) forces $S\ge 24$ and $p\ge S+8\ge 32$, hence $p\ge 37$.

Theorem 1

(Intrinsic global comparison). Assume

$$r\ge 7,s\ge 7,S\le p-4.$$

(4)

There are global functions $F_1,F_2$ on X, defined over ${\mathbf{F}}_p$, such that

$$\Gamma (X,\mathcal{O})={\mathbf{F}}_p[F_1,F_2,c^{\pm 1}],\Gamma (X_1,\mathcal{O})={\mathbf{F}}_p[F_1,F_2].$$

On the split unramified torus with ordered parameters

$$(A,B,c/B,c/A),$$

(5)

the restrictions are $F_1=A$ and $F_2=AB$. With the cocharacters of [4],

$${\beta }_0=(1,1,1,1),{\beta }_1=(1,1,0,0),{\beta }_2=(2,1,1,0),$$

the assignment

$$T_{-{\beta }_1}\mapsto F_1,T_{-{\beta }_2}\mapsto F_2,T_{{\beta }_0}\mapsto c^{-1}$$

is an isomorphism from the W-regular weight Hecke algebra to $\Gamma (X,\mathcal{O})$. Its restriction to the torus is the normalized mod-p Satake transform followed by the exceptional dual-torus identification. Central specialization $T_{{\beta }_0}=1$ gives the comparison on $X_1$.

The Hecke isomorphism in this theorem is defined by the displayed torus restriction and generator convention. Its comparison with a Hecke action on patched modules or with finite-group cosocle operators is a separate problem, not a conclusion here. This theorem uses no automorphic globalization or patched module. Its domain includes the original strongly 7-generic numeric ordinary range

$$r,s\ge 8,S\le p-8.$$

(6)

The proof uses integral functions under a 5-generic type hypothesis and component inclusion under a 3-deep weight hypothesis. We do not apply the uniform 8-deep ${\mathrm{GL}}_4$ isomorphism theorem outside its domain.

For the second statement, let $k/{\mathbf{F}}_p$ be finite. Consider the normalized Fontaine–Laffaille seed with Frobenius matrix

$$F_{\mathrm{FL}}(\alpha ,\beta )=\left(\begin{array}{cccc}1& 1& \alpha & \beta \\ 0& 1& 1& 1-\alpha \\ 0& 0& 1& 1\\ 0& 0& 0& 1\end{array}\right)$$

(7)

and jumps $a_0,a_1,a_2,a_3$, interpreted by shifting the smallest jump to zero and twisting back. Throughout the paper, $F_1,F_2,F_d$ with numeric subscripts denote the global functions of Theorem 1, $F(·)$ denotes a Serre weight, and $F_{\mathrm{FL}}$ denotes a Fontaine–Laffaille Frobenius matrix. Put

$$N=\alpha +\beta -1,Q=S\beta -s\alpha ,$$

(8)

and assume $\beta NQ\ne 0$. The unramified diagonal parameters are all 1. The resulting representation is denoted $\overline{\rho }$. In the suitable-globalization setup of [4] (Definition 4.4.1, Lemma 4.4.14), with the congruent patched pair at all patching places satisfying the axioms (PM1)–(PM6) recorded in §7.1 and specified in Lemma 12, let

$$u={\theta }_0/p,t={\theta }_2/p,v=u/t$$

be the integral divided Frobenius units on its fixed-similitude deformation germ. Their intrinsic formal realization on the selected component, with the local formulas needed here, is recalled and verified in Appendix A. Reductions use the coefficient uniformizer $\varpi$, not p when the coefficient field is ramified. Theorem 2 is conditional on this patched input; §7.1 states precisely which part is hypothesis and which is proved in [4].

Theorem 2

(Independent completed images). Assume (6), (8), and the suitable patched setup just specified at $\overline{\rho }$. On the completed fixed-similitude germ ${\widehat{X}}_{\overline{\rho }}^{\psi }$, the ring

$${\mathcal{S}}_{\mathrm{loc}}=k\left[[u-u_0,t-t_0]\right]=k\left[[u-u_0,v-v_0]\right]$$

is embedded as a ring of formal functions. Let

$$R=\Gamma (X_1,\mathcal{O})=k[F_1,F_2],\mathfrak{m}=(F_1-1,F_2-1).$$

For the continuous restriction map

$$\iota :{\widehat{R}}_{\mathfrak{m}}⟶\Gamma ({\widehat{X}}_{\overline{\rho }}^{\psi },\mathcal{O})$$

there is a continuous k-algebra left inverse, so ι is injective. Put

$$Z_i=F_i-1,U=u-u_0,V=t-t_0.$$

On the framed fixed-similitude residual ring $\mathcal{A}$ at this point, these four functions form part of a regular parameter system:

$$\mathcal{A}\simeq k\left[[Z_1,Z_2,U,V,W_1,\dots ,W_{10}]\right].$$

(9)

In particular,

$$im\left(\iota \right)\cap {\mathcal{S}}_{\mathrm{loc}}=k.$$

(10)

Every nonconstant formal series in the prescribed local spectral coordinates fails to arise from the completed intrinsic global-function ring.

Corollary 1

(Formal meromorphic separation). Under the hypotheses of Theorem 2, the subfields

$$K_{\mathrm{glob}}=Frack\left[[Z_1,Z_2]\right],K_{\mathrm{spec}}=Frack\left[[U,V]\right]$$

of $Frac\mathcal{A}$ are linearly disjoint over k. Every nonconstant member of $K_{\mathrm{spec}}$ is transcendental over $K_{\mathrm{glob}}$.

The patched hypothesis is needed for the particular old spectral germ and the joint chart in Theorem 2. The split injectivity of the intrinsic completed-global map alone follows from ordinary crystalline deformation theory at these seeds, without that hypothesis; see Lemma 15. The identification of the type residual quotient with the ordinary crystalline residual quotient (Lemma 12) is a mod-p Breuil–Mézard type statement on weight multiplicities, and this is why the patched framework of [4], which makes both sides available on a common ring, enters the proof. The points treated by Theorem 2 are the declared normalized seeds with unramified diagonal parameters 1. We do not assert that (10) holds at every point or for arbitrary diagonal characters.

Remark 1

(What Theorem 2 uses and concludes). By the Rees degeneration (11), every global function on X takes the same value on a representation and on its semisimplification; on the dense ordinary locus this is the value on the split diagonal object with parameters $(A,B,c)$. Global functions therefore see only this semisimplification, whereas the divided Frobenius coordinates $u,t$ vary with the adjacent extension classes $(\alpha ,\beta )$, as Lemma 10 shows. Read in isolation, the statement that a nonconstant series in $u,t$ is not a global function is thus close to tautological. The content of the theorem lies in three points that require the inputs assembled in §7. First, u and t are well-defined formal functions on the completed germ at all: they are divided Frobenius eigenvalues, invariant under changes of Kisin eigenbasis and frame after the p-flat cancellation recorded in §6 and §7, and their definition uses the identification of the type residual quotient with the ordinary crystalline residual quotient (Lemma 12). Second, they complete the two character directions $Z_1,Z_2$ to part of a regular parameter system (9), which is what makes the joint power-series embedding and the fraction-field statement available. Third, the two formal fraction fields separate. The coordinates $u,t$ also depend on choices that we now record: they are attached to the inertial type ${\tau }_0$ and to the numbering of its Frobenius eigenvalues, $u={\theta }_0/p$ and $t={\theta }_2/p$, the indices referring to the ordered diagonal inertia $({\omega }^{a_3},{\omega }^{a_2},{\omega }^{a_1},{\omega }^{a_0})$ through the type chart of Appendix A. With the ordered inertia fixed, as it is throughout this paper, the numbering is canonical; for a bare residual representation without the ordered flag it would not be.

The transformation law under a renumbering of the type’s Frobenius eigenvalues is as follows. At $v_{\mathrm{K}}=0$ the four eigenvalues in the eigenbasis order of [4] (§2.3) are $pu,p^2\xi /t,pt,p^2\xi /u$ (Appendix A), with p-valuations $(1,2,1,2)$; the divided units attached to the four positions are $u,\xi /t,t,\xi /u$, and the Frobenius pairing pairs the first position with the fourth and the second with the third, the divided units at paired positions multiplying to ξ. A renumbering replaces the pair $(u,t)$ by the divided units at the new ordered positions. Whenever the two new positions are not paired with each other, the new pair consists of one element of $\{u,\xi /u\}$ and one of $\{t,\xi /t\}$, hence is obtained from $(u,t)$ by swapping the two coordinates, by replacing one or both entries by their inverses up to the unit ξ, or both — each an invertible transformation of the completed two-variable germs, because $u,t,\xi$ are units on it. For every such renumbered pair the conclusions of Theorem 2 — the regular parameter system (9), the intersection (10), and the fraction-field separation — hold verbatim. A renumbering selecting the two positions of one pairing class would not give independent coordinates and is not used.

The formal-local predecessor [5] motivates the prescribed functions $u,t$. Here “nonextendable” means precisely failure to belong to the image of the specified completed-global restriction $\iota$; this meaning and the maps are defined in this paper, independently of terminology or results in that preprint. We rederive the needed local comparison, including its coefficient-ring and completion interfaces. No compatibility with an independently constructed natural finite-group cosocle operator is asserted.

2. Component Conventions and Torus Detection

We first work over $\overline{k}=\overline{{\mathbf{F}}_p}$. The construction of X in [3] (Definition 4.1.8, Theorem 4.1.10, proof of Proposition 4.1.13) uses a Siegel extension vector bundle over the ordinary ${\mathrm{GL}}_2$ component. For the weight (2), its ${\mathrm{GL}}_2$ label is

$$({\lambda }_1+{\lambda }_2+{\lambda }_3+2,{\lambda }_1+{\lambda }_3+2)=(a_3-1,a_2).$$

Its ordered inertial characters are therefore $({\omega }^{a_3},{\omega }^{a_2})$. The symplectic dual supplies the other two characters in (1). Indeed Lee’s duality map is

$$(a,b;d)⟼(a+b+d,a+d;a+b+2d),$$

and its standard entries at $\lambda +(2,1;0)$ are exactly $(a_3,a_2,a_1,a_0)$. All relevant simple differences are strictly between 0 and $p-1$. The equal-character and cyclotomic exceptional ordinary loci do not occur under (4).

Lemma 1

(Full split torus). The split rank-one $(\phi ,\Gamma )$-module family with ordered characters

$${\omega }^{a_3}{nr}_A,{\omega }^{a_2}{nr}_B,{\omega }^{a_1}{nr}_{c/B},{\omega }^{a_0}{nr}_{c/A}$$

defines a morphism $T={{\mathbf{G}}_m}^3\to X$.

Proof. 

These rank-one module families define a torus morphism to the ambient stack. To check its factorization, take any geometric triple $(A,B,c)$. It belongs to a finite coefficient field after extension. Choose $z\ne 0$ with $cz\ne 1$ and use the Fontaine–Laffaille matrix

$$F_{\mathrm{FL}}=\left(\begin{array}{cccc}c/A& 1& 0& z\\ 0& c/B& 1& 1/B\\ 0& 0& B& 1\\ 0& 0& 0& A\end{array}\right),J=antidiag(1,-AB/c,AB/c,-1).$$

Direct multiplication gives $F^{t}JF=cJ$, $\det F=c^2$, and $\det J=A^2{B}^2/c^2$. The paired jumps sum to $a_0+a_3$. After shifting by $-a_0$ the jumps lie in $[0,S]\subset [0,p-2]$. The exact field-coefficient Fontaine–Laffaille functor and its twisted duality [6, §2.1, Proposition 2.2.1, Appendix A], [4, §2.3], therefore give a symplectic representation. Its adjacent extensions are nonsplit: a filtered splitting would have to use the higher-jump line, whose Frobenius image has a nonzero adjacent coefficient. Full faithfulness detects a hypothetical Galois splitting.

Its top two characters form an ordinary nonsplit ${\mathrm{GL}}_2$ object in the specified component. The symplectic extension belongs to the Siegel family defining X, after an unramified square-root twist to match its similitude. Such a twist exists at geometric points. The interval conditions exclude every exceptional ordinary case in that construction. Consequently this representation lies in X.

In a symplectic basis adapted to its full flag, conjugation by

$$d\left(w\right)=diag(w^3,w,w^{-1},w^{-3})$$

(11)

degenerates the representation to its split diagonal at $w=0$. All upper entries acquire positive powers. The family factors through the closed component: its ideal vanishes after inverting w, and the polynomial coefficient ring has no w-torsion. Thus the chosen split torus point lies in X. The closed-substack ideal pulls back to a quasi-coherent ideal on the scheme T (equivalently, pull back its ideal on a smooth atlas and descend). Finally T is reduced and Jacobson. The pullback of the ideal of X vanishes at all geometric points, hence vanishes as an ideal. This proves factorization of the whole torus morphism.    □

Lemma 2

(Detection). Torus restriction is injective:

$$res:\Gamma (X,\mathcal{O})\hookrightarrow \overline{k}[A^{\pm 1},B^{\pm 1},c^{\pm 1}].$$

Proof. 

