Proof. The full level moduli space is a fine moduli scheme with a universal abelian scheme (Chai 1985, I, Theorem 1.4); for the complex moduli interpretation see also (Milne 2013, Theorem 4.6 and Corollary 4.7). The finite change-of-level group acts on it, and its quotient is the coarse moduli space, so the forgetful map is finite. We fix the symplectic pairing component after adjoining the relevant roots of unity. Normalize a component of its pullback dominating \(C\). Normal curves in characteristic zero are smooth; the resulting map is finite and surjective. All the schemes and morphisms in this construction descend to a number field. Fibres of a finite map are finite, so an infinite set downstairs has infinitely many lifts. For a cover with several components, at least one component contains infinitely many lifts; a dominating component of a finite cover of an irreducible curve is itself surjective.

After normalizing the base, the finite maps extend to finite maps between smooth projective completions. For such a map of generic degree \(D_0\), the degree formula for a fibre gives \([K(t):K(\pi(t))]\leq D_0\). Over the finitely many singular points of the original coarse curve, the fixed finite map has a fixed finite set of preimages; enlarge \(D\) to bound their residue-field extensions as well. Thus the constant \(D\) in (2) is uniform, but need not equal the generic degree of the map to the singular curve.

Functoriality of heights identifies a height of a pulled-back ample divisor with the height downstairs up to a bounded function. Any two ample divisors on a fixed projective curve bound each other’s heights linearly: sufficiently large positive integral multiples of either minus the other are ample and have heights bounded below. This proves (2). Replacing the fixed field \(K\) changes degrees by at most a fixed factor. None of these constants depends on the point. ◻

Proof. In a symplectic basis, the matrix of \(\psi\) is \(J=\left(\begin{smallmatrix}0&I_2\\-I_2&0\end{smallmatrix}\right)\). The equations \(T^tJ=JT\) and \(\mathop{\mathrm{Tr}}(T)=0\) give \[T=\begin{pmatrix}A&\beta J_2\\ \gamma J_2&A^t\end{pmatrix}, \qquad A=\begin{pmatrix}a&b\\c&-a\end{pmatrix}, \qquad J_2=\begin{pmatrix}0&1\\-1&0\end{pmatrix}.\] One checks that \(T^2=(a^2+bc-\beta\gamma)I_4\). Thus \[q(T)=a^2+bc-\beta\gamma,\] which has signature \((3,2)\) and is nondegenerate over \(\mathbb Z[1/2]\). The alternating form \((u,v)\mapsto\psi(Tu,v)\) identifies the self-adjoint operators with alternating two-forms. The identity operator corresponds to \(\psi\), and its orthogonal complement for the trace form is the primitive part. Taking account of the polarization twist gives \(\mathbb V\simeq(\bigwedge^2\mathbb H)(1)_{\mathrm{prim}}\) and the stated Hodge numbers.

Conjugation by a similitude preserves self-adjointness and the trace pairing. Its determinant on \(\mathbb V\) is \(1\), since \(\mathop{\mathrm{GSp}}_4\) is connected. The induced Lie algebra map \(\mathfrak{sp}_4\to\mathfrak{so}_5\) is nonzero. Simplicity and equality of dimensions make it an isomorphism. Consequently the induced map on the adjoint groups is an isomorphism; these are the split adjoint groups of types \(C_2\) and \(B_2\). Its kernel on \(\mathop{\mathrm{GSp}}_4\) is exactly the scalar torus, proving (4) and the assertion about the spin covering. These statements are over \(\mathbb Q\), not merely over \(\mathbb C\).

Put \(D=\mathop{\mathrm{End}}^0(\mathcal A_t)\). The polarization induces the positive Rosati involution on \(D\). Over \(\mathbb R\), the identification \(D\otimes\mathbb R\simeq M_2(\mathbb R)\) turns a positive involution into the adjoint for a positive definite form, and hence into transpose in a suitable basis. Its fixed subspace has dimension three. The cohomological trace is nonzero on the identity, so its kernel in that subspace has dimension two. For a nonzero symmetric endomorphism \(u\), Rosati positivity gives \(\mathop{\mathrm{Tr}}(uu^\dagger)=\mathop{\mathrm{Tr}}(u^2)>0\). Thus \(q\) is positive definite on this trace-free subspace.

The rational realization of endomorphisms on \(H^1\) is faithful and identifies endomorphisms with rational Hodge endomorphisms (Milne 1995, Theorem 4.1). In particular these vectors are Hodge vectors in \(\mathbb V_t\). Clearing denominators in a rational basis gives integral endomorphisms with the same symmetry and trace conditions. Equivalently one can first apply \(u\mapsto u+u^\dagger\) and then \(v\mapsto4v-\mathop{\mathrm{Tr}}(v)\mathrm{id}\) to integral endomorphisms; the principal polarization makes the first operation integral. Finally, intersection of an integral lattice with a rational subspace is saturated. ◻

Proof. The generic Mumford–Tate group of a Hodge-generic curve is \(\mathop{\mathrm{GSp}}_4\). The connected monodromy normality theorem for a polarizable geometric variation on a smooth algebraic base places connected monodromy normally in its derived group (André 1992, sec. 5, Theorem 1). Since \(\mathop{\mathrm{Sp}}_4\) is \(\mathbb Q\)-simple, the connected monodromy is either trivial or \(\mathop{\mathrm{Sp}}_4\). In the first case a finite cover kills monodromy, and the theorem of the fixed part makes the variation constant; the moduli map would be constant. This is impossible. This consequence is also recorded in (André 1992, sec. 6, Remark 1). A dominant pullback has the same generic Mumford–Tate group and still has a nonconstant moduli map, so the argument also applies there. The assertion for \(\mathbb V\) follows from (4).

