Proof. Choose distinct primes \(r_1,r_2\geq3\) and put \(N=r_1r_2\). These preliminary levels give a fine moduli cover. The auxiliary incidence primes will be chosen separately in 10, after the correspondence list has been fixed. After adjoining the requisite roots of unity, the fine Siegel moduli space \(\mathcal A_2(N)\) carries a universal principally polarized abelian scheme and has a finite surjective forgetful map to the coarse space \(\mathcal A_2\); its analytic uniformization and the algebraicity of the universal family are described in (Milne 2013, Theorem 4.6 and Corollary 4.7). Take an irreducible component of its pullback to \(C\) that dominates \(C\), and normalize. Normalization is finite for a curve over a field of characteristic zero. The resulting curve is smooth, and pullback supplies the abelian scheme. Removing the preimages of any finite exceptional set of coarse points leaves a finite surjective map onto the complementary open of \(C\).

All these varieties, components, morphisms, and level structures are defined over a number field after a fixed enlargement. The fibers of a fixed finite map have bounded length; on a dense open this length is its degree. The residue field degree of any point in such a fiber is at most that length. This proves (7), with the degrees of any additional fixed covers included in \(b\). It also shows that the image of every omitted finite set is finite. An infinite coarse locus therefore has infinitely many lifts after every such fixed omission, and we choose one of them as \(t_0\). Later covers may be supplied with a fixed lift of this point; we then reuse \(S,t_0\) for the refined data. Open charts and frames used at the center are chosen to contain it.

A smooth curve over a number field has a smooth projective completion. The point \(t_0\) is an interior point of the moduli family, and its divisor has positive degree on that completion. This construction uses no geometric boundary point and imposes no condition on the arithmetic reduction of \(\mathscr A_{t_0}\). ◻

Proof. Use a symplectic basis, with matrix \(J=\left(\begin{smallmatrix}0&I_2\\-I_2&0\end{smallmatrix}\right)\). The equations \(D^{\mathsf t}J=JD\) and \(\mathop{\mathrm{Tr}}D=0\) give exactly \[ D=\begin{pmatrix}A&uK_2\\vK_2&A^{\mathsf t}\end{pmatrix}, \qquad A=\begin{pmatrix}a&b\\c&-a\end{pmatrix}, \qquad K_2=\begin{pmatrix}0&1\\-1&0\end{pmatrix}. \tag{12}\] Multiplication gives \[ D^2=(a^2+bc-uv)I_4, \qquad \mathop{\mathrm{Tr}}(D^2)=4(a^2+bc-uv). \tag{13}\] Thus the form is the orthogonal sum of a nonzero one-dimensional form and two hyperbolic planes. Its Gram determinant in the five coordinates displayed above is \(64\), a unit over every \(\mathbb Z_\ell\) with \(\ell\) odd. A principal polarization makes the Tate module symplectically unimodular, so the same calculation proves (10).

Conjugation by a symplectic similitude preserves both conditions in (9) and the trace pairing. Since \(\mathop{\mathrm{GSp}}_4\) is connected, its orthogonal image has determinant one. The induced action of \(\mathop{\mathrm{Sp}}_4\) is nontrivial, as is already seen by conjugating the matrices in (12). The simple Lie algebra \(\mathfrak{sp}_4\) consequently has zero kernel in this action. Both \(\mathop{\mathrm{Sp}}_4\) and \(\mathop{\mathrm{SO}}_5\) have dimension ten, so the image is all of \(\mathop{\mathrm{SO}}_5\). The finite kernel is central in the connected group \(\mathop{\mathrm{Sp}}_4\) and is exactly \(\{1,-1\}\). Over an algebraic closure a similitude is a scalar times a symplectic element, which proves the scalar-kernel assertion for \(\mathop{\mathrm{GSp}}_4\) and identifies the restriction as the spin covering. The natural representation of \(\mathop{\mathrm{SO}}_5\) is irreducible over an algebraically closed field of characteristic zero: the orbit of any nonzero vector, whether isotropic or not, spans the entire quadratic space. This proves absolute irreducibility. Equivalently, \(D% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}\psi(D\cdot,\cdot)\) identifies the self-adjoint space with alternating two-forms, and the trace-free summand with the primitive exterior-square representation.

For the last assertion, put \(A=\mathscr A_t\). The equivalence between complex abelian varieties up to isogeny and polarizable rational Hodge structures of type \((-1,0),(0,-1)\) identifies \(\mathop{\mathrm{End}}^0(A)\) with the Hodge endomorphisms of \(H_1(A,\mathbb Q)\) (Milne 1995, Theorem 4.1). Write \(H_1(A,\mathbb Q)=H_1(E,\mathbb Q)\otimes\mathbb Q^2\) using the given unpolarized isogeny. Since \(E\) has no CM, its endomorphism algebra is \(\mathbb Q\); the Hodge endomorphism algebra of \(A\) is therefore \(\mathop{\mathrm{Mat}}_2(\mathbb Q)\). An alternating polarization on this tensor product has the form \(\psi_E\otimes B\), where \(B\) is a symmetric rational form on \(\mathbb Q^2\). Indeed, each of its four matrix entries is a Hodge alternating pairing on \(H_1(E,\mathbb Q)\), hence a scalar multiple of \(\psi_E\); alternation of the total form makes these four scalars symmetric. Positivity of the polarization, after choosing the sign of \(\psi_E\), makes \(B\) positive definite. The Rosati involution on \(\mathop{\mathrm{Mat}}_2(\mathbb Q)\) is consequently the adjoint for \(B\).

Its self-adjoint part has dimension three and contains the identity. The homology trace on this algebra is twice the ordinary matrix trace, so imposing trace zero leaves dimension two. Over \(\mathbb R\), conjugation by the positive square root of \(B\) turns a self-adjoint matrix \(D\) into a symmetric matrix \(D_0\). Hence, for \(D\ne0\), \[\mathop{\mathrm{Tr}}_{H_1(A)}(D^2)=2\mathop{\mathrm{tr}}(D_0^2)>0.\] This proves the asserted positivity for every principal polarization, without requiring the initial isogeny from \(E^2\) to preserve it. ◻

Proof. Hodge genericity in Siegel moduli means that the generic Mumford–Tate group of \(H\) is \(\mathop{\mathrm{GSp}}_4\). The variation is pure and polarizable and comes from an abelian scheme over a smooth algebraic curve. Thus it is among the good variations to which the connected-monodromy normality theorem applies: connected algebraic monodromy is a normal subgroup of the derived generic Mumford–Tate group (André 1992, sec. 5, Theorem 1). Since \(\mathop{\mathrm{Sp}}_4\) is almost simple, the connected monodromy is trivial or all of \(\mathop{\mathrm{Sp}}_4\).

Here is a direct exclusion of the trivial possibility. It would make the monodromy group finite, and a finite cover would trivialize the integral local system. The period map would then be a single-valued holomorphic map from a smooth algebraic curve to Siegel space. Realize Siegel space as a bounded domain in \(\mathbb C^3\). Each coordinate of this map is bounded near each of the finitely many missing points of the smooth projective completion, so has a removable singularity there. Every extended coordinate is constant on the projective curve. The family would have constant moduli image, contrary to its domination of \(C\). This also explains concretely the applicability of (André 1992, sec. 6, Remark 1). The monodromy conclusion is independent of the Mumford–Tate group at the chosen center. 6 now gives the assertion on \(\mathcal P\).

A nonconstant morphism between smooth algebraic curves restricts, after removal of finitely many points, to a finite etale morphism onto a dense open. Its topological fundamental group therefore has image of finite index in that of this open; the latter maps surjectively to the fundamental group of the original curve. Finite-index subgroups have the same connected Zariski closure. This proves invariance under finite covers and open restrictions. For a dominant map from a higher-dimensional smooth variety, choose an algebraic curve in that variety whose projection is nonconstant, and normalize it. Such a curve is obtained by successive general affine hyperplane sections through a point where the differential of the projection is nonzero. Its monodromy is contained in the pullback monodromy and is already full by the curve case. This proves the pullback assertion.

For completeness, full monodromy persists under field conjugation without a choice of a conjugation-compatible complex period map. Choose a prime \(\ell\). The Betti monodromy is Zariski dense in \(\mathop{\mathrm{Sp}}_4\), so its image on \(H\otimes\mathbb Q_\ell\) is Zariski dense in \(\mathop{\mathrm{Sp}}_{4,\mathbb Q_\ell}\). The comparison of topological and geometric etale fundamental groups identifies the latter image with a dense subgroup of the geometric etale monodromy image. A field conjugation identifies the corresponding geometric etale families and their symplectic Tate systems. Applying the same comparison on the conjugate complex curve proves that its Betti algebraic monodromy is again full. In particular its coarse image is Hodge generic: otherwise its monodromy would lie in the derived group of a proper generic Mumford–Tate subgroup. One can also use the preservation of Siegel special subvarieties by conjugation.

It remains to prove the assertion concerning \(W\). Work at a generic Mumford–Tate point of the direct sum of the two pulled back homology variations, in rational symplectic bases. Both projections of its connected algebraic monodromy are full, by the pullback assertion just proved. Pass to the adjoint groups \(G_1,G_2\), each isomorphic to \(\operatorname{PGSp}_4\) and to the corresponding \(\mathop{\mathrm{SO}}_5\). Let \(M\subset G_1\times G_2\) be the connected joint image. The kernel of either projection is a normal subgroup of the other adjoint simple group. Thus either one kernel is the whole factor, giving \(M=G_1\times G_2\), or both are trivial and \[ M=\{(x,\phi(x)):x\in G_1\} \tag{15}\] for a \(\mathbb Q\)-isomorphism \(\phi:G_1\to G_2\). This is the group form of Goursat’s argument; no passage to a merely real graph has been made.

The remaining task is to lift this adjoint graph to a positive rational similitude identifying the homology Hodge structures. This requires both the graph relation and the polarization characters. Let \(T\) be the generic Mumford–Tate group of the direct sum. The normality theorem applies on the smooth algebraic variety \(W\) and says that \(T\) normalizes connected monodromy. Its adjoint image therefore normalizes (15). The normalizer of this graph in \(G_1\times G_2\) is the graph itself. Indeed, normalization by \((u,v)\) says that \(v\phi(x)v^{-1}=\phi(u)\phi(x)\phi(u)^{-1}\) for all \(x\); the trivial center of \(G_2\) gives \(v=\phi(u)\).

The root diagram of type \(C_2\) has no nontrivial automorphism, so every automorphism of its adjoint split group is inner. The conjugating element is unique, hence is rational when the automorphism is rational. Finally the sequence \[1\longrightarrow\mathbb G_m\longrightarrow\mathop{\mathrm{GSp}}_4 \longrightarrow\operatorname{PGSp}_4\longrightarrow1\] and \(H^1(\mathbb Q,\mathbb G_m)=1\) lift it to a rational similitude \(g\in\mathop{\mathrm{GSp}}_4(\mathbb Q)\). Consequently the two Hodge homomorphisms, after conjugation by \(g\), agree in the adjoint group. Their remaining difference is a scalar character \(c\) of the Deligne torus. Each polarization is a morphism of Hodge structures \(\psi_i:H_i\otimes H_i\to\mathbb Q(1)\). Thus the full similitude character \(\nu\circ h_i\) is the character of the same Tate structure \(\mathbb Q(1)\) for both \(i\), on the whole Deligne torus. Since a scalar \(c\) has multiplier \(c^2\), the equality \(h_2=c\,g h_1g^{-1}\) therefore gives \(c^2=1\). The Deligne torus is connected as an algebraic group, whence \(c=1\). We have obtained a rational Hodge isomorphism on \(H_1\), not just an identification of its adjoint representation.

Its multiplier is positive. For if \(J_i\) are the complex structures at this point, then \(gJ_1=J_2g\), and positivity of the two polarization metrics in \[\psi_2(gv,J_2gv)=\nu(g)\psi_1(v,J_1v)\] forces \(\nu(g)>0\). By the rational Hodge equivalence (Milne 1995, Theorem 4.1), an integral multiple of \(g\) is a polarized isogeny.

This same \(g\) describes an analytic open of the joint image. Indeed, trivialize the rational local system in a simply connected open. For each of its countably many rational tensors, the locus where it is Hodge is closed analytic. A point outside the union of those loci that are proper has the generic Mumford–Tate group. Every tensor Hodge at that point is therefore Hodge throughout the open. By the tensor-stabilizer description of the Mumford–Tate group, all Hodge homomorphisms there factor through the transported generic group \(T\). Applying the graph relation for that group makes \(g\) a Hodge isomorphism throughout the open. The joint period map is therefore locally in the graph of \(g\), and its coarse image lies locally in the Hecke correspondence \(T_g\). This correspondence is closed and algebraic: it is the image of the finite polarized-isogeny moduli correspondence, or equivalently of the quotient for \(\Gamma\cap g^{-1}\Gamma g\), where \(\Gamma=\mathop{\mathrm{Sp}}_4(\mathbb Z)/\{1,-1\}\). An algebraic equation for \(T_g\) vanishing on a nonempty analytic open of the irreducible image vanishes on that image. Thus the entire joint image lies in \(T_g\). The claim when the second projection is constant follows already from full monodromy of the first projection. ◻

Proof. We prove finiteness, including the analytic stabilizer assertion on which it depends. Use one fixed torsion-free principal level \(N\) as in 5. There are finitely many irreducible closed curves \(D_\alpha\subset\mathcal A_2(N)\) lying above the curves \(C_i\). All of them have full connected monodromy by 7. Write \(\mathfrak H_2\) for Siegel space and \[G=\mathop{\mathrm{GSp}}_4(\mathbb R)^+/\mathbb R^\times \simeq\mathop{\mathrm{Sp}}_4(\mathbb R)/\{1,-1\}, \qquad \Gamma_N\subset G\] for the effective level group. The map \(\pi_N:\mathfrak H_2\to\mathcal A_2(N)(\mathbb C)\) is a covering: the choice of torsion-free level eliminates stabilizers.

Proof. We will use the following elementary observation about rational charts. The big Bruhat cell of the split group \(\mathop{\mathrm{Sp}}_4\) has coordinates \(\mathbb A^4\times\mathbb G_m^2\times\mathbb A^4\): multiply the four negative root subgroups, a diagonal torus element, and the four positive root subgroups, in a fixed order. Their matrices, their inverses, and their actions on \(V\) have Laurent-polynomial entries of bounded degrees in these ten coordinates, independently of \(k\). The same assertion holds for a big cell of a split parabolic, using its Levi root subgroups and unipotent radical. Conjugating a parabolic or inserting a fixed group element changes coefficients and does not change these degree bounds.

Consequently, an open condition expressed by minors of a bounded-size matrix in this representation can be tested on either chart by a polynomial of degree at most an absolute constant \(D\). If the condition holds at some geometric point, it holds on the corresponding dense chart: both the group and the parabolic are geometrically irreducible. Choose a minor that is not identically zero, clear its torus denominators, and multiply by the torus coordinates. The resulting nonzero polynomial has a degree bounded by another absolute constant. A nonzero polynomial of total degree less than \(q\) cannot vanish on all of \(k^n\). This follows by induction on \(n\), applying the univariate root bound to its nonzero coefficients. Thus every such geometrically nonempty open condition has a \(k\)-point once \(q>q_0\), for one constant \(q_0\) that works for all planes under consideration. No coefficient height or isometry type enters this bound.