Use the reduced Siegel extension-family source and its unramified twists whose scheme-theoretic image is X. Each geometric point has a full ordinary flag, and its split semisimplification belongs to the torus of Lemma 1. Every global function has the same value on the representation and its semisimplification, by the Rees family (11): it is constant on ${\mathbf{G}}_m$, so regularity gives the same value at zero. A function zero on the torus therefore pulls back to zero at all geometric points of the reduced source, hence to zero on that source. The scheme-theoretic-image property makes it zero on X. This uses density of the extension-family source; the split torus itself need not be dense in X.    □

3. Two Levi Walls

We use the following input from [1] (Proposition 5.3.3 and §5.4). This upper bound assumes only that the weight is non-Steinberg; the stronger depth bound for the full isomorphism in Theorem 5.3.4 is not used. For a non-Steinberg ordinary component with ordered unramified parameters $(a,b)$, its global functions restrict into

$$\overline{k}[a,{\left(ab\right)}^{\pm 1}]\subset \overline{k}[a^{\pm 1},b^{\pm 1}].$$

(12)

Only this necessary inclusion is needed, not a generic ${\mathrm{GL}}_2$ global isomorphism.

Lemma 3

(Siegel wall). Every Laurent monomial $A^i{B}^j{c}^k$ occurring in the restriction of a global function on X satisfies $i-j\ge 0$.

Proof. 

Let ${\mathcal{E}}_{\mathrm{top}}$ be the ${\mathrm{GL}}_2$ component with ordered inertia $(a_3,a_2)$, whose Lee label is $(a_3-1,a_2)$. The map

$${\mathcal{E}}_{\mathrm{top}}\times {{\mathbf{G}}_m}_c⟶{\mathcal{X}}_{\mathrm{Sym},\mathrm{red}},(\theta ,c)⟼\theta \oplus ({\theta }^{\vee }\otimes {\omega }^{a_0+a_3}{nr}_c)$$

is defined on module families, with its perfect evaluation pairing. Its dense ordinary source lies in the zero section of the Siegel extension family, up to unramified twist. Since the source is reduced, the pulled-back ideal of X vanishes everywhere. This gives a whole-component map to X.

The highest-weight difference is $r-1\in (0,p-1)$, so (12) applies. Pullback and torus restriction put every global restriction in $\overline{k}[A,{\left(AB\right)}^{\pm 1},c^{\pm 1}]$. Writing $A^i{B}^j=A^{i-j}{\left(AB\right)}^j$ proves the assertion.    □

Lemma 4

(Klingen wall). Every monomial in the same restriction also satisfies $j\ge 0$.

Proof. 

Use the middle ${\mathrm{GL}}_2$ component with ordered inertia $(a_2,a_1)$ and highest weight $(a_2-1,a_1)$. For any fixed unramified parameters $B,c/B$, their ratio has inertia ${\omega }^s$, neither 1 nor $\omega$. Local Tate duality and the local Euler characteristic give $H^0=H^2=0$ and $\dim H^1=1$. Thus all nonzero extensions with those characters become isomorphic after changes of the two rank-one bases.

The field objects used in Lemma 1 have a nonsplit middle extension, so can match any such ordinary middle object. Conjugation by $diag(w,1,1,w^{-1})$ has upper-entry weights $(1,1,2,0,1,1)$. Its limit retains that middle extension and kills the other upper entries. The whole degeneration lies in X. Consequently the canonical map

$$(\theta ,L_A)⟼L_A\oplus \theta \oplus (L_A^{\vee }\otimes \det \theta )$$

from the middle component times ${{\mathbf{G}}_m}_A$ lands in X on its dense ordinary locus. Reducedness extends factorization to the entire source, with its boundary and canonical determinant pairing.

Apply (12) to $(B,c/B)$. Its image is contained in $\overline{k}[B,c^{\pm 1},A^{\pm 1}]$, which imposes $j\ge 0$.    □

Distinct Laurent monomials are linearly independent. Combining Lemmas 2–4 gives the exact necessary cone

$$res\Gamma (X,\mathcal{O})\subset \overline{k}[A,AB,c^{\pm 1}].$$

(13)

No function is extended by reversing a Levi pullback.

4. Integral Prefix Functions and the Lower Inclusion

Here the convention of the accepted version [1] matters. A ${\mathrm{GL}}_4$ component with highest weight $\kappa$ has ordinary inertia $\kappa +\eta$, where $\eta =(3,2,1,0)$. Therefore set

$$I=(a_3,a_2,a_1,a_0),\kappa =I-\eta .$$

(14)

For an unambiguous type ordering we first twist by ${\omega }^{-a_0}$, so $a_0=0$, and twist back at the end. All unramified coordinates are unchanged by this fixed inertial twist.

Lemma 5

(Direct prefix construction). On the ${\mathrm{GL}}_4$ component ${\mathcal{E}}_{\kappa }$ there are global functions $g_d$, $1\le d\le 4$, with $g_4$ invertible, whose restrictions to its ordered split torus are ${\zeta }_1\dots {\zeta }_d$.

Proof. 

The six positive-root pairings of I are $r,r+s,S,s,r+s,r$; those of $\kappa$ are

$$r-1,r+s-2,S-3,s-1,r+s-2,r-1.$$

Under (4), the former are in $(3,p-3)$ and the latter in $(5,p-5)$. Thus $F\left(\kappa \right)$ has the identity lowest-alcove presentation with $\omega =I$ and is 3-deep. The adjacent highest-weight differences are $r-1,s-1,r-1$, each in $[0,p-1]$, so $\kappa \in X_1\left(T\right)$. The tame type $\tau =\tau (1,\kappa )$ is 5-generic. These are separate tests.

By [7] (Lemma 6.2.8), every ordinary point with inertia $\kappa +\eta$ has an upper triangular potentially crystalline lift of type $(\eta ,\tau )$. This lemma requires $(n-1)$-deepness, not the stronger hypotheses of the proposition following it. Hence the dense ordinary locus of ${\mathcal{E}}_{\kappa }$ lies in the residual fixed-type stack. Pulling back its closed ideal to the reduced component proves whole-component containment.

The integral interpolation theorem [1] (Theorem 4.3.2) supplies normalized functions ${\tilde{F}}_{\{1,\dots ,d\}}$ on the Breuil–Kisin stack. Its Breuil–Kisin/Galois comparison [1] (Proposition 5.1.1, Corollary 5.1.2) puts them on the potentially crystalline stack if $\tau$ is $(e_{\mathrm{K}}{h}_{\eta }+2)$-generic, where $e_{\mathrm{K}}$ is the absolute ramification index. Here $e_{\mathrm{K}}=1$ and $h_{\eta }=3$, giving exactly 5. Restrict and reduce these integral functions along the containment just proved, defining $g_d$. The integral algebra also contains the inverse of the total-product generator, so $g_4$ is invertible.

To evaluate, use the explicit split lift

$$\rho ={\displaystyle \underset{i=1}{\overset{4}{⨁}}}{\epsilon }^{4-i}{\tilde{\omega }}^{{\kappa }_i}{nr}_{{\tilde{\zeta }}_i}.$$

Its residual inertia is I. The accepted-version formula has Weil–Deligne Frobenius values $q^{i-4}{\tilde{\zeta }}_i$ and prefix normalization $q^{d(7-d)/2}$. For the first d indices,

$${\displaystyle \frac{d(7-d)}{2}}+{\displaystyle \sum _{i=1}^d}(i-4)=0.$$

It follows that $g_d={\zeta }_1\dots {\zeta }_d$ on the torus, with coefficient 1. The functions exist globally before evaluation, and reduction follows integral division. The construction uses only the lift lemma and the integral inputs just cited.    □

Remark 2

(Version conventions). In the first version of [1], the same physical component is labelled by $I+w_0\eta$; this exceeds (14) by $(3,3,3,3)$. The defining ordinary locus and the unramified prefix values agree. We use the accepted version (arXiv:2402.14011v2) and its component and integral normalization throughout; the version-specific source and its hash are identified in the accompanying source-verification note, so that every locator used here can be checked against it.

Lemma 6

(Whole-component standard morphism). The standard representation induces ${\mathrm{std}}_X:X\to {\mathcal{E}}_{\kappa }$ on the entire reduced component.

Proof. 

On the Siegel family let $\theta$ be its top ${\mathrm{GL}}_2$ object and ${\chi }_{\mathrm{sim}}$ its similitude. The extension coefficient module $\mathcal{W}={Sym}^2\theta \otimes {\chi }_{\mathrm{sim}}^{-1}$ has successive inertial exponents $S,r+s,s$. They lie in $[7,p-4]$, excluding both 0 and 1 modulo $p-1$.

Its canonical rank-one quotient of exponent s records the middle adjacent extension; the kernel has the other two exponents. Local Tate duality and devissage give $H^2$ of that kernel equal to zero. Thus $H^1\left(\mathcal{W}\right)$ surjects onto the rank-one quotient’s $H^1$, which has dimension 1. The cocycle vector bundle in [3] surjects onto $H^1\left(\mathcal{W}\right)$. The nonzero middle-extension condition is consequently nonempty open dense in every relevant fiber. The fixed inertia excludes cohomology jumps: $H^0,H^2$ vanish for all constituents, so the source’s relative Herr complexes have the stated constant cohomology ranks.

On this dense source open, the top, middle and bottom adjacent extensions are all nonsplit. The four characters are distinct, hence the standard representation has a unique complete flag with ordered inertia I. It lies in the defining ordinary open of ${\mathcal{E}}_{\kappa }$. The ideal of that closed component therefore vanishes on the scheme-theoretically dense source of X, and reducedness gives its vanishing on X. This proves factorization on the whole component, without any dominance or finiteness assertion for ${\mathrm{std}}_X$.    □

Put $F_d={\mathrm{std}}_X^{*}{g}_d$. Their torus values are $A,AB,Ac,c^2$. The similitude morphism independently supplies the invertible function c; no square root of $F_4$ is chosen. Together with (13) and injective torus detection, these lower generators prove

$$\Gamma (X_{\overline{k}},\mathcal{O})=\overline{k}[F_1,F_2,c^{\pm 1}].$$

(15)

Algebraic independence follows from the independent Laurent monomials $A,AB,c$. The same detection also proves $F_3=c{F}_1$ and $F_4=c^2$ globally.

5. Central Fiber, Rationality, and the Hecke Diagram

Lemma 7

(Actual central-fiber formula). One has $\Gamma (X_1,\mathcal{O})=\Gamma (X,\mathcal{O})/(c-1)$, with all stabilizers retained.

Proof. 

Unramified twisting by ${nr}_z$ is an action on the stack and preserves X, by its defining twisted extension family. It changes c to $c{z}^2$. The finite etale square cover $c=z^2$, of rank 2 since $p>2$, gives an equivalence

$$X{\times }_{{{\mathbf{G}}_m}_c}{{\mathbf{G}}_m}_z\simeq X_1\times {{\mathbf{G}}_m}_z.$$

The maps tensor by ${nr}_z$ and ${nr}_z^{-1}$. The deck involution on the right is $({\rho }_1,z)\mapsto ({\rho }_1\otimes {nr}_{-1},-z)$, not merely $z\mapsto -z$.

Write $R_1=\Gamma (X_1,\mathcal{O})$. Finite-free base change gives $\Gamma (X,\mathcal{O}){\otimes }_{k\left[c^{\pm 1}\right]}k\left[z^{\pm 1}\right]=R_1\left[z^{\pm 1}\right]$. The product formula follows by expanding on a finite smooth atlas and descending the finitely many Laurent coefficients; finite presentation in [3] supplies such an atlas. If $R_1=R_1^{+}\oplus R_1^{-}$ is the eigenspace decomposition of the twist involution, invariants are

$$R_1^{+}\left[c^{\pm 1}\right]\oplus z{R}_1^{-}\left[c^{\pm 1}\right].$$

Evaluation at $z=1$ is surjective: $h^{+}+h^{-}$ lifts to $h^{+}+z{h}^{-}$. Its kernel is exactly $(c-1)$ in the invariant algebra. This proves the claimed formula, and reducedness of the coordinate fiber follows from the etale cover and the product equivalence.    □

The labelled component and split torus are defined over ${\mathbf{F}}_p$: Galois preserves their defining ordinary locus. The functions in (15) are initially defined over a finite coefficient extension as allowed by the source theorem. Each Galois conjugate of $F_1,F_2$ has the same rational torus restriction, and therefore equals the original by Lemma 2. Finite Galois descent gives these functions over ${\mathbf{F}}_p$. An ${\mathbf{F}}_p$-global section has a unique finite polynomial/Laurent expansion over $\overline{k}$; Galois invariance makes every coefficient rational. Thus (15) and Lemma 7 hold over ${\mathbf{F}}_p$ and over finite coefficient extensions.