For the two systems let \(M\) be their generic Mumford–Tate group and \(H\) their connected monodromy, after fixing rational symplectic bases on a simply connected open. Both projections of \(H\) are \(\mathop{\mathrm{Sp}}_4\). Put \(G=\operatorname{PGSp}_4\). The algebraic Goursat lemma and simplicity give two possibilities for the image \(\bar H\subset G\times G\): it is the product, or it is the graph of a \(\mathbb Q\)-defined automorphism \(\alpha\) of \(G\). Indeed the kernels of the two projections are normal; in an adjoint simple group a proper such kernel is trivial. In the product case dimensions give \(H=\mathop{\mathrm{Sp}}_4\times\mathop{\mathrm{Sp}}_4\).

In the graph case, every automorphism of the split adjoint group of type \(C_2\) is inner, since its Dynkin diagram has no automorphism. As \(G\) is adjoint, a \(\mathbb Q\)-defined inner automorphism is conjugation by an element \(\bar g\in G(\mathbb Q)\). The central exact sequence \[1\longrightarrow\mathbb G_m\longrightarrow\mathop{\mathrm{GSp}}_4 \longrightarrow G\longrightarrow1\] and \(H^1(\mathbb Q,\mathbb G_m)=0\) show that \(\bar g\) lifts to \(g\in\mathop{\mathrm{GSp}}_4(\mathbb Q)\). Conjugate the second factor by this lift, so that \(\bar H\) becomes the diagonal. Normality of \(H\) in \(M\) forces the adjoint image of \(M\) into the normalizer of this diagonal. That normalizer is the diagonal itself: a normalizing pair \((g_1,g_2)\) has \(g_2^{-1}g_1\) central in the adjoint group.

The two Hodge homomorphisms therefore agree after passage to \(G\). Their discrepancy on the weight-one spaces is a real algebraic character of the Deligne torus \(\operatorname{Res}_{\mathbb C/\mathbb R}\mathbb G_m\) into the scalar \(\mathbb G_m\). Such a character is a power of the norm. The two weights are equal, so restriction to real scalars makes this power zero. The rational map \(g\) thus identifies the Hodge structures themselves. Its similitude multiplier is positive because it compares the positive definite forms supplied by their polarizations and complex structures.

This argument holds at very general points of the chosen open; holomorphicity extends the resulting fixed period relation across it. Write \(\mu>0\) for the multiplier of \(g:H^1(A_1)\to H^1(A_2)\). The homology map in the direction from \(A_1\) to \(A_2\) is \[F=\mu(g^{-1})^\vee:H_1(A_1,\mathbb Q)\longrightarrow H_1(A_2,\mathbb Q).\] It has multiplier \(\mu\) and commutes with the complex structures. Choose an integer \(n>0\) such that \(nF\) is integral. It induces a holomorphic isogeny \(\varphi:A_1\to A_2\), which is algebraic since the tori are projective. Its multiplier is the positive integer \(m=n^2\mu\): integrality follows by evaluating its polarization identity on a pair of integral homology vectors of pairing \(1\). Moreover \(\varphi^*=n\mu g^{-1}\), so the covariant cohomological transport defined above is \[T_\varphi=m(\varphi^*)^{-1}=ng.\] Thus \(\varphi\) realizes the original projective rational representative.

A fixed rational similitude defines a closed algebraic Hecke correspondence, finite over either moduli factor. Concretely, for a fixed integral multiplier \(m\), its kernels are among the finite subgroup schemes of \(A[m]\), and the principal polarizations and fixed level structures give the corresponding finite moduli problem. Closure of the local relation consequently contains the image of \(W\) in one such correspondence. Its projections to both curves are dominant by hypothesis. Finite central twists of the local systems can be killed by a finite cover and do not change this conclusion about their moduli points. ◻

Proof. We first show that the real stabilizer of a lifted curve is discrete. Finite area of the curve quotient will then make its integral stabilizer a subgroup of finite index. Maps between two lifts consequently have only finitely many cosets, and rational representatives will give the fixed isogeny multipliers.

Let \(\mathfrak H_2\) be Siegel space, let \(G_{\mathbb R}=\operatorname{PGSp}_4(\mathbb R)^+\) be its effective connected group of holomorphic isometries, and fix a torsion-free principal congruence subgroup \(\Gamma\) at fine level. For any curve in the list, choose an irreducible analytic component \(X\) of its full inverse image in \(\mathfrak H_2\). It is a closed analytic curve. Its normalization may be used wherever a smooth curve is needed. The integral stabilizer \[\Gamma_X=\{\gamma\in\Gamma:\gamma X=X\}\] contains the image of the monodromy of the normalization of the corresponding fine-level curve. In fact this image is the deck group on the lifted normalization over the smooth part. By 4, it is Zariski dense in the real adjoint group.