For the nondegenerate sum, work first over \(\bar k\). Choose a nondegenerate four-space \(M\) containing \(H_0\), and write \(M=H_0\perp K\) with \(\dim K=2\). Choose bases in which both forms are \(I_2\). The plane that is the graph of \(A=\mathop{\mathrm{diag}}(a,b):K\longrightarrow H_0\) is disjoint from \(H_0\) and has Gram matrix \[I_2+A^{\mathsf t}A=\mathop{\mathrm{diag}}(1+a^2,1+b^2).\] Choices of \(a,b\in\bar k\) give each of the ranks \(0,1,2\). Over \(\bar k\), bilinear forms on a two-space are classified by rank, so there is a graph isometric to \(H\), including when \(H\) is degenerate. Witt’s extension theorem for subspaces of a nondegenerate space extends this isometry to an orthogonal transformation of \(V_{\bar k}\). Its determinant can be changed by a reflection in a nonisotropic vector perpendicular to the graph. Such a vector exists: that perpendicular has dimension three, whereas the largest totally isotropic subspace of \(V_{\bar k}\) has dimension two. We obtain an element of \(\mathop{\mathrm{SO}}(V_{\bar k})\), which lifts to \(\mathop{\mathrm{Sp}}_4(\bar k)\) by the spin covering. Nonvanishing of the Gram determinant of the four concatenated basis vectors is an open condition and implies their independence. The chart argument just proved supplies \(g_+\in\mathop{\mathrm{Sp}}_4(k)\) uniformly.

For the degenerate sum, choose an isotropic line \(L\subset H_0^\perp\) over \(k\). The perpendicular is a nondegenerate ternary space. Every such space over a finite field of odd cardinality is isotropic: after diagonalization, the two sets \(\{ax^2:x\in k\}\) and \(\{-c-by^2:y\in k\}\) each have \((q+1)/2\) elements and therefore meet, giving an isotropic vector with third coordinate \(1\). The three-space \(H^\perp\) also has an isotropic line over \(k\). If it is nondegenerate the same argument applies; if it is degenerate its nonzero radical supplies one.

The group \(\mathop{\mathrm{Sp}}_4(k)\) is transitive on the isotropic lines of \(V_k\). Indeed, under the primitive exterior-square realization these lines are the Plücker lines of the Lagrangian two-spaces in the symplectic four-space; symplectic bases show transitivity over \(k\) itself. Move the chosen line in \(H^\perp\) to \(L\). After this preliminary move, \(H\subset L^\perp\). Let \(P\) be the stabilizer of \(L\) in \(\mathop{\mathrm{Sp}}_4\) and put \[Q=L^\perp/L.\] This is a nondegenerate ternary space. The projection of \(H_0\) is a plane \(\overline H_0\subset Q\) because \(H_0\cap L=0\). Over \(\bar k\), the Levi action of \(P\) on \(Q\) includes \(\mathop{\mathrm{SO}}(Q)\). If \(H\) contains \(L\), its projection is a line. This line can be moved outside \(\overline H_0\), since the standard representation of \(\mathop{\mathrm{SO}}(Q)\) is irreducible. Its inverse image then meets \(H_0\) trivially.

If \(H\) does not contain \(L\), its projection is a plane \(\overline H\subset Q\). Move it so that \(\overline H\ne\overline H_0\). Such a move exists since a plane cannot be invariant under the irreducible group \(\mathop{\mathrm{SO}}(Q)\). The two projected planes now meet in a line \(J\). The unipotent radical permits an arbitrary change of the lift \(Q\longrightarrow L^\perp\) by a linear map \(Q\longrightarrow L\). Explicitly, choose a hyperbolic pair \(e,f\), with \(L=\bar k e\) and \(\langle e,f\rangle=1\), and identify \(Q\) with \((\bar k e+\bar k f)^\perp\). For \(z\in Q\) the orthogonal unipotent transformation is \[e% \BeginAccSupp{method=hex,unicode,ActualText=27FC}% \longmapsto\EndAccSupp{}e,\qquad n% \BeginAccSupp{method=hex,unicode,ActualText=27FC}% \longmapsto\EndAccSupp{}n-\langle n,z\rangle e,\qquad f% \BeginAccSupp{method=hex,unicode,ActualText=27FC}% \longmapsto\EndAccSupp{}f+z-\tfrac12\langle z,z\rangle e.\] These transformations lift to the unipotent radical in \(\mathop{\mathrm{Sp}}_4\); equivalently, in its symplectic realization that radical consists of the matrices \(\left(\begin{smallmatrix}I&Z\\0&I\end{smallmatrix}\right)\) with \(Z\) symmetric. Since the form on \(Q\) is nondegenerate, \(n% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}-\langle n,z\rangle\) runs through all linear forms on \(Q\). Choose it so that the lifts of the nonzero vectors of \(J\) in the moved \(H\) and in the fixed \(H_0\) are distinct. Then \(H\cap H_0=0\).

We have exhibited a geometric point of \(P\) where a four-by-four minor of the concatenated bases of \(H_0\) and the moved \(H\) is nonzero. The parabolic chart argument gives such a point over \(k\) for the same uniform lower bound on \(q\). Their sum, contained in \(L^\perp\) and of dimension four, is exactly \(L^\perp\). Its radical is \(L\), so it is degenerate. This proves the first assertion and completes all three possible target-rank cases. ◻

Proof. Choose a symplectic integral Betti lattice. Its geometric monodromy group \(\Gamma\subset\mathop{\mathrm{Sp}}_4(\mathbb Z)\) is finitely generated, because the fundamental group of a smooth algebraic curve is finitely generated, and is Zariski dense by 7. Strong approximation for a finitely generated Zariski-dense subgroup of a connected, simply connected, absolutely almost simple rational group gives \[\Gamma\longrightarrow\mathop{\mathrm{Sp}}_4(\mathbb F_\ell) \quad\hbox{surjective for every sufficiently large prime }\ell;\] these are precisely the hypotheses of (Matthews et al. 1984, Introduction, Theorem, p. 515) for \(\mathop{\mathrm{Sp}}_4\).

Fix \(\ell_1>q_0\) outside this theorem’s finite exceptional set, the finitely many primes of degeneracy of the fixed center plane, and the primes dividing the already fixed multipliers in 8. Its reduction kernel \(\Gamma_1\) has finite index in \(\Gamma\). Schreier’s theorem gives finite generation of \(\Gamma_1\); its Zariski closure is still \(\mathop{\mathrm{Sp}}_4\), since a connected algebraic group cannot be a finite union of translates of a proper closed subgroup. Apply strong approximation to \(\Gamma_1\), and choose \(\ell_2>q_0\) distinct from \(\ell_1\), outside its exceptional set, outside the center-plane degeneracy set, and outside the residue characteristics of all non-potentially-good places of \(A_{t_0}\). The last set is finite because the fixed abelian variety has good reduction away from finitely many places. All these exclusions use fixed data.

It follows that \[ \Gamma\longrightarrow \mathop{\mathrm{Sp}}_4(\mathbb F_{\ell_1})\times\mathop{\mathrm{Sp}}_4(\mathbb F_{\ell_2}) \quad\hbox{is surjective}. \tag{23}\] To check this explicitly, first realize a prescribed element in the first factor by an element of \(\Gamma\); correct its second factor using an element of \(\Gamma_1\). The cover parametrizing the two full markings with fixed Weil pairings is therefore connected: its monodromy acts transitively on the product torsor of markings. After a fixed extension of the field of definition, it is a fixed geometrically connected finite étale cover of our curve, carrying both markings. Fix any lift of \(t_0\) to it.

For each target, 9 prescribes a marking at \(\ell_1\) and, independently, a marking at \(\ell_2\) satisfying (21) and (22). The full product torsor lies in the connected cover, so these choices give a point of that cover. We continue to write \(S,t_0,t\) for the resulting curve, center, and selected target. Selection need not be an algebraic rule: every point in a fiber of a finite morphism of fixed degree has degree over the original point bounded by that degree. Subsequent finite omissions on the curve omit only finitely many points downstairs, as in 5.

Finally, the planes come from actual rational endomorphisms. Their étale realizations, intersections with integral Tate lattices, and induced marked reductions all commute with simultaneous conjugation. A conjugation changes the pairing trivialization by the same similitude on the center and target; on the operator five-space this is an isometry. Dimensions and degeneracy are unchanged. Potential good reduction is also invariant under finite extension, so adjoining levels and a field of definition creates no new residue characteristics to be excluded from the choice of \(\ell_2\). ◻

Proof. Take \(R=\mathcal O^{\mathrm{sh}}_{X,\bar z}\). This ring is noetherian and regular, hence is an integral domain. The inverse image \(U\) of the moduli interior in \(\mathop{\mathrm{Spec}}R\) is a nonempty open in an integral scheme, hence is connected. We make this construction on the regular toroidal base before pulling back to either trait.

For every \(n\ge1\), multiplication by \(\ell^n\) on the smooth semiabelian scheme is étale: its differential is multiplication by the invertible integer \(\ell^n\). Its kernel \(\mathcal G[\ell^n]\) is consequently separated, quasi-finite and étale over \(R\). It need not be finite over \(R\). The finite-part decomposition over a henselian local base writes it as its finite open-and-closed part and a part with empty closed fiber. The finite part is a subgroup; its formation commutes with local morphisms of henselian local schemes. More generally this is the construction of the finite part and the torus part of quasi-finite flat subgroups in (Lan 2013, sec. 3.4.1, equations (3.4.1.1)–(3.4.1.4)).

Here the whole special fiber is a torus, so the finite part and the torus part coincide. Denote this subgroup by \(E_n\). It is finite étale of rank \(\ell^{2n}\), with special fiber \(\mathcal G_{\bar z}[\ell^n]\). Since \(R\) is strictly henselian, \(E_n\) is constant and each of its special points has a unique lift to an \(R\)-section. The multiplication and transition maps among the \(E_n\) are the unique lifts of those on the special torus. Thus the system is isomorphic to the standard system \((\mathbb Z/\ell^n\mathbb Z)^2\), with its usual reduction transition maps, after choosing compatible generators in the special torus.

Over \(U\), the full \(\ell\)-marking identifies \(\mathcal G[\ell]_U\) with the constant group \((\mathbb Z/\ell\mathbb Z)^4\). Every section of \(E_1\) has a constant label under this identification, because \(U\) is connected. The labels form a subgroup of dimension two; call it \(W_{\bar z}\). Lift either trait to the strict henselizations using the chosen geometric specialization. Compatibility of the finite part under these local base changes shows that its generic toric torsion has exactly these labels. This proves that the two traits have the same marked subspace. In particular, uniqueness is being used over a common strict local base, rather than separately to compare unrelated trait liftings.

On a trait, passage to the inverse limit of the generic fibers of \(E_n\) gives a rank-two lattice \(\mathcal T\subset T_\ell A\). Its reduction modulo \(\ell\) is the embedded two-space \(W_{\bar z}\). It is saturated. Indeed, lift a basis of \(W_{\bar z}\) to compatible generators of the system \(E_n\); the resulting two Tate vectors have independent reductions in the free rank-four Tate module. They extend to a \(\mathbb Z_\ell\)-basis, so their span is a direct summand. The inverse-limit construction shows that this span is precisely \(\mathcal T\).

An integral endomorphism of the generic fiber extends to the semiabelian scheme on the trait by the normal-base extension theorem above. On the special fiber it preserves the torus and its torsion. Uniqueness of the lifted torsion sections implies preservation of each \(E_n\) and therefore of \(\mathcal T\). This argument is compatible with extending the trait, even ramifiedly. Rational linear combinations preserve \(\mathcal T_{\mathbb Q_\ell}\). Since \(\mathcal T\) is saturated, \[\mathcal T=\mathcal T_{\mathbb Q_\ell}\cap T_\ell A.\] Every integral Tate operator lying in their rational span preserves this intersection. Reducing that assertion modulo \(\ell\) proves the last statement, in particular for the full saturation of an endomorphism plane even if its chosen spanning pair becomes dependent modulo \(\ell\). ◻

Proof. Write \(\psi\) for the symplectic form. Its restriction to \(W\) has rank two or zero. In the first case \(V_4=W\perp W^\perp\). A self-adjoint operator preserving \(W\) also preserves \(W^\perp\), since \(\psi(Xv,w)=\psi(v,Xw)\). A self-adjoint operator on a symplectic two-space is scalar, as is seen by solving \(D^{\mathsf t}J=JD\) for a two-by-two matrix. Our operator is consequently \(aI_W\oplus bI_{W^\perp}\), and its trace-zero condition is \(2a+2b=0\). The space of such operators has dimension one.

In the rank-zero case \(W\) is Lagrangian. Choose it as the first Lagrangian in a symplectic basis, with form matrix \(\left(\begin{smallmatrix}0&I\\-I&0\end{smallmatrix}\right)\). The self-adjoint trace-free operators preserving \(W\) are exactly \[ \left\{ X(D,J)=\begin{pmatrix}D&J\\0&D^{\mathsf t}\end{pmatrix}: \mathop{\mathrm{tr}}D=0,\quad J^{\mathsf t}=-J \right\}. \tag{24}\] This is a four-space, since \(\dim\mathfrak{sl}_2=3\) and the skew-symmetric two-by-two matrices have dimension one. Its trace pairing is \[\mathop{\mathrm{Tr}}\bigl(X(D,J)X(D',J')\bigr)=2\mathop{\mathrm{tr}}(DD').\] The trace pairing on \(\mathfrak{sl}_2\) is nondegenerate in odd characteristic: in coordinates \(D=\left(\begin{smallmatrix}a&b\\c&-a\end{smallmatrix}\right)\) it pairs \((a,b,c)\) with \((a',b',c')\) as \(2aa'+bc'+cb'\). Hence the radical of (24) is exactly the one-dimensional \(J\)-space. Any four-space of operators contained in this stabilizer equals it and is degenerate. Both cases give the required contradiction. ◻

Proof. We first identify the special group at the center. After a finite local extension the elliptic curve \(E\) in an isogeny \(A_{t_0}\sim E^2\) has semistable reduction. One may obtain this extension by choosing a full elliptic level invertible at the place and applying the same proper toroidal construction in dimension one. An elliptic curve with semistable reduction is either good or multiplicative. Toric rank is invariant under isogeny and is additive on products: extend an isogeny and a quasi-inverse, with composites multiplication by an integer, to the semiabelian models. On special fibers their restrictions map tori into tori, since a homomorphism from a torus to an abelian variety is zero. The two composites on the tori are multiplication by that integer, so these restrictions are isogenies and the torus dimensions agree. Thus the potential toric rank of \(A_{t_0}\) is zero or two. Rank zero in the semistable model is good reduction; in that case \(A_{t_0}\) would be potentially good. Our hypothesis therefore gives rank two.

Use the full \(\ell_2\)-level toroidal model, which is available at this residue characteristic by 10. Properness extends the center point to a section over the valuation ring, after a fixed local extension if needed. Its pullback semiabelian scheme is the semistable model of the center, by uniqueness of semiabelian extension over a normal trait. Its geometric special fiber is therefore a two-dimensional torus. Denote the specialization in the toroidal base by \(\bar z\).

We show that a disk of fixed radius forces this very same specialization for every target. Choose an affine open of the toroidal scheme containing \(\bar z\), and enlarge the fixed local field so that the center section and the required residue point are defined. The generic image of \(t_0\) lies in this affine open. Finitely many affine coordinate generators pull back to rational functions on the curve that are regular at \(t_0\); shrink its parameter neighborhood also to exclude their denominators and the complement of this affine open. Their values at \(t_0\) are integral. Choose a local parameter \(x\) at \(t_0\) with \(x(t_0)=0\). Continuity gives a real \(r>0\) such that \[|x(t)|<r \quad\Longrightarrow\quad |f_j(t)-f_j(t_0)|<1 \qquad\hbox{for every coordinate generator } f_j.\] The assertion uses the branch of \(x\) through \(t_0\). Each \(f_j\) has a convergent expansion on a fixed smaller disk, so the same inequalities hold in the fixed algebraic closure of the local field, for points in every finite extension. They do not involve a valuation normalization by a varying ramification index. The values \(f_j(t)\) are integral and satisfy the equations of the chosen affine chart, and have the center’s residues. They consequently define an integral section with specialization \(\bar z\). Separatedness identifies its generic point with the given moduli point of \(t\).