For completeness, put ${\lambda }_1=r+s-2$, ${\lambda }_2=r-1$, so that by (3) the weight (2) has coordinates $({\lambda }_1,{\lambda }_2;z)$ with $z={\lambda }_1+{\lambda }_2+2a_0$. The full source torus character has triple $({\lambda }_1,{\lambda }_2;{\lambda }_1+{\lambda }_2+2a_0)$. Its eight signed-permutation extremal characters are distinct modulo $p-1$. Indeed ${\lambda }_1>{\lambda }_2>0$, $0<{\lambda }_1-{\lambda }_2<p-1$, and ${\lambda }_1+{\lambda }_2=S-3<p-1$. A signed swap would require ${\lambda }_1-{\lambda }_2$ or ${\lambda }_1+{\lambda }_2$ to vanish modulo $p-1$. A single sign reversal changes the similitude exponent by ${\lambda }_1$ or ${\lambda }_2$, and reversing both changes it by ${\lambda }_1+{\lambda }_2$; these are all strictly between 0 and $p-1$. This uses the full torus, not only its derived two coordinates. Hence $\sigma$ is W-regular.

For a cocharacter $\ell =({\ell }_1,\dots ,{\ell }_4)$ with ${\ell }_1+{\ell }_4={\ell }_2+{\ell }_3$, its unique expansion in the $\beta$ basis has coefficients ${\ell }_4,{\ell }_2-{\ell }_3,{\ell }_1-{\ell }_2$. Therefore

$$X_{*}{\left(T\right)}^{-}=\mathbf{Z}{\beta }_0+\mathbf{N}(-{\beta }_1)+\mathbf{N}(-{\beta }_2).$$

Herzig’s Theorem 1.2 and Proposition 1.4 [8] give the projection-normalized monoid basis with additive convolution. Under the exceptional duality, ${\beta }_1$ is the first standard dual character, ${\beta }_2$ is the product of the first two, and ${\beta }_0$ is the dual similitude. The positive-support torus operator evaluates the inverse cocharacter. Thus for $\ell =z{\beta }_0-n{\beta }_1-m{\beta }_2$,

$${\tau }_{\ell }⟼c^{-z}{A}^n{\left(AB\right)}^m.$$

This verifies the torus diagram on every basis element and proves Theorem 1. In particular the central sign is $c^{-1}$.

6. A Constant-Character Spectral Deformation

The functions $u,t$ are divided eigenvalues on the potentially crystalline ring of Hodge type $\eta =(3,2,1,0)$ and tame type ${\tau }_0$ of [4]. They are not the diagonal characters of an ordinary crystalline flag. Indeed, at $v_{\mathrm{K}}=0$, the matrix in (A2) is upper triangular with diagonal

$$pu,p^2\xi /t,pt,p^2\xi /u.$$

Here $u,t,\xi$ are units. The normalized Frobenius valuations are $(1,2,1,2)$, whose increasing ordering is $(1,1,2,2)$, whereas the Hodge weights are $(0,1,2,3)$. Thus these characteristic-zero lifts are not ordinary crystalline lifts with that Hodge type. “Ordinary” for the residual representation does not assert ordinarity of these lifts. The ordinary crystalline comparison uses instead the Hodge tuple $(a_3,a_2,a_1,a_0)$ and trivial tame type.

Equality of the two residual quotient ideals in §7 identifies residual Galois deformation objects; it does not identify their characteristic-zero filtered Frobenius objects. On this common residual ring, the divided type eigenvalues can therefore vary with the extension parameters $(\alpha ,\beta )$ while all residual flag characters stay fixed. The formulas in Lemma 10 exhibit this variation. In contrast, $F_1,F_2$ are the ordinary character functions detected by the Rees construction. The eigenvalues are invariant under changing the Kisin eigenbasis or Galois frame. Concretely, a frame change g replaces a Frobenius matrix M by $g^{-1}M\phi \left(g\right)$, and at $v_{\mathrm{K}}=0$ this is conjugation by $g\left(0\right)$, because Frobenius fixes the coefficient ring here and therefore $\phi \left(g\right)\left(0\right)=g\left(0\right)$; the eigenvalues of $M\left(0\right)$ are unchanged. The divided versions inherit this invariance by cancelling p on the p-flat frame relation before reduction, as recorded in §7.

We now impose (6) and the normalized seed (7)–(8), with the suitable patched hypothesis of Theorem 2. Let $L=k^{\prime }\left[e\right]/e^{n+1}$ for a finite coefficient extension $k^{\prime }/k$. We will construct a separate deformation over every such L; no relative Fontaine–Laffaille theorem over a two-variable coefficient scheme is assumed.

Appendix A gives a six-free-coordinate Kisin chart, with variables denoted $a,b,h,t,m,d_2$. Assign

$$a=\beta ,b=N^{-1},h=-1+{\displaystyle \frac{\alpha (1-\alpha )}{\beta }},t={\displaystyle \frac{N}{\beta }},m=-\alpha ,d_2=1,$$

(16)

for arbitrary lifts $\alpha ,\beta$ of the seed in L. This is a chart point, hence a Galois point of the same reduced selected component. Its residual Kisin matrix, writing $v_{\mathrm{K}}$ for the variable, is

$$\mathbf{A}=\left(\begin{array}{cccc}0& 0& 0& -v_{\mathrm{K}}^2/\beta \\ 0& \beta v_{\mathrm{K}}^2/N& 0& v_{\mathrm{K}}^2/N\\ 0& -(1-\alpha (1-\alpha )/\beta )v_{\mathrm{K}}^2& N{v}_{\mathrm{K}}/\beta & -(1-\alpha )v_{\mathrm{K}}^2/\beta \\ \beta v_{\mathrm{K}}& -\alpha v_{\mathrm{K}}^2& v_{\mathrm{K}}& v_{\mathrm{K}}^2\end{array}\right).$$

(17)

This is verified by substitution in all sixteen entries of the integral normal form; the symplectic equations alone would not determine the matrix.

Lemma 8

(Exact phi-gauge). The restriction of the chart point (17) to $G_{{\mathbf{Q}}_p\left(\sqrt[p^{\infty }]{p}\right)}$ is isomorphic, with pairing and descent data, to that of the Fontaine–Laffaille deformation (7) over L.

Proof. 

Let $w_0$ be the antidiagonal permutation matrix, ${\Delta }_0=diag(1,-1,1,1)$,

$$\mathbf{B}=w_0{F}_{\mathrm{FL}}{w}_{0}diag(v_{\mathrm{K}},v_{\mathrm{K}}^2,v_{\mathrm{K}},v_{\mathrm{K}}^2),\mathbf{Y}={\Delta }_0\mathbf{A}{\Delta }_0^{-1},$$

and set

$$\mathbf{X}=\left(\begin{array}{cccc}1& (\alpha -1)/\beta & 1/\beta & -1/\beta \\ 0& 1& 1/N& -1/N\\ 0& 0& 1& (\alpha -1)/\beta \\ 0& 0& 0& 1\end{array}\right).$$

Direct multiplication gives $\mathbf{Y}=\mathbf{X}\mathbf{B}$. Put

$$J_f=antidiag(1,-1,1,-1).$$

The matrices $\mathbf{B},\mathbf{Y}$ have multiplier $v_{\mathrm{K}}^3$ for $J_f$. The matrix $\mathbf{X}$ is $J_f$-symplectic, and ${\Delta }_0^t{J}_f{\Delta }_0=J_g$.

Put

$$\Delta =(a_3-1,a_2-2,a_1-1,a_0-2),v_{\mathrm{K}}=z^{p-1},D_z=diag\left(z^{\Delta }\right),H=F_{\mathrm{FL}}^{-1}{w}_{0}diag\left(z^{-\Delta }\right).$$

Since $I-\Delta =(1,2,1,2)$,

$$H^{-1}[diag(v_{\mathrm{K}}^{a_0},\dots ,v_{\mathrm{K}}^{a_3})F_{\mathrm{FL}}]\phi \left(H\right)=D_z\mathbf{B}{D}_z^{-1}.$$

This is the natural raw Fontaine–Laffaille restriction matrix of [6] (Proposition 2.2.1, Lemma 2.2.7). Frobenius fixes every coefficient of L, including its nilpotents.

For J define $\Phi \left(J\right)=diag\left(v_{\mathrm{K}}^{\Delta }\right)J\left(v_{\mathrm{K}}^p\right)diag\left(v_{\mathrm{K}}^{-\Delta }\right)$. The least positive $\Delta$ difference is $\delta \ge 7$, and the largest is $M=S+1\le p-7$. Starting with $J_0=1$, set $J_{i+1}=\mathbf{Y}\Phi \left(J_i\right){\mathbf{B}}^{-1}$. Then $J_1=\mathbf{X}$. Since $\mathbf{X}-1$ is strictly upper triangular and $\Delta$ is decreasing, the entries of $\Phi \left(\mathbf{X}\right)-1$ have order at least $\delta$. Conjugating by $\mathbf{B}$, whose diagonal powers are $(1,2,1,2)$, costs at most one order; hence

$$J_2-J_1=\mathbf{X}\mathbf{B}(\Phi \left(\mathbf{X}\right)-1){\mathbf{B}}^{-1},{ord}_{v_{\mathrm{K}}}(J_2-J_1)\ge \delta -1\ge 6.$$

For $i\ge 1$, put $D_i=J_{i+1}-J_i$. Additivity of $\Phi$ gives

$$D_{i+1}=\mathbf{Y}\Phi \left(D_i\right){\mathbf{B}}^{-1}.$$

If ${ord}_{v_{\mathrm{K}}}{D}_i\ge q$ entrywise, evaluation at $v_{\mathrm{K}}^p$ multiplies orders by p, the diagonal twist by $\Delta$ contributes at least $-M$, and the conjugation by $\mathbf{B}$ contributes at least $-1$ because $\mathbf{Y}=\mathbf{X}\mathbf{B}$ and $\mathbf{X}$ is integral. Consequently

$${ord}_{v_{\mathrm{K}}}{D}_{i+1}\ge pq-M-1.$$

For $q\ge \delta -1\ge 6$ this lower bound exceeds q: indeed $M\le p-7$ gives $(p-1)q-M-1\ge 5p>0$. Iterating the bound makes the orders of the increments tend to infinity. Thus the sequence converges to an invertible Iwahori matrix with

$$Y=JB\Phi {\left(J\right)}^{-1}.$$

This proves an exact all-orders isomorphism. Pairing is preserved because paired $\Delta$ sums coincide. Lower Iwahori entries stay integral after conjugation by $D_z$, with worst exponent $(p-1)-M=p-S-2\ge 6$. Hence descent data are preserved as well.    □

Lemma 9

(Full Galois and symplectic comparison). The chart point has a complete free symplectic Galois flag over L with the constant ordered diagonal characters ${\omega }^{a_3},{\omega }^{a_2},{\omega }^{a_1},{\omega }^{a_0}$.

Proof. 

Forget the L structure of the FL object and apply the exact fully faithful contravariant field-coefficient functor of [6] (Theorem 2.1.3) in rank $4(n+1)$. Here the coefficient category is the field-coefficient Fontaine–Laffaille category: the rank increases, while the filtration interval stays $[0,S]\subset [0,p-2]$. Multiplication by e is an endomorphism in that category. Naturality transports this endomorphism and its restriction comparison. Contravariance reverses products, which still gives an L action because L is commutative. Exactness and dimension preservation give ${\dim }_{k^{\prime }}im\left(e^j\right)=4(n+1-j)$ for $0\le j\le n+1$, so all four Jordan blocks have length $n+1$. This gives a free rank-four L representation. Each diagonal object is scalar extension of a fixed rank-one field object. Additivity and the transported L action therefore make its Galois image the scalar extension of that fixed character; this proves constancy of the flag factors, in addition to their freeness.

For the pairing use the Frobenius-algebra functional $L\to k^{\prime }$ extracting the coefficient of $e^n$. It identifies ${Hom}_L(M,L)$ with ${Hom}_{k^{\prime }}(M,k^{\prime })$ by postcomposition; the inverse follows from the nondegenerate coefficient-extraction pairing on L. The finite-free duality argument in [4] (§2.3) uses only Hom, filtration annihilators and Frobenius evaluation, so applies in this rank as well. It transports the perfect alternating field pairing, with e self-adjoint. If this pairing is $B_0$, reconstruct the L pairing by

$$b(z,w)={\displaystyle \sum _{j=0}^n}{e}^j{B}_0(e^{n-j}z,w).$$

An index shift proves e-linearity, since $e^{n+1}=0$. For every $0\le j\le n$, self-adjointness gives $B_0(e^{n-j}w,z)=B_0(w,e^{n-j}z)=-B_0(e^{n-j}z,w)$. Thus each coefficient is skew-symmetric and has zero diagonal because $2\ne 0$, independently of the parity of n. Its $e^n$ coefficient is the perfect field form, giving zero radical and hence perfection by equal lengths. The self-dual free flag is preserved. No factorial denominators occur at $n\ge p$.