If a selected target lay in this disk, the two traits would therefore satisfy 11 with \(\ell=\ell_2\). Both \(\overline\Pi_{t_0,\ell_2}\) and \(\overline\Pi_{t,\ell_2}\) would preserve the same two-space \(W_{\bar z}\) in the marked symplectic four-space. Their sum is a nondegenerate four-space by (22), in direct contradiction to 12. This proves exclusion of the target point from the disk itself.

The center has only finitely many non-potentially-good places, and the fixed data have only finitely many conjugates. Apply the argument at each of them. Simultaneous conjugation preserves (22), so it gives finitely many fixed radii with the asserted properties. All of this argument uses integral models and prime-to-characteristic torsion. No comparison with de Rham horizontal transport at a bad center is needed. ◻

Proof. After a fixed finite theta-level cover, a symmetric ample line bundle and its theta data give an algebraic theta-null map. On that cover use the pullback of \([t_0]\) as the ample divisor; its height at a lift is \(h_{[t_0]}(t)+O(1)\). The theta-null projective height is therefore at most \(Ch\) by the preceding height comparison. Pazuki’s comparison of theta height and stable Faltings height (Pazuki 2012, Corollary 1.3(1)) gives \(h_F(\mathcal A_t)\leq Ch\). The logarithmic error in that comparison is absorbed into this upper bound. The cover has fixed degree, so using it does not impose a bound on the varying degree \(d\).

We spell out the field-degree dependence in the endomorphism estimate. The Proposition of (Masser and Wüstholz 1994, sec. 4), with its induction parameter \(m=n\) and with principal polarization, gives integral endomorphisms spanning the geometric endomorphism algebra with Rosati quadratic sizes at most \[c\max\{2,[L:\mathbb Q],h_F(\mathcal A_t)\}^{\kappa_2},\] where \(c\) and \(\kappa_2\) depend only on the dimension \(n=2\). The degree dependence is exposed in that proposition and in (Masser and Wüstholz 1994, sec. 5, equation (5.3) and Lemma 5.1); it is not an unspecified constant allowed to depend arbitrarily on \([L:\mathbb Q]\). The estimates apply to arbitrary abelian varieties and thus include the split algebra \(\mathop{\mathrm{Mat}}_2(\mathbb Q)\).

Here one can take \([L:K(t)]\) bounded. The continuous Galois action on the rank-four lattice \(\mathcal O_t\) has finite image. To see finiteness, choose an integral basis; each basis endomorphism is defined over a finite extension, so an open Galois subgroup fixes the entire basis. Finite subgroups of \(\mathop{\mathrm{GL}}_4(\mathbb Z)\) have bounded order. One elementary bound follows by reduction modulo \(3\): its kernel has no nontrivial torsion. In fact a torsion element in that kernel has a prime-order power; writing it as \(1+3^sA\), with some entry of \(A\) a \(3\)-adic unit, the binomial expansion rules out order prime to \(3\), and also rules out order \(3\) by comparing the valuations of the linear and higher terms. Thus reduction embeds the finite image in \(\mathop{\mathrm{GL}}_4(\mathbf F_3)\). Taking the fixed field of its kernel proves the assertion about \(L\). Since \(K\) is fixed, \([L:\mathbb Q]\leq Cd\), which gives the generator bound in (26).

For completeness, the generator bound controls the full-order discriminant without requiring the chosen generators to be an integral basis. Put \[b(f,g)=\mathop{\mathrm{Tr}}_{H_1}(fg),\qquad q(f,g)=b(f,g^\dagger).\] The form \(q\) is positive definite and integral. Principal polarization makes \(\dagger\) an automorphism of the integral order. Consequently, in an integral basis, the matrices of \(b\) and \(q\) differ by right multiplication by an integral matrix of determinant \(\pm1\). Their determinants have the same absolute value. Select four independent small generators and let \(\Lambda\) be their span over \(\mathbb Z\). Hadamard’s inequality gives \[\det(q|_\Lambda)\leq\prod_{i=1}^4 Q_t(f_i) \leq C(dh)^\kappa, \qquad \det(q|_{\mathcal O_t}) =\frac{\det(q|_\Lambda)}{[\mathcal O_t:\Lambda]^2}.\] On \(\mathop{\mathrm{Mat}}_2(\mathbb Q)\) the rank-four homology representation is the sum of two copies of its standard module. Thus \(\mathop{\mathrm{Tr}}_{H_1}(f)=2\mathop{\mathrm{tr}}_2(f)\), which also equals the trace of the regular representation. This proves the discriminant convention and its bound.

The \(q\)-orthogonal projection onto the symmetric trace-free subspace is \[\pi(f)=\frac{f+f^\dagger}{2} -\frac{\mathop{\mathrm{Tr}}_{H_1}(f)}4\mathop{\mathrm{id}}.\] The two projections commute, and neither increases the \(q\)-norm. Moreover \[ 4\pi(f)=2(f+f^\dagger)-\mathop{\mathrm{Tr}}_{H_1}(f)\mathop{\mathrm{id}}\in\mathcal O_t. \tag{27}\] By 6, the image has dimension two. The images of the chosen rationally spanning generators span it. Select two independent nonzero images and use (27) to obtain \(P_1,P_2\). Their quadratic sizes increase by at most \(16\). All these operations take place over \(L\). ◻

Proof. We first extend the normalized biextension, then construct its vector extension, and finally show that endomorphisms act integrally on the resulting Lie bundle. Every power \(\mathcal G^r\) is regular. A line bundle on \((\mathcal G^r)_U\) has a rational section; the closure of its divisor is a Weil divisor on \(\mathcal G^r\) and is Cartier by regularity. This extends the line bundle.

Every prime divisor of \(\mathcal G^r\) supported over \(T\setminus U\) is the reduced pullback of a unique prime divisor of \(T\). Indeed flatness implies that the image of a codimension-one point has codimension at most one. It cannot be the generic point of \(T\). The fiber above a codimension-one point is geometrically integral, so its generic point is unique; smoothness makes the pullback divisor reduced. It follows that two extensions, with their generic identification, differ by a divisor pulled back from \(T\).

Start with any extension \(\mathscr B_0\) of the Poincaré bundle and write \(p:\mathcal G^2\to T\). Replacing it by \[\mathscr B=\mathscr B_0\otimes p^*((e,e)^*\mathscr B_0)^{-1}\] gives the prescribed trivialization at the simultaneous origin. The generic axis rigidifications extend uniquely: the divisor of each rational rigidification is pulled back from \(T\), and restriction to the origin, where it is already the identity, makes that divisor zero. For the same reason, the two generic biextension isomorphisms extend on \(\mathcal G^3\). For example, the ratio line bundle in \[(m,1)^*\mathscr B\simeq p_{13}^*\mathscr B\otimes p_{23}^*\mathscr B\] has a rational trivialization whose vertical divisor is detected at \((e,e,e)\). It is therefore zero. A rational section of a line bundle on a regular scheme with zero divisor is an everywhere invertible section. Associativity, compatibility of the two laws, and the unit identities hold on the dense open, hence everywhere. The same argument proves uniqueness of a normalized extension. This is the divisorial argument used over a trait in (Edixhoven and Lido 2023, Proposition 6.7.1); the argument just given establishes the stated version over \(T\).

Let \(N\) be the first infinitesimal neighborhood of the identity in the second copy of \(\mathcal G\). Its ideal is square-zero, and \(\mathcal O_N=\mathcal O_T\oplus\omega_{\mathcal G/T}\) as an augmented \(\mathcal O_T\)-module. Over a test scheme \(V\to T\), an element of \(\mathcal E(V)\) is a point \(g\in\mathcal G(V)\) together with a trivialization of \(\mathscr B|_{\{g\}\times N_V}\) extending its given value at the origin. Such trivializations exist locally on \(V\). Two of them differ by an element of \(1+\omega_{\mathcal G/T}\otimes\mathcal O_V\), whose multiplication is addition because the ideal is square-zero. The resulting torsor is represented explicitly as follows. If \(q:\mathcal G\times_TN\to\mathcal G\) is the finite flat projection and \(\pi:\mathcal G\to T\) is the structure map, axis normalization gives an exact sequence of vector bundles \[0\longrightarrow\pi^*\omega_{\mathcal G/T} \longrightarrow q_*(\mathscr B|_{\mathcal G\times_TN}) \longrightarrow\mathcal O_{\mathcal G}\longrightarrow0.\] The inverse image of the section \(1\) is the required affine torsor. Its points are actual sections of this fixed bundle; no quotient by automorphisms of semiabelian extensions is taken. The first biextension law supplies its group law. Its fiber over \(e\) is the vector group \(\omega_{\mathcal G/T}\), with its canonical origin. This proves (28), including smoothness and compatibility with arbitrary base change. Taking Lie algebras gives an exact sequence of vector bundles of ranks \(2,4,2\).

Over the abelian open, this is the universal vector extension of \(\mathcal G_U\), expressed as infinitesimally rigidified line bundles on its dual; see (Mazur and Messing 1974, I, §2.6 and §4) and (Cais 2010, Proposition 2.2). In cohomological notation the classical Lie realization is \[\mathop{\mathrm{Lie}}E(A)=H^1_{\rm dR}(A^\vee).\] The Poincaré de Rham pairing identifies the right side with \(H_1^{\rm dR}(A)\), with the usual Tate line understood over the base field. This is the covariant realization: a map \(f:A\to A\) acts via \((f^\vee)^*\) on \(H^1_{\rm dR}(A^\vee)\) and via \(f_*\) on homology. The Hodge filtration and functoriality in this statement are made explicit in (Cais 2010, Proposition 5.1 and Corollary 5.2). Only this characteristic-zero identification is needed. On a toric special fiber we retain the constructed rank-four Lie bundle; no universal property or integral de Rham duality is asserted there.

Now pull back to a trait. The normal-base homomorphism extension theorem (Faltings and Chai 1990, I, Proposition 2.7), (Lan 2013, Proposition 3.3.1.5) applies directly to the semiabelian scheme \(\mathcal G_R\). It extends both \(f\) and \(f^\dagger=\lambda^{-1}f^\vee\lambda\) to \(\mathcal G_R\); principal polarization is what makes \(f^\dagger\) integral. The generic normalized biextension identity \[ (f,1)^*\mathscr B\simeq(1,f^\dagger)^*\mathscr B \tag{29}\] extends over \(R\) by the same divisor argument. Pulling an infinitesimal rigidification back along \(f^\dagger|_N\) therefore sends the fiber of \(\mathcal E\) over \(g\) to its fiber over \(f(g)\). It defines an \(R\)-homomorphism \(\mathcal E\to\mathcal E\). On the generic fiber it is the covariant map just described. Its differential preserves \(\mathop{\mathrm{Lie}}(\mathcal E_R)\), proving the integrality assertion. After a finite trait extension the base-changed group scheme is again semiabelian and the new trait is normal. Apply the same homomorphism extension theorem and the same normalized biextension argument there. Compatibility of the constructed torsor with base change then proves the final assertion. ◻

Proof. Choose a rational frame \(e_1,\ldots,e_4\) for \(H_{\rm dR}\) and its dual frame, and include the rational coefficient maps from \(\mathop{\mathrm{End}}(H_{\rm dR})\) to the chosen coordinates on \(\mathcal P_{\rm dR}\). For each \(i\), these form a finite list of rational sections of vector bundles on \(\mathscr X_i\). Extend each of those bundles to a coherent sheaf \(\mathscr F\) on \(\overline{\mathscr X}_i\). Such an extension exists by clearing powers of \(\ell_i\) in finite sets of local generators and relations on a finite affine cover. Removing its torsion preserves its restriction to \(\mathscr X_i\). No local freeness of \(\mathscr F\) over \(\ell_i\) is needed.

Let \(r\) be one of the rational sections. The local functions \(b\) for which \(br\) belongs to \(\mathscr F\) form a coherent nonzero ideal sheaf \(\mathscr I_r\). Its stalk at the characteristic-zero center \(t_0\) is the unit ideal, because the chosen frames and coefficient maps are regular there. For a sufficiently high power of the ample line bundle \(\mathscr L_i=\mathcal O_{\mathbf P^{M_i}}(1)|_{\overline{\mathscr X}_i}\), \(\mathscr I_r\otimes\mathscr L_i^{\otimes m}\) is generated by global sections. We may therefore choose a section nonzero at \(t_0\). Viewed as a section of \(\mathscr L_i^{\otimes m}\), it gives a fixed \(s\in H^0(\overline{\mathscr X}_i,\mathscr L_i^{\otimes m})\) such that \(sr\) is a regular section of \(\mathscr F\otimes\mathscr L_i^{\otimes m}\). The zero divisor of \(s\) is an effective Cartier divisor, since \(\overline{\mathscr X}_i\) is integral. Make these choices for the finite list of sections on both models and omit their finitely many generic-fiber zeros on the curve; the center remains in the resulting open. The same choices can be made for each of the fixed conjugate data.

At places of residue characteristic different from \(\ell_i\), equip the vector bundles with the lattice norms from \(\mathscr X_i\). Use the model metric of \(\mathscr L_i\) at all finite places, including those above \(\ell_i\), and continuous metrics at infinity. For a point off the zero divisor put \(\lambda_{s,v}(t)=-\log\|s(t)\|_v\). Regularity of \(sr\) bounds the positive logarithmic norm of \(r(t)\) by \(\lambda_{s,v}(t)\) at the available places. Every finite-place \(\lambda_{s,v}(t)\) is nonnegative, because \(s\) is an integral section on the proper model \(\overline{\mathscr X}_i\). Moreover its archimedean terms are uniformly bounded below, since \(\|s\|\) is bounded above on the compact complex curve. Consequently \[\sum_{\substack{v\nmid\infty\\v\nmid\ell_i}} n_v\lambda_{s,v}(t) \leq\sum_{v\nmid\infty}n_v\lambda_{s,v}(t) \leq h_{\mathscr L_i^{\otimes m}}(t)+C \leq Ch.\] The height in this display uses the generic-fiber line bundle and the fixed metrics just specified. In particular no vector-bundle lattice over \(\ell_i\) has been used to bound the partial sum. Adding the finitely many denominator bounds gives the required global frame cost.

If \(f\) preserves \(\mathscr H_t\), then its operator norm for that lattice is at most one. Its matrix coefficients are evaluations \(e_i^*(fe_j)\) and are bounded by the product of the norms of \(e_i^*\) and \(e_j\). The fixed coordinate maps on \(\mathcal P\) contribute their own denominator costs. Adding these finitely many terms proves (30). Applying the argument for both models and adding their bounds covers all finite places.

For the refinement, outside finitely many rational primes the center extends to the abelian interior and the rational frames, their duals, and the coordinate maps are integral and invertible throughout its formal neighborhood. Their costs are zero there. At one of the remaining potentially-good center places, first make a fixed finite local extension giving good reduction. The center then lies in the interior of a prime-to-characteristic integral model. On a sufficiently small disk in its actual branch, the family remains in that interior with the same specialization. Since the rational frames are regular and invertible at the characteristic-zero center, their matrices and inverse matrices are bounded on a smaller closed disk. This follows by evaluating their finitely many regular coordinate expressions; the disk radius and the bounds depend only on the center and the expressions. It remains valid for points over arbitrarily ramified extensions. Each such rational prime contributes a fixed constant, because its total place weight is at most one. There are only finitely many such primes and conjugate centers. ◻

Proof. At finite places use 17. At infinity combine (31) with the frame and dual-frame bounds. The logarithm of the Rosati size is at most \(C(1+\log d+\log h)\). The global frame cost is at most \(Ch\), and \(\log h\leq h\); this proves (33). Near the good center branches, the finite-place cost is bounded and the archimedean frames have bounded norms. This gives (34). The construction is invariant under simultaneous conjugation, and all constants may be maximized over the fixed finite collection of conjugate models and frames.