The full Galois comparison follows from Lemma 8 and the cyclotomic-free restriction-cocycle theorem of [9] (Definition 3.8, Proposition 3.12). The adjoint of the residual representation is triangular and has constituents with inertial exponents 0 and the signed differences of the $a_i$. None is 1 modulo $p-1$ in (6), so the theorem applies in its arbitrary-rank form. Conjugate by the given restriction isomorphism. If the two representations agree modulo $e^j$, their next discrepancy is an adjoint-valued cocycle restricting to zero, hence vanishes as an actual cocycle by Proposition 3.12 (not merely as a cohomology class). Induction gives equality over L. Pairing compatibility follows naturally; a scalar discrepancy can alternatively be removed by its residue-one square root. This proves the assertion at each finite length.    □

Lemma 10

(Functional coordinates). On (16) the formal functions satisfy

$$u={\displaystyle \frac{(S+2)\beta }{Q}},t={\displaystyle \frac{N}{\beta }}.$$

The map $(\alpha ,\beta )\mapsto (u,t)$ is an isomorphism on the indicated completed two-variable germs.

Proof. 

The divided integral relation in Appendix A, reduced only after p-flat cancellation, expresses the combination $K_0=D_0+D_1+d_2$ at the chart point as

$$K_0={\displaystyle \frac{(G_0+1)bmt+(T_0-1)d_2}{T_0+1}},u=K_0^{-1},$$

where $G_0=s-1$, $T_0=S+1$, and the appendix constant $E=p-1$ enters at its p-flat value $-1$. Since $bmt=-\alpha /\beta$ and $d_2=1$, this is $K_0=Q/\left[(S+2)\beta \right]$. It is a functional identity over L, not a differentiation of a residual-point table. The Jacobian determinant is $s(S+2)/\left(\beta Q^2\right)$, a unit. More explicitly put $H_0=S+s-st-(S+2)/u$; the inverse is

$$\beta ={\displaystyle \frac{s}{H_0}},\alpha ={\displaystyle \frac{S-(S+2)/u}{H_0}}.$$

At the base point $H_0=s/\beta$ is a unit. Every finite one-variable arc in the $(u,t)$ disk is therefore realized by the construction above.    □

7. Occurrence and the Common Residual Quotient

This section proves the two residual-level inputs of Theorem 2: that the required weight occurs in the principal series of the type (Lemma 11), and that the type and ordinary crystalline residual quotients are the same quotient of the same framed residual ring (Lemma 12). It also records the patched axioms used, and the framed versal map through which all character and spectral deformations below are pulled back. The integral chart and the precision estimates are computational and remain in Appendix A.

7.1. The Patched Axioms and Their Source

Theorem 2 uses the patched framework of [4] (§3.3, §4.4), and we record the axioms in full so that the proofs below can be checked against them; verbatim quotations of every statement used from [4], and of the weight convention of [3], are collected in Appendix B. Let $\mathcal{J}$ be the finite patching index set, $G={\mathrm{GSp}}_4\left({\mathbf{Q}}_p\right)$ and $K={\mathrm{GSp}}_4\left({\mathbf{Z}}_p\right)$. Let $({\mathcal{M}}_{\infty },{\mathcal{M}}_{\infty }^{\mathrm{cris}})$ be patched modules over the patched ambient rings

$$R_{\infty }=\left({\widehat{⨂}}_{j\in \mathcal{J},\mathcal{O}}{R}_{{\overline{\rho }}_j}^{\square ,{\psi }_{j,\mathrm{pc}}}\right)\left[[x_1,\dots ,x_h]\right],R_{\infty }^{\mathrm{cris}}=\left({\widehat{⨂}}_{j\in \mathcal{J},\mathcal{O}}{R}_{{\overline{\rho }}_j}^{\square ,{\psi }_{j,\mathrm{cris}}}\right)\left[[x_1,\dots ,x_h]\right],$$

where ${\psi }_{j,\mathrm{pc}}$ and ${\psi }_{j,\mathrm{cris}}$ are the potentially crystalline and crystalline similitude characters of [4] (§3.3), congruent modulo $\varpi$. Write $★\in \{\varnothing ,\mathrm{cris}\}$ and ${\mathcal{M}}_{\infty }^{★}\left(V\right)={Hom}_{\mathcal{O}\left[K^{\mathcal{J}}\right]}^{\mathrm{cts}}{(V,{\left({\mathcal{M}}_{\infty }^{★}\right)}^{\vee })}^{\vee }$ for a finite $\mathcal{O}$-module V with continuous $K^{\mathcal{J}}$-action. A congruent pair of patched modules is a pair satisfying the following axioms, quoted from [4] (§3.3).

(PM1)

${\mathcal{M}}_{\infty }^{★}$ is a finitely generated $R_{\infty }^{★}\left[\left[K^{\mathcal{J}}\right]\right]$-module with a compatible $R_{\infty }^{★}\left[G^{\mathcal{J}}\right]$-action.

(PM2)

${\mathcal{M}}_{\infty }^{★}$ is projective profinite over $\mathcal{O}\left[\left[K^{\mathcal{J}}\right]\right]$; consequently the functor ${\mathcal{M}}_{\infty }^{★}(-)$ is exact.

(PM3)

If ${\mathcal{M}}_{\infty }^{★}\left({\left({\sigma }^{★,\mathcal{J}}\right)}^{\circ }\right)\ne 0$ for a $K^{\mathcal{J}}$-stable lattice ${\left({\sigma }^{★,\mathcal{J}}\right)}^{\circ }$, then the $R_{\infty }^{★}$-action on it is maximal Cohen–Macaulay over $R_{\infty }^{★}\left({\sigma }^{★}\right)$, and ${\mathcal{M}}_{\infty }^{★}\left({\sigma }^{★,\mathcal{J}}\right)[1/p]$ is projective of constant rank $d\ge 1$ over $R_{\infty }^{★}\left({\sigma }^{★}\right)[1/p]$.

(PM4)

For a regular principal series type $\tau$ and $w_j\in W$, the tautological action of $\mathcal{H}{\left(\left(\sigma {\left(\tau \right)}^{w_j}\right)\right)}_{\mathcal{J}}$ on ${\mathcal{M}}_{\infty }\left(\sigma {\left(\tau \right)}^{\mathcal{J}}\right)$ factors through the eigenvalue morphisms ${\Theta }^{w_j}$ into $R_{\infty }\left(\sigma \left(\tau \right)\right)[1/p]$.

(PM5)

There is a canonical isomorphism of $R_{\infty }/\varpi =R_{\infty }^{\mathrm{cris}}/\varpi$-modules ${\mathcal{M}}_{\infty }/\varpi \simeq {\mathcal{M}}_{\infty }^{\mathrm{cris}}/\varpi$.

(PM6)

For ${\mu }_{\overline{\rho }}=(a_2-a_0-2,a_1-a_0-1;a_0+a_3-3)\in X^{*}{\left(T\right)}^{+}$, written in the coordinates (3), the weight $F\left({\mu }_{\overline{\rho }}\right)$ occurs in the crystalline patched module: $F{\left({\mu }_{\overline{\rho }}\right)}^{\mathcal{J}}\in W\left({\mathcal{M}}_{\infty }^{\mathrm{cris}}\right)$.

The elements $w_j\in W$ of (PM4) act on the type by right translation and are unrelated to the Rees parameter w of (11); the patching variables $x_1,\dots ,x_h$ come from the patching construction of [4] (§4.4) and are unrelated to any other letter of this paper. The existence of such a pair for the local representations at all places of $\mathcal{J}$ is proved in [4] (§4.4), culminating in [4] (Lemma 4.4.14), for globalizations suitable in the sense of [4] (Definition 4.4.1) and under that reference’s global hypotheses; axiom (PM6) is obtained there by a change-of-weight argument. Two further statements of [4] are used at fixed places: [4] (Theorem 2.3.10), which gives formal smoothness of relative dimension 14 over $\mathcal{O}$ for the fixed-similitude type ring at strongly 4-generic residual representations with $R_{\overline{\rho }}^{\eta ,{\tau }_0,\psi }\ne 0$, and [4] (Lemma 2.4.1, Proposition 2.4.2), which identifies the ordinary crystalline flag quotient and proves it formally smooth of relative dimension 14 over $\mathcal{O}$ (stated there as absolute dimension 15). For the seeds of this paper the local numeric inputs—strongly 7-genericity, hence strongly 4-genericity, and nonvanishing of the type and crystalline local rings—follow from (6), Lemma 5 and the two lemmas below; in particular the local hypotheses of the quoted statements are satisfied. What remains an input is the existence of a suitable globalization with the congruent pair at every patching place: Theorem 2 is conditional on that existence statement. The other statements of [4] used here are proved in that paper and quoted verbatim in Appendix B; they are cited inputs, not results proved in this paper.

The Globalization Input

Definition 4.4.1 of [4], quoted in Appendix B, imposes on a globalization $(F,\overline{r},\chi )$ of $\overline{\rho }$ the conditions (A1)–(A5): a totally real field F of even degree over $\mathbf{Q}$, oddness of $\overline{r}$, vastness and tidiness, potential diagonal automorphy of $(\overline{r},\chi )$, and strong 7-genericity of $\overline{r}{|}_{G_{F_v}}$ of weight $(a_3,a_2,a_1,a_0)$ at all $v|p$. Condition (A5) holds for our seed by the local verification just recalled; conditions (A1)–(A4) are part of the existence input. [4] (Lemma 4.4.5, Corollary 4.4.6), quoted in Appendix B, in fact constructs suitable globalizations for 7-generic ${\mathrm{GSp}}_4$ representations over ${\mathbf{Q}}_p$ when $p>7$, by producing an ordinary crystalline lift with potentially diagonalizable standard representation and applying a potential automorphy lifting theorem. As printed, however, Corollary 4.4.6 states the weight as $(a_3,a_2,a_2,a_0)$ rather than $(a_3,a_2,a_1,a_0)$, in both the arXiv source and the published version; the third entry disagrees with the weight used everywhere else in [4], including Definition 4.4.1(A5) and the type of Definition 2.3.6 on which the corollary’s proof relies. We therefore do not rely on this corollary. Whether it applies to the seeds of this paper under the evident correction is a concrete, checkable question; a positive answer, combined with Lemma 4.4.14, would make Theorem 2 unconditional at the declared points. Until that check is carried out, the existence of a suitable globalization remains an explicit hypothesis, and this is the precise sense in which Theorem 2 is conditional.

7.2. Support on the Selected Component

We place the type germ on the selected Serre-weight component; a type label is not the ordinary weight label. The ordinary weight from (2) admits its ordinary crystalline lift locus: the adjacent gaps are at least 2 and the total spread is at most $p-2$. The ordinary flag deformation statement [4] (Proposition 2.4.2) therefore shows that its dense ordinary locus lies in the residual crystalline locus for Hodge type $(a_3,a_2,a_1,a_0)$. Reducedness gives whole-component containment. At our Fontaine–Laffaille seed the framed crystalline ring is regular by the same ordinary result. Its unrestricted-similitude versal ring therefore has one branch. The crystalline special fiber and the ambient reduced symplectic stack are both equidimensional of dimension 4 [3]. The branch maps its generic point to the generic point of the selected component through the seed; the seed membership is supplied by the Siegel family in Section 2. Thus the pulled-back component ideal in the regular versal domain is zero, and fixed-similitude specialization preserves this zero ideal. This uses the stack dimension 4 after passing from the framed versal ring, not an equality between the dimension of that ring and the dimension of the stack.

7.3. Occurrence of the Required Weight

All weights in this subsection use the [4] coordinates of (3). In these coordinates the weight (2) is $F(a_{\circ }-2,b_{\circ }-2;z)$ with the shorthand

$$a_{\circ }=r+s,b_{\circ }=r+1,z=a_0+a_3-3,$$

used only in this subsection; these $a_{\circ },b_{\circ }$ are unrelated to the torus coordinates $(A,B)$ and to the chart variables of (16). The numeric hypotheses (6) become $a_{\circ }-b_{\circ }\ge 7$, $b_{\circ }\ge 9$ and $p-a_{\circ }-b_{\circ }\ge 7$.

Lemma 11

(Occurrence without a multiplicity-free hypothesis). The full ${\mathrm{GSp}}_4\left({\mathbf{F}}_p\right)$ principal series for ${\tau }_0$ contains $F(a_{\circ }-2,b_{\circ }-2;z)$ as a Jordan–Hölder factor.

Proof. 