For \(NF_j\) integral, the finite-place calculation acquires at most \(\log^+|N|_v^{-1}\). Its total weighted sum is \(\log N\), and its sum on any subcollection is no larger. The Hodge calculation applies directly to \(F_j\) with its asserted Rosati bound. This proves the bounded-denominator assertion.

Finally let \(\phi:(\mathcal A_t,\lambda_t)\to (\mathcal A_{t'},\lambda_{t'})\) have fixed multiplier \(m\), so \(\phi^\vee\lambda_{t'}\phi=m\lambda_t\) and \(\phi^{-1}=m^{-1}\phi^\dagger\). Its conjugation map on endomorphisms is a Rosati isometry. For \(P_j'\) integral, the transported endomorphism is \[\widehat P_j=\phi^{-1}P_j'\phi =m^{-1}\phi^\dagger P_j'\phi.\] Thus \(m\widehat P_j\) is integral and \(Q_t(\widehat P_j)=Q_{t'}(P_j')\). Apply the result just proved; this avoids estimating the coordinate height of a matrix of \(\phi\). There are uniformly boundedly many such \(\phi\) for fixed \(m\): its kernel is contained in \(\mathcal A_t[m]\), and, for each kernel, the polarized identifications of the quotient with \(\mathcal A_{t'}\) form a torsor under a finite polarized automorphism group. The latter has bounded order, since its faithful action on integral homology is a finite subgroup of \(\mathop{\mathrm{GL}}_4(\mathbb Z)\). The kernel choices and hence the isogeny choices are bounded in number by a constant depending only on \(m\). Their Galois orbits have that bound, so choosing \(\phi\) requires only a bounded field extension of the compositum of the two fields of data. There are only finitely many allowed \(m\). ◻

Proof. Let \(D_0=[t_0]\) on the smooth projective completion of the selected curve. Its degree is positive. Choose local Weil functions \(\lambda_{D_0,v}\) compatible with the fixed integral models away from \(\mathcal S\). Their weighted sum is a height \(h_{D_0}(t)\) up to a bounded constant. With the height convention using \(D_0\), this is \(h+O(1)\); for any other fixed ample convention there is still a constant \(c>0\) with \(h_{D_0}(t)\geq ch-O(1)\). For the latter assertion, choose a positive rational multiple of \(D_0\) whose difference with the other ample divisor has positive degree; the height of that difference is bounded below. This justifies the comparison without introducing an error that grows linearly with \(h\) on the wrong side.

On the disk through \(t_0\), the ideal of \(D_0\) is generated by \(x\). Hence \[\lambda_{D_0,v}(t)=\log\frac1{|x(t)|_v}+O_v(1).\] At the spread-out primes the error is zero in the center tube, and the local Weil function is zero outside that tube. These assertions can also be read from projective distance to the fixed point: on the smooth chart its ideal is generated by the étale coordinate, whereas outside its residue tube one of the ideal generators is a unit. This uses the ideal of \(t_0\), and never assigns proximity to another zero of \(x\) to \(D_0\).

At each of the finitely many remaining local embeddings of the fixed data, \(\lambda_{D_0,v}\) is bounded above outside any fixed small center disk. Within that disk it differs from \(-\log|x|_v\) by a fixed bound. These bounds hold on algebraic points over every finite local extension: they follow from fixed analytic units and fixed strict inequalities defining the disks. At the bad center places the exclusion in 13 therefore bounds the entire possible contribution by a constant. Shrinking the other fixed disks also loses only a constant in the weighted sum. Before imposing (35), the retained actual-branch contributions thus have sum at least \(ch-O(1)\).

It remains to bound the cost of that cutoff. Take a field of degree \(D\leq Cd^C\) containing \(a\) and the labeled target data, before taking any normal closure. We have \(h(a)\leq Ch\) by 14. Let \(\mathcal T(a)\) be the rational primes admitting an embedding with \(|a|_v<1\). If \(v\mid p\) is one such place, then, with integral valuation \(\mathop{\mathrm{ord}}_v\) and residue degree \(f_v\), \[w_v\log\frac1{|a|_v} =\frac{f_v\mathop{\mathrm{ord}}_v(a)}{D}\log p\geq\frac{\log p}{D}.\] Summing positive valuations and using the product formula gives \[ \sum_{p\in\mathcal T(a)}\log p\leq D h(a), \qquad N:=|\mathcal T(a)|\leq CDh. \tag{37}\] This argument sums the positive valuations themselves; cancellations in the rational norm of \(a\) are irrelevant.

The total weight over each rational prime is at most one. The discarded mass there is at most \(2\beta_p\). Order the \(N\) primes increasingly as \(p_1,\ldots,p_N\). The function \(\log u/(u-1)\) decreases for \(u>1\), and \(p_i\geq i+1\). Consequently \[ \sum_{p\in\mathcal T(a)}\beta_p \leq\sum_{i=1}^N\frac{\log(i+1)}i \leq C(1+\log(N+1))^2 \leq C(1+\log(dh))^2. \tag{38}\] This proves (36). Both the support bound and the weight identity survive field extension, so no ramification degree or degree of a normal closure enters the estimate. ◻

Proof. Outside \(\mathcal S\) the abelian family and the common specialization are part of the integral chart already chosen. At a retained prime in \(\mathcal S\), the center is potentially good. Choose an auxiliary fine level invertible in that residue characteristic and a finite local extension over which the center has good reduction. The center then lies in the interior of the smooth integral moduli space. A sufficiently small fixed analytic neighborhood maps into the tube of its special point. Every algebraic point in that neighborhood, even over a further ramified extension, therefore gives a good-reduction lift with that same special fiber. The disks defining \(V_0\) have been chosen inside these neighborhoods.

For clarity, the comparison used here is a comparison of rational cohomology spaces. For a complete mixed-characteristic DVR with perfect residue field \(k\), fraction field \(L\), and a smooth proper lift \(\mathscr X\) of \(X_0\), it has the form \[ H^1_{\mathrm{crys}}(X_0/W(k))\otimes_{W(k)}L \simeq H^1_{\mathrm{dR}}(\mathscr X_L/L). \tag{40}\] The rational comparison of Berthelot–Ogus (Berthelot and Ogus 1983) is functorial and allows every ramification index; this form is stated in (Berthelot 1981--1982, sec. 1.2) and explicitly distinguished from integral comparison in (Abhinandan and Youcis 2025, Introduction, equations (2)–(3)). It does not assert preservation of integral lattices. We use its relative realization to compare the two lifts, as follows.

Embed the common proper special fiber \(\overline A\) as the closed fiber over the base special point in the smooth formal total family. Its rigid cohomology is computed by the de Rham complex on its tube; properness eliminates a compactification boundary for this special fiber. Independence of the smooth embedding, functoriality and extension of coefficients identify restriction to each smooth proper lift with (40); these are the tube constructions of (Berthelot 1981--1982, secs. 1.4–1.5 and 2.5(c)). One may work on a smaller base disk, on which the relative Gauss–Manin connection is ordinary. An absolute de Rham hypercohomology class on the total tube maps to a horizontal relative class: the connecting morphism obtained by filtering the absolute de Rham complex by base differential forms is precisely the Gauss–Manin connection, and its composition with the absolute-to-relative map is zero. The evaluation isomorphism with the cohomology of a proper lifted fiber then shows that a basis of the common rigid cohomology gives a basis of horizontal relative classes on this disk. The change between two evaluations is therefore the Taylor isomorphism; the convergent Taylor formalism is described in (Berthelot 1981--1982, sec. 4.1). All these operations commute with extension of the coefficient field, so the same diagram holds for ramified evaluation fields. The unramified version of this diagram is displayed in (Lawrence and Venkatesh 2020, sec. 3.3, diagram (3.9)).

After pullback to the parameter disk, the change of comparisons is therefore horizontal and has identity value at \(a=0\). Uniqueness for the ordinary differential equation identifies it with the Taylor solution throughout the retained convergence disk. Dualizing gives the statement for covariant homology, and taking endomorphisms gives the stated action on \(\mathcal P\). Functoriality identifies an endomorphism of either lift with its specialization, and likewise identifies an isogeny with its specialized map. The polarization and trace pairing are preserved in these identifications. Formula (39) follows.

This argument uses a common good-reduction family. The disks at a center which is not potentially good have already been excluded by 13. Also, horizontal transport here identifies cohomology and its tensors; no assertion about horizontal constancy of the Hodge filtration is needed. ◻

Proof. A polarized isogeny \(\phi\) of multiplier \(m_*\) satisfies \(\phi^*\lambda'=m_*\lambda\). Since the fibers have dimension two, \(\deg\phi=m_*^2\), and its kernel lies in \(A[m_*]\). Away from primes dividing \(m_*\), kernels are among a fixed finite set of subgroup schemes of this finite étale torsion scheme. For each such kernel, the quotient is fixed. Polarized identifications of that quotient with the second center form either an empty set or a torsor under its finite polarized automorphism group. This proves finiteness in characteristic zero.

Here is a finite-type construction that also includes all special-fiber possibilities. After a fixed extension and localization, choose relatively ample symmetric line bundles \(L_0,L_0'\) inducing \(N\lambda_0,N\lambda_0'\) for the same fixed integer \(N>0\). On the graph of any isogeny of multiplier \(m_*\), the product bundle restricts to \(L_0\otimes\phi^*L_0'\), numerically equivalent to \(L_0^{\otimes(1+m_*)}\). Riemann–Roch on the abelian surface gives the fixed Hilbert polynomial \[\chi\bigl(A_{t_0},(L_0\otimes\phi^*L_0')^{\otimes n}\bigr) =N^2(1+m_*)^2 n^2.\] All graphs therefore lie in one finite-type relative Hilbert scheme of the product of the two fixed abelian schemes. The locus on which the first projection of the graph is an isomorphism is open. On this graph locus, preservation of the origin is closed and makes the resulting morphism a homomorphism by rigidity for abelian schemes. The condition \(\phi^\vee\lambda_0'\phi=m_*\lambda_0\) is a closed equality of homomorphisms to the dual abelian scheme. This constructs a separated finite-type scheme of the desired polarized homomorphisms; the polarization identity forces each of them to be an isogeny. The construction uses the same Hilbert polynomial in every residue characteristic, so it includes any additional special-fiber polarized automorphisms as well.

After a fixed extension making its finitely many geometric generic points rational, finite presentation spreads each of them to a section over an open arithmetic base. A common localization works for all of them. Each section has closed image by separatedness. The complement of the finite union of these images is consequently open, hence constructible, and has empty generic fiber. Its image in the one-dimensional arithmetic base is constructible and misses the generic point, hence consists of finitely many closed points. Removing their residue characteristics proves the assertion. One can obtain the same conclusion by spreading out the finite kernel list and the polarized-identification schemes. Both constructions involve fixed centers and fixed multipliers, so the removed primes are independent of \(t\) and of any target isogeny subsequently chosen. ◻

Proof. At a retained finite place, (39) identifies the left-hand entries with traces of products of actual specialized endomorphisms. For an integral endomorphism \(f\) of an abelian variety, the degree polynomial \(\deg(n-f)\) is integral and equals the characteristic polynomial on every prime-to-characteristic rational Tate module; see (Milne 1986, Proposition 12.9 and the preceding paragraph). We explain the comparison with rational crystalline homology, since the product \(\overline B_i\overline P_j\) need not lift to either characteristic-zero fiber. The cup-product map \(\bigwedge^r H^1_{\mathrm{crys}}(\overline A)\to H^r_{\mathrm{crys}}(\overline A)\) is an isomorphism. Indeed, the rational comparison with the center’s good lift respects cup product: the restrictions and base-change maps of the de Rham complexes in the tube construction are multiplicative. It therefore identifies the displayed map, after a faithful scalar extension, with the de Rham cup-product map; that map is an isomorphism by comparison with the cohomology of a complex torus. Being defined by cup product, the crystalline isomorphism is natural for every endomorphism of \(\overline A\), including those that do not lift.

For an actual integral endomorphism \(u\) of \(\overline A\), the map \([n]-u\) is an isogeny for every sufficiently large integer \(n\). Its pullback on first cohomology is \(n-u^*\), by the Künneth description of first cohomology and the group law. On top cohomology its action is multiplication by \(\deg([n]-u)\). One may verify this last assertion in rigid cohomology, using the trace fundamental class and compatibility of cycle classes with proper pushforward (Petrequin 2003, sec. 2.1 and Proposition 6.4), and then use smooth-proper crystalline–rigid comparison (Berthelot 1981--1982, sec. 1.5, Proposition 2). The degree statement includes inseparable isogenies: it is the degree of a finite flat map, not just the number of geometric kernel points; see also (Petrequin 2003, Lemma 6.5). Taking the top exterior power therefore gives \(\det(n-u^*\mid H^1_{\mathrm{crys}})=\deg([n]-u)\) for infinitely many \(n\), hence as a polynomial identity. Dualization preserves the characteristic polynomial. Combining this with the Tate-module identity proves that all the homological traces used here agree. Thus these traces are rational and are integers when both factors are integral; fixed rational denominators introduce only fixed denominators in the trace.

All specialized operators here are symmetric for the common specialized polarization. Rosati positivity, valid in every characteristic (Milne 1986, Theorem 17.3), and Cauchy–Schwarz give \[ \left|\mathop{\mathrm{Tr}}(\overline B_i\overline P_j)\right|^2 \leq\mathop{\mathrm{Tr}}(\overline B_i^2)\mathop{\mathrm{Tr}}(\overline P_j^2) =\mathop{\mathrm{Tr}}(B_i^2)\mathop{\mathrm{Tr}}(P_j^2)\leq C(dh)^C. \tag{42}\] The absolute value in this inequality is the ordinary real absolute value of the rational trace. It is unrelated to the residue characteristic norm. At infinity, analytic Betti transport on the fixed simply connected center disk identifies integral homology lattices. The analogous cross traces are again rational with fixed denominators. The Hodge metrics on this fixed disk, in a flat trivialization, are uniformly comparable. Cauchy–Schwarz for the corresponding operator norms and the target Rosati bound give the same polynomial size bound. These observations establish [loc:single] and its coefficient-height estimate at all of its assigned places.

Now let \(p\notin\mathcal S\). Take a normal closure of a field containing the labeled data, embed it at the assigned place, and choose a lift of residue \(p\)-power Frobenius in its decomposition group. The lift preserves this valuation. Apply it simultaneously to the center, parameter, target, frames, polarization, levels and both endomorphism labels; denote the result by primes. This gives \(|a'|_v=|a|_v\). Absolute heights are invariant under conjugation, so \[h(R)=h(a'/a)\leq h(a')+h(a)=2h(a)\leq Ch.\] The primed special fiber is the \(p\)-power twist of the unprimed one. Functoriality of relative Frobenius gives, for the same isogeny \(F\), \[\overline B_i'=F\overline B_iF^{-1},\qquad \overline P_j'=F\overline P_jF^{-1}.\] These are conjugations in rational endomorphism algebras; no inverse of Frobenius on an integral crystalline lattice is asserted. Taking the trace of the product removes \(F\). Applying 20 on each of the two actual common reductions gives exactly [loc:paired].

At these primes all fixed levels are invertible. Integral smooth proper specialization identifies the \(\ell_1\)-adic homology lattices of each pair of lifts with the lattice of its common reduction, compatibly with the specified level marking. For either plane \(H\) this transports the intersection of \(H\otimes\mathbb Q_{\ell_1}\) with the ambient integral operator lattice to the corresponding intersection after specialization. Thus the separately saturated lattices, and their marked reductions, retain the incidence in 10. This statement concerns the rational planes and their intersections with lattices. It does not require the chosen small vectors \(B_i,P_j\) themselves to be bases after reduction modulo \(\ell_1\).