Write $\chi ({\lambda }_1,{\lambda }_2;d)$ for a Weyl character of highest weight $({\lambda }_1,{\lambda }_2;d)$ in the coordinates (3). The general character formula [4] (3.2.6), explicitly expanded in [10] (proof of Proposition 4.11, p. 16), gives the sum of the following twelve dominant Weyl characters. Each entry $({\lambda }_1,{\lambda }_2;d)$ denotes $\chi ({\lambda }_1,{\lambda }_2;d)$, and every coefficient is $+1$:

$$\begin{array}{cc}(a_{\circ }-2,b_{\circ };z)& (a_{\circ },b_{\circ };z)\\ (p-a_{\circ }-3,b_{\circ };z+p-1)& (p-a_{\circ }-1,b_{\circ };z+p-1)\\ (p-b_{\circ }-2,a_{\circ }-1;z+p-1)& (p-b_{\circ }-2,p-a_{\circ }-2;z)\\ (p-b_{\circ }-1,a_{\circ };z+p-1)& (p-b_{\circ }-1,p-a_{\circ }-1;z)\\ (p+b_{\circ }-1,a_{\circ };z+p-1)& (p+b_{\circ }-1,p-a_{\circ }-1;z)\\ (p+a_{\circ }-1,p-b_{\circ }-1;z)& (2p-a_{\circ }-2,p-b_{\circ }-1;z+p-1).\end{array}$$

This is the Weyl-character formula, before any tensor-product reduction or multiplicity claim. Dominance follows from $a_{\circ }-b_{\circ }\ge 7$, $b_{\circ }\ge 9$ and $p-a_{\circ }-b_{\circ }\ge 7$, so none of these summands is a virtual negative character. For $\lambda =(p-b_{\circ }-1,p-a_{\circ }-1;z)$, the alcove inequalities in [4] (Definition 3.2.1, Table A.1) put $\lambda$ in $C_1$. Reflection across $x+y+3=p$ sends it to $(a_{\circ }-2,b_{\circ }-2;z)\in C_0$. The $C_1$ row of [4] (Lemma 3.2.7, Table A.3) therefore places $F(a_{\circ }-2,b_{\circ }-2;z)$ in this Weyl summand, hence in the principal series. The argument uses neither the stronger 7-deep hypothesis of [4] (Lemma 3.2.10) nor any statement about all twenty factors.    □

7.4. Equality of the Residual Quotients

Lemma 12

(Equality and contraction of residual quotient ideals). In the congruent patched setup, the type and ordinary crystalline residual rings at the distinguished place are equal as quotients of the same framed residual deformation ring. The universal residual representation agrees under this equality.

Proof. 

Let $\mathcal{J}$ be the finite patching index set. The setup used here is that of §7.1, with each ${\overline{\rho }}_j$ strongly 7-generic of the prescribed weight and the congruent pair satisfying (PM1)–(PM6). Enlarge the common coefficient field if necessary. Write ${\lambda }_{\mathrm{cr}}=(a_3,a_2,a_1,a_0)$ and set

$${\mathcal{S}}_0=R_{\infty }/\varpi =R_{\infty }^{\mathrm{cris}}/\varpi ,\mathcal{T}={\mathcal{S}}_0/I_{\mathcal{T}},\mathcal{D}={\mathcal{S}}_0/I_{\mathcal{D}},$$

where

$$\begin{array}{cc}\hfill \mathcal{T}& =\left({\widehat{⨂}}_{j\in \mathcal{J},\mathcal{O}}{R}_{{\overline{\rho }}_j}^{\eta ,{\tau }_0,{\psi }_{j,\mathrm{pc}}}\right)\left[[x_1,\dots ,x_h]\right]/\varpi ,\hfill \\ \hfill \mathcal{D}& =\left({\widehat{⨂}}_{j\in \mathcal{J},\mathcal{O}}{R}_{{\overline{\rho }}_j}^{{\lambda }_{\mathrm{cr}},1,{\psi }_{j,\mathrm{cr}}}\right)\left[[x_1,\dots ,x_h]\right]/\varpi .\hfill \end{array}$$

The two ambient rings agree residually because ${\psi }_{j,\mathrm{pc}}mod\varpi ={\psi }_{j,\mathrm{cr}}mod\varpi$. This is the canonical ambient identification of (PM5).

Put $\mathcal{N}=M_{\infty }\left(F_{\sigma }^{\mathcal{J}}\right)$, where $\sigma =F(a_{\circ }-2,b_{\circ }-2;z)$ is the weight of Lemma 11. The positive-coroot pairings of the algebraic highest weight plus $\rho$ are $a_{\circ }-b_{\circ }+1,b_{\circ }-1,a_{\circ }+b_{\circ }-1,a_{\circ }$, all in $(0,p)$. Its Weyl lattice therefore reduces to the single $F_{\sigma }$, by [4] (Lemma 3.2.7). (PM6) makes its crystalline patched module nonzero, and (PM5) identifies its reduction with $\mathcal{N}$, including the ${\mathcal{S}}_0$ action. The crystalline local rings are formally smooth of relative dimension 14 by [4] (Proposition 2.4.2). By (PM3) and Auslander–Buchsbaum their patched lattice module is free of positive rank. Thus $\mathcal{N}$ is free of positive rank over $\mathcal{D}$ and ${Ann}_{{\mathcal{S}}_0}\mathcal{N}=I_{\mathcal{D}}$.

Choose any $K^{\mathcal{J}}$-stable $\mathcal{O}$-lattice ${\Lambda }_{\tau }$ in the full type $\sigma \left({\tau }_0\right)$. By Lemma 11 and (PM2), $\mathcal{N}$ is a subquotient of $M_{\infty }\left({({\Lambda }_{\tau }/\varpi )}^{\mathcal{J}}\right)$: apply the exact covariant functor to a composition series. In particular that type module is nonzero. Exactness also identifies it with $M_{\infty }\left({\Lambda }_{\tau }^{\mathcal{J}}\right)/\varpi$. By (PM3) its action factors through $\mathcal{T}$; hence $I_{\mathcal{T}}\mathcal{N}=0$ and $I_{\mathcal{T}}\subseteq I_{\mathcal{D}}$. This also proves nonvanishing of every local type factor without using the support equality. Each such factor is formally smooth of relative dimension 14, by [4] (Theorem 2.3.10) (or the chart calculation of Appendix A, with this nonvanishing input). Consequently

$$\mathcal{T}\simeq k\left[[y_1,\dots ,y_d]\right],\mathcal{D}\simeq k\left[[y_1,\dots ,y_d]\right],d=14\left|\mathcal{J}\right|+h.$$

In particular $\mathcal{T}$ is a local domain, although the entire type special fiber can have many components elsewhere. The quotient $\mathcal{D}=\mathcal{T}/(I_{\mathcal{D}}/I_{\mathcal{T}})$ has the same dimension as $\mathcal{T}$. Every nonzero ideal of this regular local domain lowers dimension, so $I_{\mathcal{T}}=I_{\mathcal{D}}$. The proof uses only the occurrence lemma, exactness of the functor of (PM2), and formal smoothness; it requires neither an injection of socle lattices nor a residual multiplicity-freeness statement.

For clarity, let ${\mathcal{S}}_w=R_{{\overline{\rho }}_w}^{\square ,\overline{\psi }}$ be the characteristic-p framed fixed-similitude ambient ring at the distinguished place, and ${\mathcal{T}}_w,{\mathcal{D}}_w$ its two residual quotients. The maps are

$$\begin{array}{ccc}{\mathcal{S}}_w& ⟶& {\mathcal{S}}_0\\ \downarrow & & \downarrow \\ {\mathcal{T}}_w& ⟶& \mathcal{T}={\mathcal{T}}_w\left[\left[\mathrm{other}\text{-}\mathrm{place}\mathrm{and}\mathrm{patch}\mathrm{variables}\right]\right].\end{array}$$

and the identical diagram with ${\mathcal{D}}_w,\mathcal{D},I_{\mathcal{D}}$. Here the right-hand identification uses the formal smoothness of all other local factors. Thus ${\mathcal{T}}_w\to \mathcal{T}$ and ${\mathcal{D}}_w\to \mathcal{D}$ are injective, and

$$\ker ({\mathcal{S}}_w\to {\mathcal{T}}_w)=\ker ({\mathcal{S}}_w\to \mathcal{T})=\ker ({\mathcal{S}}_w\to \mathcal{D})=\ker ({\mathcal{S}}_w\to {\mathcal{D}}_w).$$

Set $\mathcal{A}={\mathcal{T}}_w={\mathcal{D}}_w$ as this quotient of ${\mathcal{S}}_w$. This is equality of the residual deformation subfunctors, not just an abstract isomorphism. Adjoining the same unramified similitude variable preserves it.    □

7.5. The Framed Versal Map and Descent

The map $Spf\mathcal{A}\to {\widehat{X}}_{\overline{\rho }}^{\psi }$ is obtained from the universal residual representation over ${\mathcal{S}}_w$ by this quotient: the crystalline component-ideal calculation of §7.2 makes the map factor through X, and Lemma 12 gives the same factorization on the type side. In the stack deformation problem this is the framed versal presentation; changes of frame give its smooth relation. All pullbacks of $Z_i$, all character deformations and all spectral deformations below use this one map and this one universal residual object; no step passes through a freely chosen isomorphism of regular rings.

The divided functions descend through changes of frame. The eigenvalues are invariants of the underlying paired Kisin object. The residual centralizer is the scalar subgroup: distinct inertial characters force diagonal entries, and the three nonsplit adjacent extensions force those diagonal entries equal. In the fixed symplectic group it is ${\mu }_2$. Its residue-one relation is smooth, and the framed relation is the usual power-series frame torsor. Equality of the integral eigenvalues on this relation allows cancellation of p before reduction because the relation is p-flat. Faithful flat descent therefore puts the divided units on the completed stack germ. The same argument with the scalar centralizer treats the fixed-similitude representation convention. All reductions are modulo $\varpi$.

8. Ordinary Character Directions and Completed Images

Lemma 13

(Exact Artin Rees constancy). Let L be a finite local Artin coefficient algebra and let ρ be a deformation in $X_1$ with a full free self-dual ordinary flag. For every $f\in \Gamma (X_1,\mathcal{O})$ one has

$$f\left(\rho \right)=f\left({\rho }^{\mathrm{diag}}\right)\mathrm{in}L,$$

where ${\rho }^{\mathrm{diag}}$ is its split diagonal family. If the ordered diagonal characters are constant over the coefficient field, this common value belongs to that field.

Proof. 

Choose an adapted symplectic basis. Conjugation by (11) extends to matrices over $L\left[w\right]$, with constant diagonal at $w=0$. All group identities hold because they hold after inverting w and $L\left[w\right]\hookrightarrow L[w,w^{-1}]$. The initial representation has finite image; the polynomial representation factors through the same finite quotient. Put $\Lambda =L\left[w\right]$, a finite-type ${\mathbf{F}}_p$-algebra. Over the period ring $W{\left({\mathbf{C}}_p^{♭}\right)}_{\Lambda }$, form its tensor product with ${\Lambda }^4$, with Frobenius on the first factor and the diagonal $G_{{\mathbf{Q}}_p}$ action. The latter is continuous because the matrix action factors through that finite quotient. This is a free étale $(\phi ,G_{{\mathbf{Q}}_p})$-module. The coefficient equivalence [11] (Definitions 2.7.6–2.7.7, Proposition 2.7.8(1)) gives the projective étale $(\phi ,\Gamma )$-module, with its pairing. This construction is functorial under $w=0,1$ and does not require the coefficient algebra to be finite.

Over $w\ne 0$ it is isomorphic to the given arc in $X_1$. The closed ideal of $X_1$ vanishes after inverting w, so vanishes over $L\left[w\right]$, which is w-torsion-free even though L is nonreduced. A global function pulls back to the constant polynomial given by its value on the arc. At $w=0$ this equals its value on the split diagonal object. If the diagonal characters are constant over the coefficient field, this value lies in that field. This proves the stated equality over the whole Artin algebra, including its nilpotents.    □

Lemma 14

(Prescribed diagonal lifting). At the seed $\overline{\rho }$, the functor of paired upper triangular deformations with fixed similitude is formally smooth over its two unramified diagonal parameters. In particular, for every m there are compatible deformations over $L_m=k[z_1,z_2]/{(z_1,z_2)}^{m+1}$ with ordered diagonal characters

$${\omega }^{a_3}{nr}_{1+z_1},{\omega }^{a_2}{nr}_{1+z_2},{\omega }^{a_1}{nr}_{{(1+z_2)}^{-1}},{\omega }^{a_0}{nr}_{{(1+z_1)}^{-1}}.$$

Proof. 

Write the paired upper Borel as $\mathcal{B}=T⋉N^{\mathrm{u}}$. For a small extension $L^{\prime }\to L$ with kernel J killed by the maximal ideal, fix the lifted torus characters. Smoothness of $\mathcal{B}$ supplies continuous pointwise lifts with exactly this diagonal. For continuity choose a common finite quotient through which both the original representation and the lifted diagonal factor. Their multiplicative defect is a continuous 2-cocycle with values in $J{\otimes }_{k}Lie{N}^{\mathrm{u}}$. The defect is additive because $J^2=0$; a 1-cochain correction removes it precisely when its cohomology class vanishes.

The positive-root height filtration is stable under $\mathcal{B}$. The four graded characters of $Lie{N}^{\mathrm{u}}$ have inertial exponents $r,s,r+s,S$. They lie in $[8,p-8]$ and none is cyclotomic. Tate local duality for finite modules [12] (Corollary I.2.3) gives $H^2=0$ for each constituent: the dual invariant group is zero already on inertia. The trace $k/{\mathbf{F}}_p$ identifies the Tate dual with the cyclotomic-twisted k-linear dual. Devissage then gives $H^2(G_{{\mathbf{Q}}_p},Lie{N}^{\mathrm{u}})=0$. The obstruction therefore vanishes for every small extension, with no restriction on the newly prescribed unramified parameters. Iterating the small extensions $L_{m+1}\to L_m$ proves the assertion. This uses continuous Galois cochains, not a coefficient-family Fontaine–Laffaille functor.    □

Lemma 15

(Split injectivity). Assume (6) and (8) at the declared unit-diagonal seed. The intrinsic completed-global restriction ι has a continuous left inverse. This assertion does not require the suitable patched setup.