If the coarse target pair belongs to the closed exceptional list, choose a polarized isogeny \(\phi_t:A_t\to A_{t'}\) of one of its fixed multipliers \(m_*\). The list is fixed before the auxiliary levels; only its pullback to those levels is being tested here. After a local extension, \(\phi_t\) extends between the good-reduction abelian schemes. Its reduction is an isogeny of multiplier \(m_*\) between the same two special fibers as the centers. By 21, it equals the reduction of one of the fixed center isogenies \(\phi_0\). Conjugation by a polarized isogeny preserves trace and Rosati symmetry, hence induces the isometries \(I_t,I_0\). Applying the functorial comparisons to their identical reductions proves (41). Substitution into [loc:paired] gives [loc:exception].

The identity \(\phi_t^{-1}=m_*^{-1}\phi_t^\dagger\) shows that \(m_*\widehat P_j=m_*\phi_t^{-1}P_j'\phi_t\) is integral. Since conjugation preserves the Rosati form, the norms of \(\widehat P_j\) are bounded by the same polynomial as those of \(P_j'\). Apply 18 to these actual rational endomorphisms of \(A_t\) with that fixed denominator. This supplies all their required coordinate bounds directly; no height estimate for the entries of a chosen isogeny matrix is needed. Finally \(p\), \(m_*\) and \(\ell_1\) are pairwise coprime in the required positions, so the Tate-module integrality and invertibility in (iv) follow from the isogeny degrees \(p^2,m_*^2\). ◻

Proof. Counting the groups. Let \(\tau\) comprise the fixed data, \(t\) and the labeled endomorphism coordinates, and choose a field of definition with absolute degree \(D\leq Cd^C\), as supplied by 15. There are at most \(D\) conjugate tuples. At infinity and at the finitely many primes in \(\mathcal S\), group by the embedded tuple, the fixed branch and its resulting rational matrix \(\mathbf r\). Each embedding gives one such matrix, hence this uses at most \(CD\) groups. This counts actual embeddings at fixed places; counting all rational matrices of the allowed size would give an unnecessary dependence on \(h\).

For the Frobenius equations, group by the ordered pair \((\tau,\tau')\). There are at most \(D^2\) values. The normal closure in which Frobenius was chosen can have much larger degree, but only these ordered tuple values enter an equation. The field generated by a pair has degree at most \(D^2\). The primed fixed family belongs to the finite list of conjugates of the fixed family; its germs are independent of the varying target.

Split these groups according to whether the coarse pair is on the closed exceptional list. On it, refine by a choice of its fixed multiplier, an isogeny \(\phi_t\), and its specializing center isogeny \(\phi_0\). The multiplier and center choices range over finite fixed sets. For a fixed multiplier, a target isogeny is specified by a subgroup of bounded torsion and a polarized identification of the quotient with the second target. The number of subgroups is bounded in dimension two and \(m_*\). The order of a polarized automorphism group is uniformly bounded in fixed dimension: it is finite by positivity, and acts faithfully on an integral homology lattice of fixed rank, whose finite subgroups have bounded order. Thus only a fixed bounded relative extension is needed to choose \(\phi_t\), and the refinement still has at most \(CD^2\) groups. The transported tuple also has polynomial degree. Enlarging fields and lifting the embedding weights in making this partition preserves their sum.

Selecting mass. By 19, the total mass is at least \(ch-C(1+\log(dh))^2\). Either \(h\leq Cd^C\), or this is at least \(ch/2\), after increasing the fixed constants. Indeed \((\log h)^2\leq\varepsilon h+C_\varepsilon\) and \((1+\log d)^2\leq Cd^2\) absorb the error. In the latter case one of the at most \(Cd^C\) groups has mass at least \(h/(Cd^C)\). Fix that group and call its embeddings \(V\). Its algebraic tuple and constants give common equations at all of them.

Coordinate heights. Use the coordinate tuple \(U\) specified above for the chosen system. By 18, the endomorphism coordinates have global affine height \(C(h+\log d)\) and weighted positive local logs at most \(C(1+\log h+\log d)\) at the chosen nearby embeddings. This also applies to \(\widehat{\mathbf P}\) by 22. Base-coordinate heights are \(O(h)\), and their positive local logs are bounded on the fixed center disks, with zero cost away from the fixed primes. We have already proved \(h(R)\leq Ch\), while \(\log^+|R|_v=0\) on its group. These facts give (44) and the first bound in (45).

Fixed germs and their coefficients. The matrices \(Y,Y'\) solve fixed ordinary algebraic differential systems in the selected regular frames. Dual or inverse matrices, when needed to realize an operator system, satisfy fixed differential systems as well. Each \(f_\nu\) is an algebraic regular germ on the fixed chart. There are only finitely many conjugate fixed charts and centers. Thus the number of systems, their ranks and the degrees of polynomial combinations are fixed. In particular, “ordinary” here refers to the absence of a pole of the differential equation at its center; it imposes no ordinarity condition on an abelian special fiber.

For completeness, consider a local system \(Z'=A(x)Z\) at a spread-out prime, with \(A(x)=\sum_{i\geq0}A_i x^i\) integral and \(Z(0)\) integral. Writing \(Z(x)=\sum_{n\geq0}Z_nx^n\) gives \[(n+1)Z_{n+1}=\sum_{i=0}^n A_iZ_{n-i}.\] Induction proves \(n!Z_n\) integral, since \(n!/(n-i)!\) is an integer. In the row convention for back-transport the same argument applies with multiplication on the other side. Therefore \[ \|Z_n\|_v\leq |n!|_v^{-1} \leq \exp(n\beta_p). \tag{48}\] The last estimate follows from \(v_p(n!)=\sum_{r\geq1}\lfloor n/p^r\rfloor\leq n/(p-1)\). Regular algebraic coordinate germs are integral in the étale chart. A product of a fixed number of these series retains the factorial bound: multiplying its coefficient of degree \(n\) by \(n!\) expresses it as a sum with the integral factors \(n!/(n_1!\cdots n_k!)\).

At each of the finitely many other places, ordinary-point convergence gives fixed exponential coefficient bounds. At the archimedean embeddings this is Cauchy’s estimate on a fixed smaller disk. At a fixed finite place it follows from the recurrence after rescaling the parameter to make the analytic connection bounded. Summing all local coefficient bounds yields, for each fixed germ, \[h_{\mathrm{aff}}(Z_n)\leq C(1+n\log(n+1)).\] Here the contribution from all factorial denominators is at most \(\log(n!)\), and the exceptional-place contribution is \(O(n+1)\). For the vector of coefficients up to index \(j\) the same reasoning gives a polynomial bound in \(1+j\). Fixed products and sums preserve such a bound.

In \(Y'(aR)\) the coefficient of \(a^j\) is \(Y_j'R^j\). Thus the degree in the polynomial variables grows at most linearly in \(j\), without altering its coefficient heights. The same holds for \(f_\nu'(aR)\). The other coordinate factors occur with fixed bounded degree. The only varying constants of type [loc:single] are \(\mathbf r\), whose coefficient heights are \(C(1+\log h+\log d)\); the constants in type [loc:exception] are fixed centered isogeny data. This proves the remaining degree bound and (46).

Tails at the target. At \(p\notin\mathcal S\), let \(M_v\geq1\) bound all fixed-degree coordinate and constant factors in the bounded collection of expressions. We may choose it so that \[\log M_v\leq C\log^+\|U\|_v+C_v,\] where \(C_v=0\) outside fixed primes; a larger fixed bound includes constant terms when \(l=0\). The factorial argument gives \(|e_j(U)|_v\leq M_v\exp(j\beta_p)\). The substitution \(aR\) introduces no further factor because \(|R|_v=1\). By (35), \(|a|_v\exp(\beta_p)<1\), and the ultrametric triangle inequality yields \[\left|\sum_{j\geq l}a^j e_j(U)\right|_v \leq M_v\sup_{j\geq l}(|a|_v e^{\beta_p})^j =M_v|a|_v^l e^{l\beta_p}.\] Thus \(b_v=\beta_p+\log M_v\) works, including \(l=0\). At the fixed places choose the evaluation disk with radius at most one half of a fixed convergence radius in the archimedean case, and strictly smaller in the nonarchimedean case. The geometric-series estimate gives the same bound with \(b_v=C_v+C\log^+\|U\|_v\) and, where necessary, an additional \(C\log^+\|\mathbf r\|_v\). The sum of the latter contributions is bounded by the coefficient height of \(\mathbf r\) and the fixed number of places. The positive coordinate logs have already been bounded. Finally (38) bounds the sum of \(\beta_p\) over every prime in this group. Enlarging \(b_v\) to the maximum for the bounded number of expressions proves (47).

The entire construction uses absolute weights. Selecting a subgroup of embeddings only decreases the nonnegative coordinate and tail costs; field extension preserves the previous sums. None of these estimates requires a bound in terms of \(d\) alone on the number of supersingular, or of any other, proximity primes. This completes the numerical verification and leaves precisely the nonidentity condition to 27. ◻

Proof. At a general complex point of \(D\) the normalization and \(D\) are smooth. Take local coordinates \((u,v_2,\ldots,v_k)\) there with \(D=\{u=0\}\). The regular function \(a\) is \(u^e\) times a unit, for some \(e>0\). All affine coordinates of \(W\) pull back to holomorphic functions; in particular \(R\) is finite there. Consequently both \(a\) and \(aR\) tend to zero, and the ordinary-point series defining our expressions converge on a sufficiently small neighborhood. An infinite-order zero along \(D\) makes every coefficient of their convergent expansion in \(u\) zero. Thus it gives an identity on the whole local branch. This is stronger than an identity obtained by setting \(u=0\).

Let \(c_\alpha(x)\) be the algebraic coordinate germs of the first curve chart. Its added equations are \(s_\alpha-c_\alpha(a)=0\). Every polynomial relation defining the chart, and the relation \(x(s)=a\), therefore holds on the analytic branch. Such a relation is an algebraic identity on \(W\), since a nonempty analytic open of an irreducible complex algebraic variety is Zariski dense. This constructs the asserted projection to the actual chart; the same argument applies to \(s'_\alpha-c'_\alpha(aR)\) when present. Relations involving regular functions on these affine charts can equivalently be checked after clearing denominators that are invertible on the chart. The target belongs to these charts by 22, so the algebraic relations also hold at the specified target. Nonconstancy of \(a\) implies nonconstancy of \(s\).

Remove singularities and the proper closed sets where the chosen frames or projections are undefined. Shrink further to avoid the branch loci of any parameter charts used to describe the germs. A nonempty part of our analytic branch lies over the resulting smooth open: normalization is an isomorphism over the smooth locus of \(W\). This open is connected. Solutions of the pulled-back connections continue along its paths, and the identity theorem preserves the equations under this continuation. All their algebraic coordinate functions are single-valued there. This proves (ii), without requiring the transports themselves to be single-valued.

Finally, an independent pair at \(z\) has a two-by-two coordinate minor nonzero at \(z\). That minor is a nonzero regular function on \(W\). Deleting its zero set for each of the finitely many pairs proves (iii). No Hodge condition on these varying coordinates has entered the proof. ◻

Proof. For (i), let \(V'\) be the span of \((g-1)v\) for \(g\in G\). It is invariant, since \[h(g-1)v=(hg-1)v-(h-1)v.\] It is nonzero, as the nontrivial irreducible representation \(V\) has no fixed nonzero vector. Irreducibility gives \(V'=V\). Constancy of the displayed coefficient would make \(b\) perpendicular to \(V'\), a contradiction.

Assertion (ii) is the matrix-span conclusion of Burnside’s theorem; in this representation it also has the following direct verification. Choose an orthonormal basis. The diagonal matrices \(D_\eta=\mathop{\mathrm{diag}}(\eta_1,\ldots,\eta_5)\) with \(\eta_i\in\{1,-1\}\) and \(\prod_i\eta_i=1\) lie in \(G\). Their coordinate characters are distinct, so \[E_{ii}=\frac1{16}\sum_{\prod_k\eta_k=1}\eta_iD_\eta\] lies in their span. The span of a matrix group is an algebra. Signed permutation matrices of determinant one, together with the \(E_{ii}\), therefore give every matrix unit \(E_{ij}\) in this algebra. This proves (ii). The displayed identity is a linear functional of \(gM_0\); nondegeneracy of the tensor–matrix pairing proves the tensor conclusion. Polynomial identities on a Zariski-dense subgroup extend to \(G\). ◻

Proof. Choose any place in this group, of residue characteristic \(p\). 22 gives a common unprimed polarized reduction \(A_0\) and its primed reduction \(A'_0\), a relative Frobenius \(F:A_0\to A'_0\) of multiplier \(p\), and a polarized isogeny \(\phi:A_0\to A'_0\) of multiplier \(m_*\) which is simultaneously the specialization of the center and target isogenies. In particular, \(p\nmid m_*\) and \(p\ne \ell_1\). Set \[ \psi=\phi^{-1}F\in\mathop{\mathrm{End}}(A_0)\otimes\mathbb Q, \qquad \nu(\psi)=\frac{p}{m_*}, \qquad g=\mathop{\mathrm{Ad}}(\psi)\big|_{\mathcal P_{\mathbb Q_{\ell_1}}}. \tag{50}\] Functoriality of specialization preserves (49). Moreover \(F\) conjugates each unprimed labeled endomorphism to its primed specialization, for both the center and target labels using this same \(F\). Composing with \(\phi^{-1}\) therefore shows that \(g\) acts by \(\epsilon\) on each of the two specialized rational endomorphism planes.

Let \(H_0,H_t\) be these planes in \(\mathop{\mathrm{End}}(A_0)\otimes\mathbb Q\) and put \(H=H_0+H_t\). On the common integral Tate module, write \[\begin{gathered} \Lambda=\{D\in\mathop{\mathrm{End}}_{\mathbb Z_{\ell_1}}(T_{\ell_1}A_0): D^\dagger=D,\ \mathop{\mathrm{Tr}}(D)=0\},\\ L_0=(H_0\otimes\mathbb Q_{\ell_1})\cap\Lambda, \qquad L_t=(H_t\otimes\mathbb Q_{\ell_1})\cap\Lambda. \end{gathered}\] The two plane lattices here are individually saturated in \(\Lambda\). Their reductions inject into \(\Lambda/\ell_1\Lambda\) and, by [inc:markings,loc:equations], are independent planes whose sum is a degenerate four-dimensional subspace. In particular the two \(\mathbb Q_{\ell_1}\)-planes have four-dimensional sum, and \(\dim_\mathbb QH=4\). These statements use the saturated planes, regardless of whether the chosen small pairs \(\mathbf B,\mathbf P\) are integral bases at \(\ell_1\).

Every element of \(H\) is an actual rational endomorphism of \(A_0\), self-adjoint for its Rosati involution. The endomorphism trace is rational and agrees with its Tate-module trace (Milne 1986, Proposition 12.9). Rosati positivity in arbitrary characteristic (Milne 1986, Theorem 17.3) gives \[\mathop{\mathrm{Tr}}(D^2)=\mathop{\mathrm{Tr}}(DD^\dagger)>0\qquad(0\ne D\in H).\] Thus the Gram matrix of a rational basis of \(H\) is positive definite over \(\mathbb R\), with nonzero rational determinant. It remains nonsingular over \(\mathbb Q_{\ell_1}\). Write \(H_{\ell_1}=H\otimes\mathbb Q_{\ell_1}\). This rational nondegeneracy is compatible with the prescribed degeneracy of the reduced four-space: its Gram determinant may be divisible by \(\ell_1\).

Conjugation by a symplectic similitude acts through \(\mathop{\mathrm{SO}}\) on \(\mathcal P\), with scalar kernel on the four-dimensional first homology space, by 6. Since \(H_{\ell_1}\) is nondegenerate, \(g\) preserves its one-dimensional orthogonal complement. On \(H_{\ell_1}\) its determinant is \(\epsilon^4=1\); hence \(\det g=1\) makes its action on the orthogonal complement \(+1\).

If \(\epsilon=1\), this proves \(g=1\) on all of \(\mathcal P_{\mathbb Q_{\ell_1}}\). The scalar-kernel assertion makes the action of \(\psi\) on \(T_{\ell_1}A_0\otimes\mathbb Q_{\ell_1}\) a scalar \(c\). This scalar is rational: the actual rational quasi-endomorphism \(\psi\) has rational trace and \(4c=\mathop{\mathrm{Tr}}(\psi)\), by (Milne 1986, Proposition 12.9). Its symplectic multiplier is consequently \(c^2\). Equation (50) would give \(c^2=p/m_*\), impossible because the \(p\)-adic valuation of the right side is one.

Suppose instead that \(\epsilon=-1\). Then \(g^2=1\) and its minus eigenspace is exactly \(H_{\ell_1}\). Both \(F\) and \(\phi\) have degrees prime to \(\ell_1\): their degrees are \(p^2\) and \(m_*^2\), respectively. They induce integral isomorphisms on the \(\ell_1\)-adic Tate modules. Consequently \(\psi\), its inverse, and their conjugation action preserve \(\Lambda\). The trace pairing on \(\Lambda\) is unimodular, as in 6 and the choice of \(\ell_1\) in 10. Since \(\ell_1\) is odd, the integral idempotents \[e_+=\frac{1+g}{2},\qquad e_-=\frac{1-g}{2}\] give an orthogonal direct sum \[ \Lambda=\Lambda_+\mathbin{\perp}\Lambda_-, \qquad \Lambda_\pm=e_\pm\Lambda, \qquad \mathop{\mathrm{rank}}\Lambda_-=4. \tag{51}\] Orthogonality follows from \(\langle v_+,v_-\rangle=\langle gv_+,gv_-\rangle =-\langle v_+,v_-\rangle\). In bases of these two direct summands the Gram determinant of \(\Lambda\) is the product of their integral Gram determinants. It is a unit, so both determinants are units. In particular \(\Lambda_-/\ell_1\Lambda_-\) is nondegenerate.

Because \(g=-1\) on each rational plane, \(L_0\) and \(L_t\) lie in \(\Lambda_-\). Their independent reductions have total dimension four. The direct-summand property in (51) identifies \(\Lambda_-/\ell_1\Lambda_-\) with its image in \(\Lambda/\ell_1\Lambda\). Those two reductions therefore exhaust this image. Their sum must be nondegenerate, contrary to its imposed degenerate incidence. This excludes the second sign as well. ◻

Proof. Suppose all equations were identities on one such branch. Apply 24, and fix a general point \(w\in W^\circ\) at which the indicated operator pairs are independent. All vectors and coordinates at \(w\) are now fixed while we vary a continuation loop. The continued first transport \(M:\mathcal P_{s(w)}\to\mathcal P_{t_0}\) ranges through the orbit of an invertible initial transport under monodromy whose connected Zariski closure is the full special orthogonal group.

Proof. Put \(d=\max\{2,[K(t):K]\}\) and \(h=h(t)\ge2\). All fixed omitted points may be absorbed by increasing \(C\). For the remaining targets, 23 either already gives a polynomial bound for \(h\) in \(d\), or supplies a target \(z=(a,U)\), a weighted group \(V\), and common equations to which we apply 4.

We match its hypotheses explicitly. The target has \(a\ne0\) and \(|a|_v<1\) on \(V\). The global height, proximity mass, and positive coordinate bounds in (44)–(45) give condition (i). The coefficient bounds and common tail estimate in (46)–(47), together with the degree bound in (45), give (ii). The expressions are polynomial combinations of fixed bounded degree of ordinary horizontal systems and regular algebraic germs, evaluated at \(a\) and, in the paired case, at \(aR\); this is (iii). Here \(|R|_v=1\) on the selected group. The base-coordinate equations are included in this system.

These estimates use the same \(d=\max\{2,[K(t):K]\}\) for the fixed field \(K\). The selected labeled data and isogenies have a field of definition of degree polynomial in \(d\). The normalized weighted sums, including embedding multiplicities, are preserved by every further field extension. The three equation types involve only finitely many fixed choices of ambient dimension, germs, conjugate centers, and center isogenies. We may therefore choose the constants in 4 uniformly across them.

Finally, 27 gives condition (iv) for every irreducible algebraic variety through \(z\) and every normalized boundary branch, including those arising in the dimension descent. Varieties with empty boundary are already handled in the proof of 4. That theorem gives \(h\le Cd^\kappa\); taking the largest constants over the fixed alternatives proves (53). ◻

Proof. If this locus is finite, the assertion follows by taking the minimum of finitely many positive ratios; the empty case is vacuous. Otherwise, make the fixed choices in [geo:setup,inc:markings]. All the resulting covers, auxiliary primes, and the center are fixed. Enlarge the fixed number field \(K\) to contain their fields of definition. After removing a finite set \(F\subset C(\overline{\mathbb Q})\), their composite gives a finite cover \(q:S\longrightarrow C\setminus F\) of fixed degree \(N\). Further fixed deletions in the preceding sections are included in \(F\) by removing their images. We can restrict to the locus over which the cover has constant degree, and include its remaining exceptional points in \(F\) as well.

For \(s\in\Sigma_{E^2}(C)\setminus F\), choose the lift \(t\in S(\overline{\mathbb Q})\) with the two markings used in 28. Write \(r_s=[K(s):K]\) and \(d_t=\max\{2,[K(t):K]\}\). A closed point in a fiber of a finite map of degree \(N\) has residue degree at most \(N\). Thus, for every such chosen lift, \[ [K(t):K]\leq N r_s,\qquad d_t\leq 2N r_s. \tag{56}\] This bound uses the degree of the fixed cover even though the chosen point in its fiber depends on \(s\).

The height bound in 28 and the full endomorphism estimate in 15 give fixed positive constants \(C_1,C_2,u,v\) such that \[ h(t)\leq C_1 d_t^u,\qquad \Delta_s=\Delta_t\leq C_2\bigl(d_t h(t)\bigr)^v. \tag{57}\] Here forgetting a level structure leaves the geometric principally polarized abelian variety unchanged, so it leaves its endomorphism ring unchanged. The second inequality is the Masser–Wüstholz estimate established in 15; in particular its constants have polynomial dependence on the varying field degree. It estimates the full geometric order after an extension of bounded relative degree, which has already been accounted for there.

For clarity, the trace conventions agree exactly in this case. For \(D_s\simeq\mathop{\mathrm{Mat}}_2(\mathbb Q)\), its action on \(H_1(A_s,\mathbb Q)\) is a sum of two standard two-dimensional modules. Left multiplication on \(D_s\) also acts on its two columns as two standard modules. Consequently \[ \mathop{\mathrm{Tr}}_{H_1}(x)=2\mathop{\mathrm{tr}}(x)=\mathop{\mathrm{Tr}}_{D_s/\mathbb Q}(x). \tag{58}\] Thus (54) is precisely the regular-trace discriminant used in 15 and in (Daw and Orr 2022, sec. 3.1 and 5.4). Bounding a smaller lattice generated by the two symmetric endomorphisms alone would not be this assertion.

Combining (56) and (57), and increasing constants, yields \[\Delta_s\leq C_3 r_s^{v(u+1)}.\] Each point of the finite set \(F\cap\Sigma_{E^2}(C)\) has a finite positive discriminant. Increasing \(C_3\) includes these points too. Taking \(\delta=1/(v(u+1))\) and \(c=C_3^{-\delta}\) proves the lower bound in (55).

Finally, for an algebraic point of a variety defined over the number field \(K\), its distinct \(K\)-embeddings are precisely its Galois conjugates. Every embedding of its residue field into \(\mathbb C\) extends to an automorphism of \(\mathbb C\) over \(K\). Hence their number is \([K(s):K]\), proving the asserted orbit equality. ◻

Proof. Fix an embedding into \(\mathbb C\), put \(V=H_1(E,\mathbb Q)\), and let \(W=\mathbb Q^2\) with its trivial Hodge structure. The given unpolarized isogeny provides a rational Hodge isomorphism \[ H_1(A_s,\mathbb Q)\simeq V\otimes W. \tag{59}\] Choose an elliptic polarization \(\psi_E\) on \(V\). Since \(E\) is non-CM, \(\mathop{\mathrm{End}}_{\mathrm{Hdg}}(V)=\mathbb Q\). The polarization \(\psi_E\) identifies \(V\) with its appropriately twisted dual, so every Hodge pairing of two copies of \(V\) into \(\mathbb Q(1)\) is a rational multiple of \(\psi_E\). Apply this to the four matrix entries, on the two copies of \(V\), of the alternating form \(\psi_s\) defined by the actual polarization \(\lambda_s\). There is a rational bilinear form \(b\) on \(W\) with \[ \psi_s=\psi_E\otimes b. \tag{60}\] Alternation of \(\psi_s\) and \(\psi_E\) makes \(b\) symmetric. Let \(J\) be the complex structure on \(V_{\mathbb R}\), choosing the common sign convention for which \(\psi_E(v,Jv)>0\) for \(v\ne0\). The positivity of \(\psi_s\) gives \[0<\psi_s(v\otimes w,Jv\otimes w) =\psi_E(v,Jv)b(w,w) \quad (v\ne0,\ w\ne0),\] so \(b\) is positive definite over \(\mathbb R\).

Let \(G\) be the image of \(\mathop{\mathrm{GL}}(V)=\mathop{\mathrm{GSp}}(V,\psi_E)\) acting on \(V\otimes W\) through the first factor. In dimension two, \(g\) multiplies \(\psi_E\) by \(\det(g)\); therefore (60) gives a rational inclusion \[G\ \subset\ \mathop{\mathrm{GSp}}(V\otimes W,\psi_s).\] The elliptic Hodge homomorphism, acting trivially on \(W\), is the Hodge homomorphism of \(A_s\). Its \(G(\mathbb R)\)-conjugacy class is the elliptic upper and lower half-plane datum. Positivity of \(b\) ensures that this class lies in the Siegel datum for \(\psi_s\). Thus it gives a one-dimensional rational Shimura subdatum through the period point of \(A_s\).

Retain the actual lattice \(\Lambda=H_1(A_s,\mathbb Z)\) inside \(V\otimes W\). The principal polarization makes \((\Lambda,\psi_s)\) unimodular; a symplectic basis identifies it with the standard integral Siegel lattice. Intersecting its adelic stabilizer with \(G(\mathbb A_f)\) gives a compact open subgroup of \(G(\mathbb A_f)\). The associated morphism of Shimura varieties therefore has a special curve as the image of the component through our point. This construction uses \(\Lambda\) itself: neither a product lattice nor a product principal polarization has been substituted for it. Varying the rational identifications and the lattices also allows every Hecke translate.

The Mumford–Tate group of a non-CM elliptic curve is \(\mathop{\mathrm{GL}}(V)\). One can see the relevant dichotomy directly: its connected Hodge group is a reductive subgroup of \(\operatorname{SL}(V)\), and a proper such subgroup containing the elliptic Hodge circle is a torus. That torus case has an imaginary quadratic commutant and is precisely CM. Therefore the point in (59) has Mumford–Tate group \(G\), and is Hodge generic on the special curve just constructed. The commutant of \(G\) on \(V\otimes W\) is \(\mathop{\mathrm{id}}_V\otimes\mathop{\mathrm{End}}(W)\simeq\mathop{\mathrm{Mat}}_2(\mathbb Q)\), so this is an \(E^2\)-special curve.

Conversely, the defining rational representation of an \(E^2\)-special curve has this same two-copy elliptic form. At a Hodge generic point its commutant is \(\mathop{\mathrm{Mat}}_2(\mathbb Q)\) and its elliptic factor is non-CM. Equivalently, the rational matrix idempotents split its abelian surface, up to isogeny, into two isogenous elliptic factors; the off-diagonal matrix units identify their rational Hodge structures. The equivalence between abelian varieties up to isogeny and polarizable rational Hodge structures of type \(\{(-1,0),(0,-1)\}\) makes this a complex isogeny (Milne 1995, Theorem 4.1). For an algebraic point, all these homomorphisms and abelian subvarieties descend to \(\overline{\mathbb Q}\): each is defined over a finitely generated extension of \(\overline{\mathbb Q}\), which is regular, and (Milne 1986, Corollary 20.4) applies. Thus the isogeny and the elliptic factors can be taken over \(\overline{\mathbb Q}\).

It remains to check the order, rather than only its rational algebra. For the rational subgroup \(G\) defining the curve, its integral commutant is \[ R=\mathop{\mathrm{End}}_G(H_1(A_s,\mathbb Q))\cap\mathop{\mathrm{End}}_{\mathbb Z}(\Lambda). \tag{61}\] At a Hodge generic point, commuting with \(G\) is equivalent to being a Hodge endomorphism. To check the integral refinement, let \(u\in R\). The rational Hodge equivalence writes \(u=f/n\) for an endomorphism \(f\) of \(A_s\) and a positive integer \(n\). The inclusion \(u\Lambda\subset\Lambda\) means \(f\Lambda\subset n\Lambda\). Under \(A_s[n](\mathbb C)\simeq n^{-1}\Lambda/\Lambda\), it follows that \(f\) annihilates \(A_s[n]\). Factoring \(f\) through the quotient by \(A_s[n]\), namely multiplication by \(n\) on \(A_s\), makes \(u\) an actual endomorphism. Conversely every abelian-variety endomorphism preserves the integral Hodge structure. The same descent argument identifies its complex and geometric endomorphism rings. It follows that \(R=\mathcal O_s\). Daw–Orr define \(\Delta(Z_s)\) as the absolute discriminant of this integral commutant (Daw and Orr 2022, sec. 6.3); (58) proves \(\Delta(Z_s)=\Delta_s\). ◻

Proof of 1. Every special curve is defined over \(\overline{\mathbb Q}\). Since the closed Hodge-generic curve \(C\) is not contained in any such curve, its intersection with each one is a zero-dimensional algebraic set over \(\overline{\mathbb Q}\). Consequently every point of \(C(\mathbb C)\cap\Sigma^{\mathrm{DO}}_{E^2}\) is algebraic. By 30, this intersection is exactly \(\Sigma_{E^2}(C)\), and its complexity is \(\Delta_s\). 29 supplies (62) with \(L=K\), a number field and therefore finitely generated. The stated counting implication proves that \(\Sigma_{E^2}(C)\) is finite. In particular, in the infinite case used to choose the fixed center, it gives the required contradiction. No companion branch theorem is used. ◻

Proof. The Hodge group is the smallest connected \(\mathbb Q\)-algebraic subgroup whose real points contain the Hodge circle; adjoining the weight homotheties gives the Mumford–Tate group. At a Hodge-generic point of a special subvariety, its associated Hermitian domain has the dimension of that subvariety. We apply this with dimension one, retaining the actual rational Hodge structure and polarization of \(A_s\).

Suppose first that \(A_s\) is simple up to isogeny. The surface classification in (Moonen and Zarhin 1999, sec. 2(2.2)) gives four possibilities for its full rational endomorphism algebra: \(\mathbb Q\), a real quadratic field, an indefinite quaternion division algebra, or a quartic CM field. The corresponding Hodge groups are respectively \(\mathop{\mathrm{Sp}}_4\), a restriction of scalars of \(\mathop{\mathrm{SL}}_2\) from the real quadratic field, the norm-one quaternion group, and a torus. Their Hermitian domains have dimensions \(3,2,1,0\): respectively the genus-two Siegel space, two copies of the upper half-plane, one upper half-plane, and a point. Thus only the quaternionic case can occur on a special curve at a Hodge-generic point. This gives (iii) as an equality of the full endomorphism algebra.