Proof. 

The deformations in Lemma 14 belong to the ordinary crystalline flag functor of Hodge tuple $(a_3,a_2,a_1,a_0)$: the inertia of ${\epsilon }^{a_i}$ over every k-algebra is exactly ${\omega }^{a_i}$. By [4] (Lemma 2.4.1, Proposition 2.4.2), that functor is the framed crystalline deformation ring. The adjacent-gap and spread hypotheses hold because $r,s\ge 8$ and $S\le p-8$.

These are points of the selected component with their nilpotent coefficients. Indeed its dense ordinary locus admits the same crystalline lifts, so the whole reduced component lies in the crystalline special fiber. At the seed the unrestricted-similitude crystalline versal ring is a regular domain. The seed belongs to X by the Siegel construction in Section 2: its top Lagrangian plane is the prescribed nonsplit ${\mathrm{GL}}_2$ object, and its symmetric extension lies in the defining family. The interval bounds exclude the exceptional cases. Equidimensionality and the generic-point argument in §7.2 then identify the crystalline fiber’s unique component through this point with X and make the pulled-back ideal of X zero. Specializing the unramified similitude variable to zero preserves this zero ideal. This is the crystalline part of that argument, before any patched comparison of two types.

Evaluate intrinsic global functions on the compatible deformations. Lemma 13 applies to the finite Artin algebra $L_m$. Its split diagonal conclusion and the split torus morphism give, exactly over $L_m$,

$$F_1=1+z_1,F_2=(1+z_1)(1+z_2).$$

Thus evaluation on the inverse system defines a continuous map E from formal functions on the fixed-similitude germ to $k\left[[z_1,z_2]\right]$, with

$$E\iota \left(Z_1\right)=z_1,E\iota \left(Z_2\right)=z_1+z_2+z_1{z}_2.$$

This is an isomorphism between the two power-series rings, with inverse $z_1=Z_1$, $z_2=(Z_2-Z_1)/(1+Z_1)$. Composing E with that inverse gives the desired left inverse. Choices of lifts enter E; no canonical retraction is asserted.    □

Proof of Theorem 2. 

Split injectivity follows from Lemma 15. Retain the suitable patched hypothesis for the old spectral functions. Lemma 12 identifies the type residual ring $\mathcal{A}$ with the same quotient of the framed deformation ring as the ordinary crystalline residual ring, and proves that it is regular of dimension 14. The map $Spf\mathcal{A}\to {\widehat{X}}_{\overline{\rho }}^{\psi }$ is the framed versal presentation described in §7.4; in particular $Z_i$ are pulled back along that map. Consequently both the prescribed-character directions of Lemma 14 and the spectral directions from Lemma 10 are tangent vectors of this same $\mathcal{A}$. An abstract isomorphism of two rings would not suffice for this step.

On the two independent spectral directions, Lemma 13 gives $d{Z}_1=d{Z}_2=0$, while $(dU,dV)$ have rank 2 by the explicit functional inverse. On the two character directions, $(d{Z}_1,d{Z}_2)$ have rank 2 by their displayed values. The values of $dU,dV$ on the latter directions need not vanish. The resulting 4 by 4 matrix is block triangular with invertible diagonal blocks. The four cotangent vectors are therefore independent in ${\mathfrak{m}}_{\mathcal{A}}/{\mathfrak{m}}_{\mathcal{A}}^2$.

Extend them to a basis of that 14-dimensional space. The complete local inverse theorem for the regular ring $\mathcal{A}$ gives (9). It follows in particular that $k\left[[Z_1,Z_2,U,V]\right]\to \mathcal{A}$ is injective. These functions descend from the completed stack germ as shown in §7.4. A relation between them as stack functions would pull back to a relation in $\mathcal{A}$, so their four-variable map to the formal-function ring is also injective. In the joint power-series ring the two subrings $k\left[[Z_1,Z_2]\right]$ and $k\left[\right[U,V\left]\right]$ intersect exactly in k, proving (10). This identifies a subring of stack functions and part of a coordinate system on its framed chart, not the entire stack function ring with the framed ring.    □

Proof of Corollary 1. 

Take finitely many k-linearly independent elements $b_1,\dots ,b_n$ of $K_{\mathrm{spec}}$. Clear a common denominator in $k\left[\right[U,V\left]\right]$, obtaining independent series $c_1,\dots ,c_n$. There are n coefficient-extraction rows in $U,V$ whose coefficient matrix on these series is invertible over k. Indeed enumerate the monomials in $U,V$. The kernels of the first m coefficient rows form a descending chain of subspaces of the finite-dimensional space $k^n$. It stabilizes; its intersection is zero by independence of the series. Some finite coefficient matrix therefore has rank n, and n of its rows are independent.

A putative $K_{\mathrm{glob}}$-linear relation among the $b_i$, after clearing denominators on both sides, is an equality in the injected ring $k\left[[Z_1,Z_2,U,V]\right]$. Extracting those n coefficients forces all its $k\left[[Z_1,Z_2]\right]$ coefficients to vanish. Hence $K_{\mathrm{glob}}{\otimes }_k{K}_{\mathrm{spec}}\to Frac\mathcal{A}$ is injective, proving linear disjointness.

Finally k is algebraically closed inside $K_{\mathrm{spec}}$. An element algebraic over finite k satisfies $f^{q^m}=f$. It is integral over the normal ring $k\left[\right[U,V\left]\right]$, hence lies in that ring. Its constant coefficient $a\in k$ satisfies the same equation. If $f-a\ne 0$, let $d\ge 1$ be its least nonzero total homogeneous degree. In characteristic p, the least degree of ${(f-a)}^{q^m}$ is $d{q}^m>d$, contradicting ${(f-a)}^{q^m}=f-a$. Thus $f=a$. Every nonconstant $f\in K_{\mathrm{spec}}$ has k-linearly independent powers; linear disjointness makes them independent over $K_{\mathrm{glob}}$. This is the asserted transcendence.    □

9. Scope and Comparison with Earlier Results

Theorem 1 implements the global-function strategy of [2] using the symplectic component geometry, two paired Levi morphisms, and Lee’s integral ${\mathrm{GL}}_4$ generators. The weight and type bounds used for the latter are checked separately in Lemma 5; we do not invoke the 11-deep theorem of HJKLW or the 8-deep isomorphism theorem of Lee outside their domains. The component maps, central specialization and stated symplectic torus diagram are the comparisons established here.

Theorem 2 concerns a prescribed formal coordinate ring on suitable patched points. It does not contradict the intrinsic global isomorphism. It proves that no formal reparametrization preserving the old spectral image can identify that image with a nonconstant part of the completed intrinsic global image. The four functions extend to regular coordinates on the framed chart. We do not identify the entire formal-function ring of the stack with that chart or assert that these four functions generate it. Corollary 1 excludes algebraic recovery of a nonconstant old spectral meromorphic function from the intrinsic completed-global field inside that chart field. It makes no assertion about all finite covers of the nonaffine stack.

The parameter bounds are sufficient. Their failure is not a nonexistence result. Removing the patched hypothesis for the old formal germ, treating arbitrary diagonal parameters in the obstruction, ramified base fields, and less generic components are separate questions. No cohomological or torsion comparison is used here.    

Authorship and use of AI: The author directed the research programme and takes responsibility for the manuscript. AI systems performed the mathematical derivations, exact computations, source verification, internal audits and manuscript revision; these tasks are not represented as personal checks by the author. Internal AI audits do not constitute external expert peer review.

Appendix A.1. Generation of the Integral Normal Form

The initial ordinary degree ansatz in [4] (proof of Proposition 2.3.9), before its coefficient elimination, is

$$\left(\begin{array}{cccc}c& {\pi }_0+{\pi }_{1}q& w& j+kq+n{q}^2\\ 0& {ϵ}_{11}+{\delta }_{11}q+r{q}^2& {ϵ}_{12}& {ϵ}_{13}+{\delta }_{13}q+b{q}^2\\ 0& ({ϵ}_{21}+hq)(q-p)& {ϵ}_{22}+tq& {ϵ}_{23}+fq+g{q}^2\\ a(q-p)& (l+mq)(q-p)& z(q-p)& d_0+d_{1}q+d_2{q}^2\end{array}\right).$$

We work in its p-flat coefficient quotient. All 2-minors are divisible by q and all 3-minors by $q^3$. The constant coefficients of the 2-minors with rows $(i,3)$ and columns $(0,j)$, for $i=1,2$ and $j=1,2,3$, are respectively $ap{ϵ}_{11},ap{ϵ}_{12},ap{ϵ}_{13},-a{p}^2{ϵ}_{21},ap{ϵ}_{22},ap{ϵ}_{23}$. Hence all six $ϵ$ vanish. The $q^2$ coefficients of the 3-minors on rows $(1,2,3)$ and columns $(0,1,2)$ or $(0,2,3)$ are $-apt{\delta }_{11}$ and $apt{\delta }_{13}$. These force the two $\delta$ to vanish.

The six column-pairing equations for $M^t{J}_{g}M=\xi q^3{J}_g$ now give

$${\pi }_0=cl/a,{\pi }_1=cm/a,r=\xi /t,z=abt/\xi ,w=cbt/\xi ,$$

$$n=-\xi /a,k=(c{d}_2-p\xi )/a,j=(c{d}_1+pc{d}_2-p^2\xi )/a,$$

$$f=-t(l/a+pbh/\xi ),g=t(bh/\xi -m/a),$$

and the single scalar relation

$$c(d_0+p{d}_1+p^2{d}_2)=\xi p^3.$$

(A1)

For example pairing columns $0,1$ gives $(q-p)[c(l+mq)-a({\pi }_0+{\pi }_{1}q)]$; pairing columns $1,2$ after this substitution gives $rt{q}^3$. Cancellation of $q-p$ is valid because it is monic, and cancellation of p uses p-flatness. No inverse of $q-p$ is introduced. First impose the scalar relation (A1); substitution then verifies all six pairings and all minors, proving the converse. The resulting matrix is

$$\left(\begin{array}{cccc}c& c(l+mq)/a& bct/\xi & [c{d}_1+c{d}_{2}p-p^2\xi -q^2\xi +q(c{d}_2-p\xi )]/a\\ 0& q^2\xi /t& 0& b{q}^2\\ 0& hq(q-p)& qt& t[q^2(bh/\xi -m/a)+q(-bhp/\xi -l/a)]\\ a(q-p)& (l+mq)(q-p)& abt(q-p)/\xi & d_0+d_{1}q+d_2{q}^2\end{array}\right).$$

(A2)

It has determinant ${\xi }^2{q}^6$. Thus these ten coefficients generate the actual gauge ring before monodromy, with precisely the indicated Hodge and pairing conditions. In particular the two coefficients in the $(2,3)$ entry are fixed by the pairing equations themselves.

Appendix A.2. The Full Higher-Frobenius Precision

We use the canonical monodromy expansion of [7,13], rederiving its bound rather than imposing a stronger genericity assumption. Write $\mathbf{A}$ for (A2), $\Delta =(T_0,A_0,B_0,0)$ and

$${\mathbf{A}}_{†}=-E{v}_{\mathrm{K}}{\mathbf{A}}^{\prime }-[diag\Delta ,\mathbf{A}],Q_0=q^3{\mathbf{A}}^{-1}={\xi }^{-1}{J}_g^{-1}{\mathbf{A}}^t{J}_g.$$

Both $\mathbf{A},Q_0$ are integral and upper triangular modulo $v_{\mathrm{K}}$; ${\mathbf{A}}_{†}$ is strictly upper triangular modulo $v_{\mathrm{K}}$. Frobenius fixes the coefficient ring. Put $L_i=(p^i-1)/(p-1)$ and

$${\mathcal{F}}_i\left(M\right)=diag\left(v_{\mathrm{K}}^{L_i\Delta }\right)M\left(v_{\mathrm{K}}^{p^i}\right)diag\left(v_{\mathrm{K}}^{-L_i\Delta }\right).$$

Let $\delta$ be the least positive difference of $\Delta$, and $\gamma =p-1-T_0$. Here $\delta \ge 7$, $\gamma \ge 6$, and $p-T_0\ge 7$. Upper entries of ${\mathcal{F}}_i\left(\mathbf{A}\right)$ and ${\mathcal{F}}_i\left(Q_0\right)$ have order at least $\delta L_i$; lower entries have order at least $1+\gamma L_i$. They are integral for every i. For the middle factor,

$${\mathcal{F}}_1\left({\mathbf{A}}_{†}\right)\in v_{\mathrm{K}}^7{Mat}_4\left(R\left[\left[v_{\mathrm{K}}\right]\right]\right),{\mathcal{F}}_i\left({\mathbf{A}}_{†}\right)\in v_{\mathrm{K}}^{6p^{i-1}}{Mat}_4\left(R\left[\left[v_{\mathrm{K}}\right]\right]\right)(i\ge 2).$$

These follow entrywise, including the diagonal bound $p^i$.