If \(A_s\) is not simple, Poincaré reducibility gives an isogeny \(A_s\sim E_1\times E_2\). When the two elliptic factors are isogenous, their product has the same Hodge group as either factor (Moonen and Zarhin 1999, sec. 1(1.2)). For a non-CM factor this group is \(\mathop{\mathrm{SL}}_2\), with one-dimensional domain, whereas a CM factor has a torus Hodge group and a zero-dimensional domain. This gives (ii). When the factors are nonisogenous, their Hodge group is the product of the two elliptic Hodge groups (Moonen and Zarhin 1999, Corollary (3.9)). Each non-CM factor contributes one domain dimension and each CM factor contributes zero. Dimension one therefore means precisely one CM factor, giving (i). The alternatives are disjoint, either by simplicity or by the full endomorphism algebras \(\mathbb Q\times F\) and \(M_2(\mathbb Q)\).

These assertions are invariant under rational Hodge isomorphism, so they apply to every polarization and Hecke translate. The elliptic factors and isogenies at an algebraic point can be defined over \(\overline{\mathbb Q}\), by the descent explained in 30. Finally the special closure of any point on \(Z\) is a special subvariety contained in \(Z\). If it is proper, it has dimension zero and is a special point. This proves the last assertion. ◻

Proof of 3. First the set of special points on \(C\) is finite. These are the CM points of \(\mathcal A_2\). If there were infinitely many, their Zariski closure would be all of the irreducible curve \(C\), since a proper closed subset of an irreducible algebraic curve is finite. The André–Oort theorem for the coarse moduli space of principally polarized abelian surfaces would then make \(C\) special (Pila and Tsimerman 2013, Theorem 1.1). This contradicts its Hodge genericity.

By 31, after removing the already finite set of special points, every element of \(\Sigma(C)\) belongs to one of the three indicated loci. The classification applies to all Hecke translates and to the original coarse moduli space.

Theorem 1.1 of the \(E\times\mathrm{CM}\) companion (OpenAI 2026b), restated as 2(A), gives finiteness of the entire first locus, with arbitrary CM fields, orders, and unpolarized isogenies. 1, proved above, gives finiteness of the second locus. Theorem 1.1 of the quaternionic companion (OpenAI 2026a), restated as 2(B), gives finiteness of the third locus: its full endomorphism algebra is an indefinite quaternion division algebra over \(\mathbb Q\), exactly as the companion theorem requires. The union of these three finite sets and the finite set of special points contains \(\Sigma(C)\), which is therefore finite.

All these conclusions concern the original coarse curve. The normalizations and fixed covers used for the second locus have already been descended in 29, and every fixed omission has been restored there. A singular original curve has only finitely many points removed in this process. The argument is also valid when \(C\) is complete, because none of these steps requires a point of its closure on a compactification boundary. ◻

Proof. Write \(d=\dim S_C\in\{1,2,3\}\). If \(d=3\), then \(C\) is Hodge generic in \(\mathcal A_2\), and the conclusion is 3. If \(d=2\), the union consists only of special points. Infinitely many such points on \(C\) would be Zariski dense, so André–Oort (Pila and Tsimerman 2013, Theorem 1.1) would make \(C\) special, contradicting \(\dim S_C=2\). Finally, if \(d=1\), the union is empty. ◻

Proof. The fixed-variable degree inputs are the affine Gröbner basis degree bound of (Dubé 1990, Corollary 8.3) and the polynomial ideal-membership degree bound of (Aschenbrenner 2004, Theorem 3.4), which gives a proof of the Hermann–Seidenberg method. Their degree bounds become polynomials in \(D\) when \(M\) is fixed. For example, using one additional homogenizing variable, a larger permissible bound for the first input is \[\Gamma_M(D)=\left\lceil2(D^2/2+D)^{2^M}\right\rceil.\] Only this polynomial character is needed here.

We spell out the height consequence, since it must not introduce the degree or discriminant of a field of definition. A polynomial identity of a prescribed degree becomes a linear system by equating coefficients. There are at most \(\binom{T+M}{M}\) monomials of degree at most \(T\). After the degree bounds just quoted, the number of entries of every such system is polynomial in \(D+T\). For a matrix with \(u\) entries, each of height at most \(H_1\), the affine height of the full matrix is at most \(uH_1\). A determinant of size \(r\) then has height at most \(r uH_1+\log(r!)\): this follows place by place from the determinant expansion, using the ultrametric inequality at finite places. Choosing a nonzero pivot minor and applying Cramer’s rule gives solution and kernel bases of height polynomial in the matrix size times \(1+H_1\). The product formula for the pivot denominator proves these assertions with normalized absolute heights. There is no factor \([F:\mathbb Q]\).

The reduced Gröbner basis can be recovered by these linear systems: each required polynomial has bounded degree, and membership in the generator ideal can be imposed with the bounded-degree certificates. Leading coefficients and the coefficients of forbidden leading monomials are fixed by linear equations. This gives the stated height bounds for the basis and its representations. Fix those representations. With a degree-compatible order, division of a monomial of degree at most \(T\) uses multiples of basis elements of degree at most \(T\). Replacing a basis element by its fixed representation increases this degree by at most a fixed additive amount \(\Delta\). Reducing each monomial and extending linearly gives the required linear reduction map. This proves (a); the same coefficient argument and the membership degree bound prove (b).

For (c), pass to projective closures. Follow a component of interest by proper hypersurface cuts, starting with projective space. Whenever the current component is larger than the desired one, some given defining polynomial is nonzero on the current component; cut by it and choose a component still containing the desired one. There are at most \(M\) proper cuts. Geometric Bézout bounds the degrees at each step. For heights, apply the global intersection bound (Krick et al. 2001, Corollary 2.11) to the affine parts of these cuts. It bounds the height of each affine intersection by a polynomial in the degrees times one plus the preceding height and the coefficient heights of the cutting polynomials. The height of an affine variety in that result is defined through its projective closure (Krick et al. 2001, sec. 1.2.4). This absolute Chow height is nonnegative and additive on components, so it bounds each retained component separately. Its comparison with the projective coefficient height of a Chow form costs a polynomial in the degree; this follows also by comparing the coefficient norm with the logarithmic integral norm in the definition of Chow height. Components supported at infinity do not enter these affine intersections.

Here is an explicit way to return to equations. Let \(X\) be the projective closure of a component, of dimension \(r\) and degree \(\delta\), and let \(\Phi_X\) be its Chow form. In each of its \(r+1\) hyperplane groups substitute the coefficients of a hyperplane through \([1:x]\): \[(u_{i0},u_{i1},\ldots,u_{iM}) =\left(-\sum_{j=1}^M u_{ij}x_j,u_{i1},\ldots,u_{iM}\right).\] Set every coefficient in the remaining \(u_{ij}\) equal to zero. These are polynomials in \(x\), of degree at most \((r+1)\delta\), with polynomially many coefficients. They all vanish if \([1:x]\in X\). If \([1:x]\notin X\), projection from this point has image of dimension at most \(r\); \(r+1\) general hyperplanes through the point have no common intersection with \(X\). The substituted Chow form is then not the zero polynomial. Thus the coefficient equations define exactly the affine component. Normalize a nonzero coefficient of \(\Phi_X\) to one. Its affine coefficient height is its projective coefficient height by the product formula, and the indicated substitutions increase height by a polynomial in \(\delta\). This proves the equation-height and equation-count assertions, including after a field extension needed to define the component. ◻

Proof. Put \(k=\dim W\) and let \(\nu:\widetilde W\to W\) be its finite normalization. Since \(a\) is a nonzero nonunit whenever the boundary is nonempty, its zero set is pure of codimension one. Finiteness of \(\nu\) implies that every divisor of \(a\) upstairs maps onto a component of this zero set. At a general point of such a divisor \(D_0\), both \(\widetilde W\) and \(D_0\) are smooth, and the map on \(D_0\) has rank \(k-1\). Indeed the induced extension of function fields is finite and separable in characteristic zero.

Choose \(k-1\) general affine hyperplanes in the original ambient space. Their pullbacks cut smooth curve germs transverse to \(D_0\) at its general intersection points. They can also be chosen to avoid the zero set of the first nonzero transverse coefficient of any nonzero \(E_\alpha\). To see the latter assertion explicitly, use local coordinates \((t,s)\) with \(D_0=\{t=0\}\). A nonzero germ has the expansion \(t^r u(s)+O(t^{r+1})\) with \(u\) nonzero on an open subset of \(D_0\). A transverse curve through a point of that open subset preserves its order \(r\). Infinite order would instead mean that all transverse coefficients vanish on a local open set; analyticity would make the germ identically zero. Thus the hypothesis supplies a finite first order on each branch to be tested. The excluded set may be taken Zariski closed: once this finite order is known, choose a truncation longer than its ratio to \(\mathop{\mathrm{ord}}_{D_0}a\). That algebraic truncation has the same leading transverse coefficient. Its nonvanishing is the nonvanishing of an algebraic section of a power of the conormal line bundle on a smooth open subset of \(D_0\). This justifies using general algebraic hyperplanes to make all these choices simultaneously.

The sliced curve has degree at most \(D\), after removing components that do not meet the chosen open sets. The bound on the number of branches can be seen directly on the finite normalization \(\overline\nu\) of the projective closure of \(W\). Put \(L=\overline\nu^*\mathcal O(1)\). The homogeneous coordinate defining \(a=0\) is a nonzero section of \(L\); if its divisor is \(\sum_T m_TT\), then \[\deg W=L^k =\sum_T m_T\,[\overline{\mathbb Q}(T):\overline{\mathbb Q}(\overline\nu(T))] \deg\overline\nu(T).\] Each term is a positive integer. Every affine normalized divisor under consideration occurs in this sum, so there are at most \(D\) of them. Equivalently the transverse curve branches are counted with their positive hyperplane-intersection multiplicities. This also explains why no bound for equations of \(\widetilde W\) is needed.

On a tested curve, adjoin the algebraic curve coordinates implicit in the fixed germs. A parameter on each fixed curve gives a fixed finite map of its smooth projective completion to \(\mathbb P^1\). Pulling these maps back by the functions \(a\) or \(aR\) and taking the component of the specified germ increases degree by a fixed factor. Repeating this for the fixed finite list still gives an algebraic curve of degree polynomial in \(D\) in a fixed product of projective spaces. A fixed Segre embedding and normalization produce a smooth projective curve \(X\) with genus polynomial in \(D\). For example a general plane projection bounds its genus by the arithmetic genus of a plane curve of polynomial degree. The degrees of \(a\), \(R\), the original coordinate functions, and each adjoined coordinate on \(X\) are polynomial in \(D\). The chosen points of \(X\) lie over the specified ordinary centers. Ramification in these maps does not destroy regularity of the pulled back connections. If an argument is constant, its germ contributes a constant solution vector instead.

Direct sums, tensor products, and duals encode every degree-\(b\) polynomial combination of the fixed germs as \[ f=r\,y, \tag{64}\] where \(r\) is a rational row on \(X\) and \(y\) is a horizontal analytic vector for one algebraic connection of rank at most a fixed integer \(r_*\). The coordinates and the target-dependent scalar coefficients occur in \(r\). The integer \(r_*\) is determined before \(W,d,h\) are chosen: it is the sum of the ranks of the finitely many tensor and dual systems needed for all monomials of degree at most \(b\). Choose the tensor rational frame obtained from the specified fixed frames. It is regular and invertible at the tested point, and the connection is ordinary there. The polar degrees of its rational connection matrix and of \(r\) are polynomial in \(D\), since all constituent rational maps and functions have those degree bounds.

We give the zero estimate for (64). Let \(\delta=d/da\) on the function field of \(X\). If \(g\) is rational, then \[(\delta g)_\infty \leq (g)_\infty+((g)_\infty)_{\mathrm{red}}+(da)_0, \qquad \deg(da)_0\leq 2g(X)+2\deg(a).\] The first inequality follows by writing \(\delta g=dg/da\) in a local parameter; the second uses the canonical divisor degree and the bound \(\deg(da)_\infty\leq2\deg(a)\). Write the horizontal equation as \(\delta y=B y\) and define derivative rows \[r_0=r,\qquad r_{j+1}=\delta r_j+r_jB.\] Then \(r_jy=\delta^jf\). The entries of \(B\) have a common polar divisor of polynomial degree: divide the rational connection one-form by \(da\), whose zero divisor has just been bounded. If \(P_j\) bounds the poles of the entries of \(r_j\) and \(Q\) bounds the poles of \(B\), then \[P_{j+1}=2P_j+(da)_0+Q\] is a valid common bound. For \(j\leq r_*\) all these divisors therefore have polynomial degree, independently of coefficient sizes.

Let \(s'\) be the first index for which \(r_{s'}\) lies in the function-field span of \(r_0,\ldots,r_{s'-1}\). Unless \(r=0\), we have \(1\leq s'\leq r_*\). Differentiating this dependence shows that the span is stable under covariant differentiation; all subsequent rows lie in it. Let \(K_\eta\) be its common kernel. At the tested point take the local ring of convergent germs \(\mathcal O=\mathbb C\{t\}\), the ordinary frame \(\mathcal O^{r_*'}\) of the actual rank \(r_*'\leq r_*\), and the saturation \[K=\bigl(K_\eta\otimes_{\overline{\mathbb Q}(X)}\operatorname{Frac}\mathcal O\bigr) \cap\mathcal O^{r_*'}.\] This is a direct summand, since the quotient is torsion free over a discrete valuation ring. It is preserved by the regular connection with respect to \(d/dt\): generic preservation follows from that for \(\delta\), and the derivative of a regular section is regular. Thus the quotient has an ordinary connection. If \(f\) is nonzero, the projection of \(y\) to the quotient is nonzero and cannot vanish at \(t=0\), by uniqueness for an ordinary linear differential equation. It is therefore primitive in the quotient lattice.

This primitivity can be used in the original frame, without a height or pole bound on any quotient frame. Let \(A\) be the matrix with rows \(r_0,\ldots,r_{s'-1}\), and let \(P\) be the degree of a common polar divisor for its entries. Its entries have order at least \(-P\) at \(t=0\). Some \(s'\times s'\) minor \(\Delta\) is nonzero, with polar degree at most \(s'P\), and hence with \(\mathop{\mathrm{ord}}_0\Delta\leq s'P\). Smith normal form over \(\mathcal O\), after clearing the entry poles, has nonzero exponents \(\sigma_1\leq\cdots\leq\sigma_{s'}\) satisfying \[\sigma_i\geq-P,\qquad \sum_{i=1}^{s'}\sigma_i =\min_{\Delta'\ne0}\mathop{\mathrm{ord}}_0\Delta' \leq s'P.\] Consequently \(\sigma_{s'}\leq(2s'-1)P\). The invertible row and column matrices in Smith normal form preserve the local lattices and their maximal ideals. The saturated kernel corresponds to the zero columns, so primitivity modulo \(K\) means that some coordinate in a nonzero column is a unit. It follows that \[ \min_{0\leq j<s'}\mathop{\mathrm{ord}}_0(\delta^j f) \leq (2s'-1)P. \tag{65}\]

If \(e=\mathop{\mathrm{ord}}_0 a\geq1\) and \(n_0=\mathop{\mathrm{ord}}_0 f\), local differentiation gives \[\mathop{\mathrm{ord}}_0(\delta^j f)\geq n_0-je.\] Cancellation of leading terms only increases the order, so the inequality also covers that case. Combining it with (65) bounds \(n_0/e\) by a polynomial in \(D\). Finite pullback multiplies both orders by its ramification index; the same bound holds on the original normalized divisor. The transverse slice was chosen to preserve its first nonzero coefficient, so this proves the desired order bound for each nonzero original expression on each divisor.

For each divisor take the least order among the nonzero \(E_\alpha\). Cancellation at that order in \(\sum c_\alpha E_\alpha\) is a proper linear condition on \((c_1,\ldots,c_s)\), with coefficients in a residue function field. There are at most \(D\) such conditions. A proper hyperplane meets the grid \(\{0,\ldots,D\}^s\) in at most \((D+1)^{s-1}\) points: fix all coordinates except one whose coefficient is nonzero. The union of the forbidden hyperplanes cannot fill the grid. Choose a vector outside it. The resulting \(E\) has the stated polynomial order bound on every divisor. Choose an integer \(m\) larger than all the bounded order ratios. All coordinate functions are regular at these divisors, and \(E-E_{<m}=a^m\sum_{j\geq m}a^{j-m}e_j(U)\) is an analytic multiple of \(a^m\). Its order is at least \(m\mathop{\mathrm{ord}}a\). The smaller order of \(E\) therefore equals that of \(E_{<m}\), proving (63) and the lemma. ◻

Proof. The graph is irreducible of dimension \(k\). Over \(a\ne0\) it is the whole zero set of \(a^mq-F\) in \(W\times\mathbb A^1\), and its closure is an irreducible component of that zero set. This proves the complexity assertion by 33.

Let \(\nu:\widetilde W\to W\) be the finite normalization and let \(\widetilde\Gamma\) be the closure of the same rational graph in \(\widetilde W\times\mathbb A^1\). Its map to \(W\times\mathbb A^1\) is finite, so its image is closed. That image equals \(W'\): it contains the dense original graph and is contained in its closure. At the generic point of each normalized boundary divisor \(D\), the reciprocal \(g^{-1}\) is regular and vanishes, because \(g\) has a pole. On a neighborhood of that point, the upstairs graph closure satisfies \[qg^{-1}=1.\] It therefore has no point above a dense open subset of \(D\).

Fix a boundary component \(B\subset W\). All points of the finite normalization above its generic point lie on the finitely many divisors dominating \(B\). For each of these divisors remove the proper closed subset not covered by the neighborhood just used. The images of the removed subsets are closed under the finite map and have dimension at most \(k-2\). Outside their union, and outside intersections with the other boundary components, no upstairs graph point lies over \(B\). The finite surjection \(\widetilde\Gamma\to W'\) then gives the same exclusion downstairs. Doing this for every \(B\) proves (66).

On \(q+f(U)=0\) the projection has inverse \((a,U)% \BeginAccSupp{method=hex,unicode,ActualText=21A6}% \mapsto\EndAccSupp{}(a,U,-f(U))\). Thus the set in (67) is isomorphic to a closed subvariety of the projected boundary and has its asserted dimension. This step fixes the graph coordinate; no bound on the dimension of \(W'\cap\{a=0\}\) itself was used. ◻

Proof. Fix a degree-compatible Gröbner basis of \(J\) and bounded-degree representations of its elements in \(I_1,\ldots,I_r,a,f_0\) using 33. Reducing each monomial and extending linearly gives fixed linear maps, with \[ D_1=\operatorname{NF}_J(D_1) +\sum_{\rho=1}^r I_\rho A_\rho(D_1) +aA(D_1)+f_0H(D_1). \tag{70}\] If \(\deg D_1\leq T\), every coefficient polynomial on the right has degree at most \(T+\Delta\), where \(\Delta\) is polynomial in \(D\) and independent of \(T\). The normal form has degree at most \(T\).

Start with a general polynomial \(P\) of degree at most \(L\). Set the residual in layer zero to \(D_0=P\) and the other residuals to zero. For \(i=0,\ldots,n-1\), impose the linear condition \[\operatorname{NF}_J(D_i)=0.\] Use (70) for \(D_i\). Record \(a^i\sum I_\rho A_\rho(D_i)\) in the certificate. The term \(a^{i+1}A(D_i)\) is added to the residual in layer \(i+1\), or recorded as a multiple of \(a^n\) if \(i+1=n\). In the other term use the exact identity \[a^i f_0H(D_i) =a^i F_nH(D_i) -\sum_{j=1}^{n-1}a^{i+j}e_{m+j}(U)H(D_i).\] Record the first summand as a multiple of \(F_n\). For \(i+j<n\) add \(-e_{m+j}H(D_i)\) to layer \(i+j\); for \(i+j\geq n\) record the term as a multiple of \(a^n\). After the last layer no residual remains, so these conditions imply the actual membership (69). This procedure never takes a radical or raises \(P\) to a power.

All the residuals and conditions depend linearly on the coefficients of \(P\). Choose \(B\) so large that \[\Delta+C_0(1+m+j)\leq Bj\quad(j\geq1), \qquad \Delta\leq B.\] Such \(B\) is polynomial in \(D\). A contribution advancing by \(j\) layers then increases residual degree by at most \(Bj\); advancement using the generator \(a\) has the same bound with \(j=1\). Induction gives \[ \deg D_i\leq L+Bi\quad(0\leq i<n). \tag{71}\] Every recorded term in the certificate also has degree polynomial in \(D+L+n\), including the terms advanced past the last layer.

To count the conditions, let \(G\) be a polynomial bound for the degrees of generators of the initial monomial ideal of \(J\), and put \(r_0=\dim V(J)\). In a surviving monomial, let \(S\) be the set of variables whose exponents are at least \(G\). No generator of the initial ideal can be supported in \(S\), since it would then divide the monomial. The coordinate space in the variables \(S\) is therefore contained in the zero set of that monomial ideal, so \(|S|\leq r_0\). There are at most \(2^M\) choices of \(S\), at most \(G^M\) choices for the remaining bounded exponents, and at most \((1+T)^{r_0}\) choices of total degree at most \(T\) in the unbounded exponents. Hence the space of normal forms of degree at most \(T\) has dimension at most \[2^MG^M(1+T)^{r_0} \leq B'(1+T)^{k-2}.\] The equality of the dimensions of an ideal and its initial ideal follows from equality of their degree filtrations under a degree-compatible order. This count applies to nonradical ideals as well. If \(J\) is the unit ideal its normal-form space is zero. Combining the count with (71), the entire construction imposes at most the left side of (68) independent linear conditions.

On the other hand, the restrictions of degree-\(\leq L\) polynomials to \(W'\) span a space of dimension at least \(\binom{L+k}{k}\). Indeed, \(k\) generic linear coordinates are algebraically independent on \(W'\); their monomials of total degree at most \(L\) are linearly independent. If every polynomial satisfying the layer conditions vanished on \(W'\), the codimension of the solution space in the degree-\(\leq L\) polynomial space would be at least this number. That contradicts (68). A polynomial nonzero on \(W'\) satisfying (69) therefore exists.

For the asserted height control, let \(T\) be the polynomial certificate degree bound just proved. Introduce unknown coefficients for \(P\) of degree at most \(L\) and for the certificate polynomials of degree at most \(T\), and equate coefficients in \[P=\sum_\rho B_\rho I_\rho+B_a a^n+B_f F_n.\] This is a homogeneous linear system of polynomial size in \(D+L+n\). The height of its coefficient matrix is polynomial in these quantities times the stated input heights. The minors argument in 33 supplies a basis of solutions with the claimed height bound. At least one basis vector has its \(P\) component nonzero on \(W'\); otherwise every solution would vanish there, contradicting the existence already established.

Finally set \(n=cL\). For \(k\geq2\) and \(L\geq1\), the left side of (68) is at most \[B'c(2+Bc+B)^{k-2}L^{k-1},\] whereas the right side is at least \(L^k/k!\). Thus it suffices to choose the integer \[L>k!B'c(2+Bc+B)^{k-2}.\] This is polynomial in \(D+c\), because \(k\leq M\) is fixed. For \(k=1\) there are no conditions and any \(L\geq1\) suffices. This proves all assertions. ◻

Proof of 4. All implicit constants in this proof depend on the fixed data in the theorem. We keep an irreducible closed \(W\ni z\) whose defining equation count and degrees are bounded by a polynomial in \(d\) and whose defining coefficient heights are bounded by \[ D_i(d)(1+\log h)^{B_i}, \tag{72}\] where \(D_i\) is a fixed polynomial and \(B_i\) is a fixed nonnegative integer for the current descent step \(i\). Its degree has a polynomial bound as well. Initially \(W=\mathbb A^N\) and these assertions hold. At most \(N\) strict dimension drops will be made. Consequently enlarging the polynomials and exponents at each step still yields constants depending only on the initial fixed data.

First suppose \(W\cap\{a=0\}\) is empty. If \(F_1,\ldots,F_r\) define \(W\) set theoretically, the Nullstellensatz says \((F_1,\ldots,F_r,a)=(1)\). This assertion is true for the displayed ideal itself: an ideal with empty zero set is the unit ideal. By 33 there is an identity \[1=aA+\sum_{\rho=1}^r F_\rho A_\rho\] whose degrees are polynomial in \(d\) and whose coefficient heights have the form (72). At \(z\) we get \(1=aA(z)\). If \(\tau_v\) is one at an archimedean place and zero otherwise, a polynomial \(Q\) of degree \(T\) in \(M\) variables satisfies \[ \log|Q(x)|_v \leq \log\max(1,\|Q\|_v) +T\log\max(1,\|x\|_v) +\tau_v\log\binom{T+M}{M}, \tag{73}\] where \(\|Q\|_v\) denotes its maximum coefficient absolute value. On \(V\), \(|a|_v<1\), so only the positive logarithms of \(U\) enter this bound. Evaluation of \(A\) and (3) give \[\frac{h}{C_0d^{C_0}} \leq\sum_{v\in V}w_v\log|a|_v^{-1} =\sum_{v\in V}w_v\log|A(z)|_v \leq D_*(d)(1+\log h)^{B_*}.\] An inequality \(h\leq D(d)(1+\log h)^B\) implies a polynomial bound for \(h\): for fixed \(B\) there is a constant \(c_B\) such that \((1+\log h)^B\leq c_B\sqrt h\) for all \(h\geq2\), and hence \(h\leq c_B^2D(d)^2\). This proves the theorem in the present case. It includes every case where \(a\) is constant on \(W\), since its value is the nonzero target value, and every zero-dimensional \(W\).

We may therefore suppose \(a\) varies on \(W\) and its boundary is nonempty. Condition (iv) and 34 give an integer combination \(E\) and an integer \(m\), with bounds polynomial in \(d\), satisfying (63). Its coefficient and tail bounds retain the forms required in the theorem. More explicitly, if \(c_{\max}=\max_\alpha c_\alpha\), the ultrametric inequality and integrality of the \(c_\alpha\) preserve the original tail bound at finite places. At infinity replace \(b_v\) by \[b_v+\tau_v\log\max(1,s c_{\max}).\] The total extra weighted cost is at most \(\log\max(1,s c_{\max})\), since the total archimedean weight is one. Combining the coefficient tuples likewise preserves their bounds. We use \(b_v\) for these enlarged errors.

Put \(F=E_{<m}\) and form the graph variety \(W'\) of 35. Its dimension is \(k=\dim W\) and its equation count, degrees, and heights have the bounds (72), by 33. Lift the target to \(z'=(a,U,q)\), with \(q=a^{-m}F(z)\). The rational expression for \(q\) gives \[h_{\mathrm{aff}}(z') \leq C_1(d)h+C_2(d)(1+\log h)^{B_2} \leq C_3(d)h,\] where \(C_3\) is polynomial in \(d\). For the second inequality use \((1+\log h)^{B_2}\leq c_{B_2}h\); its constant is fixed at this descent step. The polynomial \(C_3\) is chosen before the interpolation degrees \(n,L\).

The local bound for \(q\) is much smaller. Since \(E(z)=0\) on \(V\), \[q=-a^{-m}\sum_{j\geq m}a^je_j(U), \qquad \log^+|q|_v\leq(m+1)b_v.\] Consequently \[ \sum_{v\in V}w_v\log\max(1,\|z'\|_v) \leq C_4(d)(1+\log h)^{B_4}. \tag{74}\] The exact equation at the lifted target is \[q+\sum_{j\geq0}a^je_{m+j}(U)=0.\]

Take set-theoretic defining equations for \(W'\) and let \(I\) be the ideal they generate. The ideal is kept as given, without assuming it radical. By 35, \[J=(I,a,q+e_m(U)) \quad\text{satisfies}\quad\dim V(J)\leq k-2.\] Set \(A_0(d)=C_0d^{C_0}\), and first choose an integer \[ c\geq 2A_0(d)(C_3(d)+1). \tag{75}\] Choose \(L\) by 36 and set \(n=cL\). Both choices are polynomial in \(d\), independent of \(h\). That lemma gives \(P\) of degree at most \(L\), nonzero on \(W'\), with a certificate \[ P=\sum_\rho B_\rho I_\rho+B_a a^n +B_f\left(q+\sum_{j=0}^{n-1}a^je_{m+j}(U)\right). \tag{76}\] Its degrees are polynomial in \(d\). The coefficient heights of \(P\) and every \(B_\rho,B_a,B_f\) are at most \(C_5(d)(1+\log h)^{B_5}\). To apply the lemma with many coefficients, bound the height of their joint tuple by the sum of the individual heights in (4); there are polynomially many indices \(m,\ldots,m+n-1\), so this still has the asserted form.

We now test whether \(P(z')\) can be nonzero. At a selected place, the parenthesized expression in (76) is \[-a^{-m}\sum_{j\geq m+n}a^je_j(U).\] Its absolute value is at most \(|a|_v^n\exp((m+n+1)b_v)\). Evaluate the two remaining certificate terms by (73), use (74), and include \(\log2\) at infinity for their sum. If \(P(z')\ne0\), the result is \[ \sum_{v\in V}w_v\log|P(z')|_v \leq -n\sum_{v\in V}w_v\log|a|_v^{-1} +C_6(d)(1+\log h)^{B_6}. \tag{77}\] Here polynomial certificate degrees multiply only the small positive local logarithms, and \((m+n+1)\sum_Vw_vb_v\) is of the same permitted size. This explains precisely where (3) is needed in addition to the global height bound.

At all other places evaluate \(P\) itself, of degree at most \(L\), using (73). Its certificate is unnecessary for this estimate. Summing gives \[ \sum_{v\notin V}w_v\log|P(z')|_v \leq L C_3(d)h+C_7(d)(1+\log h)^{B_7}. \tag{78}\] Enlarge the number field to contain \(W\), \(W'\), their chosen components, and all coefficients in (76); the normalized sums and all previously established estimates are unchanged. The product formula applied to the nonzero algebraic number \(P(z')\) combines (77) and (78) into \[0\leq\left(-\frac n{A_0(d)}+LC_3(d)\right)h +C_8(d)(1+\log h)^{B_8}.\] By (75) the coefficient of \(h\) is at most \(-L\). It follows that \(h\leq C_8(d)(1+\log h)^{B_8}\) and hence that \(h\) is polynomially bounded in \(d\) as above.

If this polynomial height bound does not hold, we have proved \(P(z')=0\). Substitute \(q=F/a^m\) and clear denominators: \[H(a,U)=a^{mL}P(a,U,F(a,U)/a^m).\] This is a polynomial. It vanishes at \(z\) because \(a(z)\ne0\), and it is not identically zero on \(W\), because \(P\) is nonzero on the irreducible graph closure and its graph over \(a\ne0\) is dense. Its degree is at most \(mL+L\max(1,\deg F)\); expanding the substitution bounds its coefficient height by \(h_{\mathrm{aff}}(P)+Lh_{\mathrm{aff}}(F)\) plus a polynomial degree error. Thus it has exactly the controls (72). Choose an irreducible component containing \(z\) of the proper intersection \(W\cap\{H=0\}\). By 33 its defining equations have the same types of degree and height bounds, and its dimension is strictly smaller.

Repeat with this component as \(W\). Condition (iv) applies to it because it is an irreducible algebraic variety containing the same target. After at most \(N\) drops one either has obtained the polynomial height bound or reaches the empty-boundary case, which gives that bound. There are finitely many fixed-step polynomial bounds in this procedure, so their maximum is bounded by \(Cd^C\) for a constant depending only on the fixed data. This completes the proof. ◻