For completeness the exact normalization of the error is as follows. Set $\lambda ={\prod }_{j\ge 0}{\phi }^j(q/p)$. Removing descent from the canonical expansion and dividing its leading unit $\phi {\left(\lambda \right)}^3$ gives $-{\mathbf{A}}_{†}{Q}_0+M$, where

$$M=\sum _{i\ge 1}{r}_i{Z}_i,Z_i=-p^{-2i}\left({\displaystyle \prod _{j=0}^{i-1}}{\mathcal{F}}_j\left(\mathbf{A}\right)\right){\mathcal{F}}_i\left({\mathbf{A}}_{†}\right)\left({\displaystyle \prod _{j=i}^0}{\mathcal{F}}_j\left(Q_0\right)\right),r_i={\displaystyle \frac{{\phi }^{i+1}{\left(\lambda \right)}^3}{\phi {\left(\lambda \right)}^3}}.$$

The last product is in descending order. The denominator $p^{2i}$ follows by using $q^3{\mathbf{A}}^{-1}$ in every inverse and the identity $\lambda =\left({\prod }_{j<i}{\phi }^j(q/p)\right){\phi }^i\left(\lambda \right)$. Near $v_{\mathrm{K}}=-p$,

$$r_i={\displaystyle \prod _{j=1}^i}{(1+v_{\mathrm{K}}^{p^j}/p)}^{-3},r_i(-p)\in {\mathbf{Z}}_p^{\times },r_i^{\prime }(-p)\in p^{p-1}{\mathbf{Z}}_p.$$

The first summand therefore has value in $p^5$ and derivative in $p^4$. For $i\ge 2$ the bounds are $6p^{i-1}-2i$ and $6p^{i-1}-2i-1$, tending strictly to infinity. Thus the entire actual normalized error satisfies

$$M(-p)\in p^5{Mat}_4\left(R\right),M^{\prime }(-p)\in p^4{Mat}_4\left(R\right).$$

(A3)

This covers all entries and the infinite tail, including the endpoint cases $\delta =7$ or $p-T_0=7$. We require no leading coefficient to be nonzero.

Appendix A.3. Selected Equations and the Actual Smooth Chart

Let $PN={\mathbf{A}}_{†}{Q}_0$. Direct multiplication gives

$$P{N}_{31}^{\prime }(-p)=pt{\xi }^{-2}{f}_L,P{N}_{30}(-p)=ap{\xi }^{-2}{f}_0,P{N}_{30}^{\prime }(-p)=-a{\xi }^{-2}{f}_1,$$

where

$$\begin{array}{cc}\hfill f_L& =B_0\xi l+p(-Em\xi +A_{0}abh),\hfill \\ \hfill f_0& =(E-T_0)\xi d_0+Ep\xi d_1+Ebmt{p}^2+G_{0}blpt,\hfill \\ \hfill f_1& =(E-T_0)\xi d_0+(E+T_0)p\xi d_1\hfill \\ \hfill & +[(3E-G_0)bmt-2E{d}_2\xi ]p^2+2G_{0}blpt.\hfill \end{array}$$

The double-vanishing monodromy condition and (A3) imply successively $l=pL$, $d_1=p{D}_1$, $d_0=p^2{D}_0$. The scalar relation then gives $c=pu$ and $u(D_0+D_1+d_2)=\xi$. Division always occurs in the p-flat integral ring. The necessary normalized equations are

$$\begin{array}{cc}\hfill 0& =B_0\xi L-Em\xi +A_{0}abh+p{H}_1,\hfill \\ \hfill 0& =(E-T_0)\xi D_0+E\xi D_1+Ebmt+G_{0}bLt+p{H}_2,\hfill \\ \hfill 0& =(E-T_0)\xi D_0+(E+T_0)\xi D_1+(3E-G_0)bmt-2E{d}_2\xi +2G_{0}bLt+p{H}_3,\hfill \\ \hfill 0& =u(D_0+D_1+d_2)-\xi .\hfill \end{array}$$

Here $H_i$ lift the higher-Frobenius errors of §A.2, rather than being set to zero. At the residual point set $K_0=D_0+D_1+d_2$. The determinant with respect to $(L,D_1,D_0,u)$ is

$$-B_0{T}_0(E-T_0){\xi }^3{K}_0,$$

a unit. The corresponding necessary-equation ring C is formally smooth of relative dimension 6 over $\mathcal{O}$, with free coordinates $a,b,h,t,m,d_2$. Eliminating the residual linear equations gives

$$K_0={\displaystyle \frac{(E-G_0)bmt-(E+T_0)\xi d_2}{\xi (E-T_0)}},u=\xi /K_0.$$

(A4)

To show sufficiency we use the comparison diagram of [4] (§2.3). The adjoint residual representation has inertial constituents 0 and signed differences of the $a_i$, none equal to 1 modulo $p-1$. The cyclotomic-free restriction result [9] (Definition 3.8, Proposition 3.12) therefore applies to the closed immersion of deformation problems. If R is the gauge ring, its common comparison ring is $R\left[\right[10\mathrm{framing}\mathrm{variables}\left]\right]$ and also ${\mathcal{A}}_G\left[\left[2\mathrm{gauge}\mathrm{variables}\right]\right]$, where ${\mathcal{A}}_G$ is the framed potentially crystalline ring with fixed similitude. There is a surjection from $C\left[\right[10\left]\right]$ to this ring. In the suitable setup its nonzero generic fiber has dimension 14, by the ordinary deformation dimension statement in [4]. The common ring has dimension at least 17, whereas $C\left[\right[10\left]\right]$ is a regular local domain of dimension 17. Its kernel is zero: any nonzero ideal in a regular local domain lowers dimension. Consequently $C=R$ and ${\mathcal{A}}_G$ is formally smooth of relative dimension 14. This counts the framed fixed-similitude ring, not the dimension of the stack.

The type Hecke interpolation on the generic fiber identifies the eigenvalues ${\theta }_0=c$ and ${\theta }_2=pt$ by the triangular Frobenius matrix. Equality extends to the integral ring by p-flatness. Faithful flatness gives $p^n{\mathcal{B}}_G\cap {\mathcal{A}}_G=p^n{\mathcal{A}}_G$ for the common ring ${\mathcal{B}}_G$. Thus ${\theta }_0/p=u$ and ${\theta }_2/p=t$ are integral units on ${\mathcal{A}}_G$; these are the eigenvalues used in the text, defined by the type chart rather than by any residual-point table.

Substitution (16) into the chart of §A.3 proves (17) in all sixteen entries. Equation (A4) with $\xi =1$ supplies the divided relation used in Lemma 10, whose inverse realizes every finite spectral arc and embeds the completed two-coordinate ring. This completes the computational input used in Theorem 2.

Appendix B.1. The Patched Axioms and the Congruent Pair ([4], §3.3)

Source: main.tex L1020–1077; SHA-256:

1bf8e677818f4c5275c3a82ca10da8006

a4cc1605f5b57fcad5d12c94a75cb9e

published as §3.3 of [4]. The lead-in sentences and the axiom list, quoted in full:

 For each $j\in \mathcal{J}$, fix continuous representation ${\overline{\rho }}_j:G_{{\mathbf{Q}}_p}\to {\mathrm{GSp}}_4\left(\mathbf{F}\right)$ of weight $(a_3,a_2,a_1,a_0)$ and similitude character ${\psi }_{j,\mathrm{pc}}={\tilde{\omega }}^{b-3}{ϵ}^3{nr}_{{\xi }_j}$ as in §2.3 [of the source] so that $R_{{\overline{\rho }}_j}^{\eta ,\tau ,{\psi }_{j,\mathrm{pc}}}\ne 0$. Also we need a crystalline character ${\psi }_{j,\mathrm{cris}}={ϵ}^b{nr}_{{\xi }_j}$ as in §2.4 [of the source]. […] Let $R_{\infty }$ (resp. $R_{\infty }^{\mathrm{cris}}$) denote $\left({\widehat{⨂}}_{j\in \mathcal{J}}{R}_{{\overline{\rho }}_j}^{\square ,{\psi }_{j,\mathrm{pc}}}\right)\left[[x_1,\dots ,x_h]\right]$ (resp. $\left({\widehat{⨂}}_{j\in \mathcal{J}}{R}_{{\overline{\rho }}_j}^{\square ,{\psi }_{j,\mathrm{cris}}}\right)\left[[x_1,\dots ,x_h]\right]$) for some $h\ge 0$ and write ${\mathfrak{m}}_{\infty }$ (resp. ${\mathfrak{m}}_{\infty }^{\mathrm{cris}}$) for its maximal ideal. Let $★\in \{\varnothing ,\mathrm{cris}\}$, so that for example $R_{\infty }^{★}=R_{\infty }$ or $R_{\infty }^{\mathrm{cris}}$.

 We say that a pair of a $R_{\infty }$-module and a $R_{\infty }^{\mathrm{cris}}$-module $(M_{\infty },M_{\infty }^{\mathrm{cris}})$ is a congruent pair of patched modules for ${({\overline{\rho }}_j,{\psi }_{j,\mathrm{pc}},{\psi }_{j,\mathrm{cris}})}_{j\in \mathcal{J}}$ if it satisfies axioms (PM1)–(PM6) below.

(PM1) 

$M_{\infty }^{★}$ is a finitely generated $R_{\infty }^{★}\left[\left[K^{\mathcal{J}}\right]\right]$-module with a compatible $R_{\infty }^{★}\left[G^{\mathcal{J}}\right]$-action.

(PM2) 

$M_{\infty }^{★}$ is a projective profinite $\mathcal{O}\left[\left[K^{\mathcal{J}}\right]\right]$-module.

If V is a finitely generated $\mathcal{O}$-module with a continuous $K^{\mathcal{J}}$-action, we define the $R_{\infty }^{★}$-module $M_{\infty }^{★}\left(V\right)={Hom}_{\mathcal{O}\left[K^{\mathcal{J}}\right]}^{\mathrm{cts}}{(V,{\left(M_{\infty }^{★}\right)}^{\vee })}^{\vee }$. By (PM2) $M_{\infty }^{★}(-)$ is an exact functor. […]

 If $\tau$ is a regular principal series type we set $R_{\infty }\left(\sigma \left(\tau \right)\right):=\left({\widehat{⨂}}_{j\in \mathcal{J}}{R}_{{\overline{\rho }}_j}^{\eta ,\tau ,{\psi }_{j,\mathrm{pc}}}\right)\left[[x_1,\dots ,x_h]\right]$.

 If $\mu \in X^{*}{\left(T\right)}^{+}$ and $\sigma =V{\left(\mu \right)}_{{\mathbf{Q}}_p}{\otimes E|}_K$ is the restriction to K of a Weyl module for G of highest weight $\mu$ with E-coefficients then we define $R_{\infty }^{\mathrm{cris}}\left(\sigma \right):=\left({\widehat{⨂}}_{j\in \mathcal{J}}{R}_{{\overline{\rho }}_j}^{\overline{\mu }+\eta ,{\psi }_{j,\mathrm{cris}}}\right)\left[[x_1,\dots ,x_h]\right]$.

 For any K-representation $\sigma$, we write ${\sigma }^{\mathcal{J}}$ for the $K^{\mathcal{J}}$-representation ${\otimes }_{\mathcal{J}}\sigma$.

(PM3) 

Let $\sigma =\sigma \left(\tau \right)$ and ${\sigma }^{\mathrm{cris}}=V{\left(\mu \right)}_{{\mathbf{Q}}_p}{\otimes E|}_K$. Let ${\left({\sigma }^{★,\mathcal{J}}\right)}^{\circ }\subset {\sigma }^{★,\mathcal{J}}$ be a $K^{\mathcal{J}}$-equivariant $\mathcal{O}$-lattice. If $M_{\infty }^{★}\left({\left({\sigma }^{★,\mathcal{J}}\right)}^{\circ }\right)\ne 0$ then the action of $R_{\infty }^{★}$ on $M_{\infty }^{★}\left({\left({\sigma }^{★,\mathcal{J}}\right)}^{\circ }\right)$ is maximal Cohen–Macaulay over $R_{\infty }^{★}\left({\sigma }^{★}\right)$. Furthermore $M_{\infty }^{★}\left({\sigma }^{★,\mathcal{J}}\right):=M_{\infty }^{★}\left({\left({\sigma }^{★,\mathcal{J}}\right)}^{\circ }\right)[1/p]$ is projective of (constant) rank $d\ge 1$ over $R_{\infty }^{★}\left({\sigma }^{★}\right)[1/p]$.

(PM4) 

Let $\tau$ be a regular principal series type and $w_j\in W$ for each $j\in \mathcal{J}$. The tautological action of $\mathcal{H}{\left(\left(\sigma {\left(\tau \right)}^{w_j}\right)\right)}_{\mathcal{J}}:={\otimes }_{j\in \mathcal{J}}\mathcal{H}\left(\sigma {\left(\tau \right)}^{w_j}\right)$ on $M_{\infty }\left(\sigma {\left(\tau \right)}^{\mathcal{J}}\right)$ is through

$${\otimes }_{j\in \mathcal{J}}\mathcal{H}\left(\sigma {\left(\tau \right)}^{w_j}\right)\stackrel{{\otimes }_{\mathcal{J}}{\Theta }^{w_j}}{\to }{\widehat{⨂}}_{\mathcal{J}}{R}_{\overline{\rho }}^{\eta ,\tau }[1/p]\to R_{\infty }\left(\sigma \left(\tau \right)\right)[1/p].$$

(PM5) 

There is a canonical isomorphism between $R_{\infty }/\varpi =R_{\infty }^{\mathrm{cris}}/\varpi$-modules $M_{\infty }/\varpi \simeq M_{\infty }^{\mathrm{cris}}/\varpi$.

[…] We define $W\left(M_{\infty }^{★}\right)$ to be the set of irreducible $\mathbf{F}\left[K^{\mathcal{J}}\right]$-representations (Serre weights) F such that $M_{\infty }^{★}\left(F\right)\ne 0$.

 Axioms (PM1)–(PM5) make sense for any $\overline{\rho }$, but the next axiom is specialized to our particular choice.

(PM6) 

Let ${\mu }_{\overline{\rho }}:=(a_2-a_0-2,a_1-a_0-1;a_0+a_3-3)\in X^{*}{\left(T\right)}^{+}$. We assume $F{\left({\mu }_{\overline{\rho }}\right)}^{\mathcal{J}}\in W\left(M_{\infty }^{\mathrm{cris}}\right)$.

The omitted passages […] are the tautological-identification sentence following (PM2) (with a reference of the source to CEG+) and the paragraph fixing a place $w\in \mathcal{J}$ between (PM5) and the definition of $W\left(M_{\infty }^{★}\right)$.

Appendix B.2. The Two Fixed-Place Statements ([4], §2.3, §2.4)

Source: main.tex L795–818 and L860–880; published as Theorem 2.3.10 and §2.4 of [4].

As in the previous section we write $\lambda =(0,0,-1,-1)\in X_{*}\left(T\right)$.

Theorem 2.3.10. Assume that $\overline{\rho }$ is strongly 4-generic and $R_{\overline{\rho }}^{\eta ,{\tau }_0,\psi }\ne 0$. Then $R_{\overline{\rho }}^{\eta ,{\tau }_0,\psi }$ is formally smooth over $\mathcal{O}$, and for $w\in W$ the image of $T_{\lambda }{ϵ}_{w{\tilde{\mu }}_0}$ under the morphism (2.2.10) is equal to $p^{k_w}{r}_w$, where $k_w\in \mathbf{Z}$ and $r_w\in R_{\overline{\rho }}^{\eta ,{\tau }_0,\psi }$ is a unit whose reduction modulo the maximal ideal is given by the following table.

where

$$\begin{array}{cc}\hfill {\zeta }_1& ={\displaystyle \frac{1}{\xi (a_3-a_0+2)}}\left[(a_3-a_0){\xi }_0-(a_2-a_1){\displaystyle \frac{{\xi }_0{x}_{02}}{{\xi }_2{x}_{03}}}\right]\hfill \\ \hfill {\zeta }_2& ={\xi }_1-{\displaystyle \frac{x_{13}}{x_{03}}}.\hfill \end{array}$$

 Let $\lambda =(a_3,a_2,a_1,a_0)\in {\mathbf{Z}}^4$ and let $\overline{\rho }:G_{{\mathbf{Q}}_p}\to {\mathrm{GSp}}_4\left(\mathbf{F}\right)$ be as in (2.3.1) [of the source]. Assume that $a_{i+1}-a_i\ge 2$ for $i=0,1,2$ and $a_3-a_0\le p-2$. […the source defines the flag functor ${\mathcal{D}}^{{\Delta }_{\lambda }}$ of symplectic deformations preserving the ordinary full flag with graded inertia ${ϵ}^{a_i}$, representable by $R_{\overline{\rho }}^{{\Delta }_{\lambda }}$.]

Lemma 2.4.1. $R_{\overline{\rho }}^{{\Delta }_{\lambda }}$ is formally smooth of dimension 15.

Proposition 2.4.2. Under the conditions on $\lambda$ above, the map $R_{\overline{\rho }}^{\lambda ,\psi }\to R_{\overline{\rho }}^{{\Delta }_{\lambda }}$ is an isomorphism. In particular $R_{\overline{\rho }}^{\lambda ,\psi }$ is formally smooth of dimension 15.

Here $R_{\overline{\rho }}^{\lambda ,\psi }$ is the crystalline symplectic deformation ring of Hodge type $\lambda$ and fixed similitude character $\psi$, and $\psi ={ϵ}^b{nr}_{\xi }$ lifts the similitude character of $\overline{\rho }$.

Appendix B.3. The Globalization Statements ([4], §4.4)

Source: main.tex L1474–1497, L1528–1567, L1668–1671; published as Definition 4.4.1, Remark 4.4.2, Lemma 4.4.5, Corollary 4.4.6 and Lemma 4.4.14 of [4].

 Definition 4.4.1. A triple $(F,\overline{r},\chi )$ of a totally real field F, a continuous representation $\overline{r}:G_F\to {\mathrm{GSp}}_4\left(\mathbf{F}\right)$, and a Hecke character $\chi :{\mathbf{A}}_F^{\times }/F^{\times }\to {\mathbf{C}}^{\times }$ is called suitable of weight $(a_3,a_2,a_1,a_0)$ if it satisfies the followings:

(A1)

F has even degree over $\mathbf{Q}$,

(A2)

$\overline{r}$ is odd,

(A3)

$\overline{r}$ is vast and tidy in the sense of [BCGP, §7.5] (this implies that $\overline{r}$ is absolutely irreducible),

(A4)

$(\overline{r},\chi )$ is potentially diagonalizably automorphic of some weight ${\xi }^{\prime }$ and level unramified away from p,

(A5)

$\overline{r}{|}_{G_{F_v}}$ is strongly 7-generic of weight $(a_3,a_2,a_1,a_0)$ as in Definition 2.3.7(ii) [of the source] for each $v|p$.

Let $\overline{\rho }:G_{{\mathbf{Q}}_p}\to {\mathrm{GSp}}_4\left(\mathbf{F}\right)$ be strongly 7-generic of weight $(a_3,a_2,a_1,a_0)$. We say that a triple $(F,\overline{r},\chi )$ is a suitable globalization of $\overline{\rho }$ if it is suitable of weight $(a_3,a_2,a_1,a_0)$, and there is a place $w|p$ of F such that $\overline{r}{|}_{G_{F_w}}\simeq \overline{\rho }$.

Remark 4.4.2. In (A4), we assume $(\overline{r},\chi )$ to be potentially diagonalizably automorphic as opposed to being modular. This is used to show that $(\overline{r},\chi )$ is modular of the “obvious” Serre weight $F\left(\mu \right)$ in Corollary 4.4.7 [of the source] by applying the automorphy lifting result for potentially diagonalizable lifts ([BLGGT, 4.2.1]). The unramifiedness in (A4) is assumed for simplicity in the construction of patched modules. Assumption (A5) is not important for the construction of patching modules.

 Lemma 4.4.5. Assume that $p>7$. Let $\overline{\rho }:G_K\to {\mathrm{GSp}}_4\left(\mathbf{F}\right)$ be a continuous representation. Suppose there is a lift $\rho :G_K\to {\mathrm{GSp}}_4\left(E\right)$ such that $std\left(\rho \right)$ is potentially diagonalizable. Then there is a totally real field F of even degree over $\mathbf{Q}$ such that $F_v\simeq K$ for all $v|p$ and a continuous representation $r:G_F\to {\mathrm{GSp}}_4\left(E\right)$, with its reduction modulo $\varpi$ denoted by $\overline{r}$, satisfying that

1.

$\overline{r}$ is odd,

2.

$\overline{r}$ is vast and tidy,

3.

$\overline{r}{|}_{G_{F_v}}\simeq \overline{\rho }$ for all $v|p$,

4.

$\overline{r}$ is unramified at all finite places not dividing p,

5.

${r|}_{G_{F_v}}$ and $\rho$ correspond to points contained in the same component of a potentially crystalline deformation ring for all $v|p$ (in particular, $std\left(r{|}_{G_{F_v}}\right)$ is potentially diagonalizable), and

6.

there exists a regular algebraic cuspidal automorphic representation $\pi$ of ${\mathrm{GSp}}_4$$\left({\mathbf{A}}_F\right)$ such that $r\simeq r_{\pi ,p,\iota }$.

Corollary 4.4.6. Suppose $\overline{\rho }:G_{{\mathbf{Q}}_p}\to {\mathrm{GSp}}_4\left(\mathbf{F}\right)$ is 7-generic of weight $(a_3,a_2,a_2,a_0)$. Then there is a suitable globalization $(F,\overline{r},{\chi }_{\mathrm{cris}})$ of $\overline{\rho }$ such that $({\chi }_{\mathrm{cris},p,\iota }\otimes {ϵ}^{-3}){|}_{G_{F_v}}={ϵ}^b{nr}_{{\xi }_v}$.

The weight in Corollary 4.4.6 is discussed in §7.1 of the main text; its third entry reads $a_2$ in both the arXiv source (main.tex L1566) and the published version, while the weight used everywhere else in [4] — in Definition 4.4.1(A5), in the inertial type of Definition 2.3.6 below, and in the corollary’s own proof, which appeals to an ordinary crystalline lift of weight ${\overline{\mu }}_{\overline{\rho },v}+\eta$ — involves $a_1$ (the third entry of the weight, and the second entry of the type ${\mu }_0$).

We now take $\mathcal{J}=S_p$ and choose an isomorphism $K^{\mathcal{J}}\simeq K_p$. The following is the main result of this section [of the source].

Lemma 4.4.14. The pair $(M_{\infty },M_{\infty }^{\mathrm{cris}})$ is a congruent patched module pair for ${(\overline{r}{|}_{G_{F_v}},{\psi }_{v,\mathrm{pc}},{\psi }_{v,\mathrm{cris}})}_{v\in S_p}$ satisfying (PM1)–(PM6), and we have $d=\left|W\right|=8$ in (PM3).

Appendix B.4. The Type and the Weight Occurrence ([4], §2.3, §3.2)

Source: main.tex L746–749 and L988–991 with the appendix table at L1855–1857; published as Definition 2.3.6 and Lemma 3.2.7 of [4].

 Definition 2.3.6. Associated with $\overline{\rho }$ [as in (2.3.1) of the source] as above, we define the symplectic inertial type ${\tau }_0=\tau (id,\overline{{\mu }_0})$ where ${\mu }_0=(a_2-a_0,a_1-a_0+1;a_0+a_3-3)$. This is a lowest alcove presentation.

Lemma 3.2.7. For $\lambda$ lying inside the stated alcoves, $\chi \left(\lambda \right)$ decomposes into irreducibles ${\chi }_p(-)$ as in Table A.3 [of the published source].

Table numbering. In the published version of [4] the decomposition table just quoted is Table A.3 (the appendix numbers the alcove table A.1, a figure A.2, this table A.3 and the graded-piece table A.4). In the arXiv v4 source the same table carries the label table:Weyl decomposition and is the second appendix table. The main text cites the published numbering. The row of that table used in Lemma 11 is the $C_1$ row, quoted here from main.tex L1836–1837 with the table’s caption (L1855):

$C_1$   ${\chi }_p\left(\lambda \right)+{\chi }_p\left(C_0\right)$,

caption: “This table gives the Jordan–Hölder factors of Weyl modules for ${\mathrm{GSp}}_4$. In it, we write ${\chi }_p\left(A\right)$ instead of ${\chi }_p\left({\lambda }_A\right)$, where ${\lambda }_A$ is the unique weight linked to $\lambda$ in alcove A.”

Appendix B.5. The Weight Convention of [3]

Source: typeweight.tex L13–33 of arXiv:2304.13879v3; §2 of [3].

Let $G={\mathrm{GSp}}_4$ be the split reductive group over $\mathbf{Z}$ defined by

$${\mathrm{GSp}}_4\left(R\right)=\{g\in {\mathrm{GL}}_4\left(R\right){\mid }^{t}gJg=sim\left(g\right)J\mathrm{for}\mathrm{some}sim\left(g\right)\in R^{\times }\}$$

for any commutative ring R, where $J=antidiag(1,1,-1,-1)$. Then $sim:g\mapsto sim\left(g\right)$ defines a character of ${\mathrm{GSp}}_4$, called the similitude character. […] We identify the character group $X^{*}\left(T\right)$ with ${\mathbf{Z}}^3$ by defining the character corresponding to $(a,b;c)\in {\mathbf{Z}}^3$ by

$$(a,b;c):diag(x,y,z{y}^{-1},z{x}^{-1})\mapsto x^a{y}^b{z}^c.$$

This is the convention $(x,y;c)$ of the paragraph “Weight coordinates” in §1; the conversion (3) to the $(x,y;z)$ convention of [4] follows by reading off the central exponent.