We verify that the real stabilizer \[H_X=\{g\in G_{\mathbb R}:gX=X\}\] is discrete. It is closed: if \(g_n\to g\) with \(g_nX=X\), closedness of \(X\) gives \(gX\subset X\), and applying the same argument to \(g_n^{-1}\) gives equality. Thus \(H_X\) is a Lie subgroup. Its Lie algebra is invariant under \(\Gamma_X\). Stabilizing a fixed real vector subspace of the adjoint representation is an algebraic condition, so Zariski density makes this Lie algebra an ideal of \(\mathfrak{sp}_4(\mathbb R)\). Simplicity leaves zero or the full Lie algebra. The latter would give \(H_X=G_{\mathbb R}\), forcing \(X=\mathfrak H_2\) by transitivity. The dimensions differ, so \(H_X\) is discrete.

Discreteness alone does not bound the number of cosets of \(\Gamma_X\). For this we use that \(\Gamma_X\backslash X\), with normalization understood, has finite area for the metric induced from an invariant Hermitian metric on \(\mathfrak H_2\). To check both the quotient and the effect of singularities, let \(S_X\) be the normalization of the corresponding closed fine-level curve and form \[Y=S_X\mathbin{\times}_{\Gamma\backslash\mathfrak H_2}\mathfrak H_2.\] The normalization map from \(S_X\) to its closed fine-level image is finite, so base change makes \(Y\to\mathfrak H_2\) finite, with image the full inverse image of that curve. On the other hand, \(Y\to S_X\) is a covering and locally a biholomorphism, so \(Y\) is smooth. The finite map to that inverse image is an isomorphism above the smooth locus of the curve; hence it is its normalization.

The component \(Y_0\) normalizing \(X\) has stabilizer \(\Gamma_X\); this group acts transitively on every fibre over \(S_X\), since a deck transformation carrying two points of the same fibre to one another preserves their common connected component, and \(\Gamma_X\backslash Y_0\simeq S_X\). It therefore suffices to bound the pullback metric at the finitely many punctures of \(S_X\).

For completeness, any holomorphic map from the unit disk to \(\mathfrak H_2\) contracts its invariant metric up to a fixed constant relative to the Poincaré metric. Realize \(\mathfrak H_2\) as a bounded domain. An automorphism moving the image of the origin to a fixed origin, followed by the Cauchy derivative estimate in bounded coordinates, bounds the derivative at zero uniformly. Disk automorphisms give the pointwise assertion. Apply it on the universal cover of a punctured parameter disk. The descended upper bound is a constant multiple of \[\frac{|du|^2}{|u|^2\log^2(1/|u|)},\] whose area on \(0<|u|<\varepsilon\) is finite. On the remaining compact part the period map is holomorphic and has finite area. Zeros of its derivative or singular branches only decrease this upper bound.

We now deduce the precise group-theoretic consequence \[ [H_X:\Gamma_X]<\infty. \tag{6}\] The action of \(G_{\mathbb R}\) on \(\mathfrak H_2=G_{\mathbb R}/K_{\mathrm{max}}\) is proper, since \(K_{\mathrm{max}}\) is compact. Therefore the discrete group \(H_X\) acts properly on \(X\), with finite point stabilizers. Choose a regular point \(z\in X\) and a sufficiently small \(H_{X,z}\)-invariant disk \(D\subset X\) of positive induced area such that \(gD\cap D\) is empty for \(g\notin H_{X,z}\). Distinct double cosets in \(\Gamma_X\backslash H_X/H_{X,z}\) give disjoint images of translates of \(D\) in \(\Gamma_X\backslash X\). Each translate embeds there: a further identification would give a nonidentity element of the torsion-free group \(\Gamma_X\) conjugate to an element of the finite group \(H_{X,z}\). All these disks have the same positive area. Finite total area thus bounds the number of double cosets, and finiteness of \(H_{X,z}\) bounds the number of left cosets. This proves (6).

Choose lifts \(X_1,X_2\) for two curves. If a rational positive projective similitude \(g_0\) carries \(X_1\) onto \(X_2\), the set of all real such maps is \(g_0H_{X_1}\). By (6) this set has only finitely many classes modulo right multiplication by \(\Gamma_{X_1}\). Every class containing a rational map admits a rational representative. Integral changes of the chosen lifts add no classes: at coarse level all lifts are conjugate under the full integral symplectic group; at the chosen fine level there are only finitely many resulting classes. Thus there are finitely many rational maps for the whole fixed list.

A dominating component of a Hecke correspondence determines a local equality between a rational translate of one lifted curve and another lifted curve. Both are irreducible closed analytic curves, so this local equality extends to equality of those lifts. It is therefore represented by one of the finite maps just obtained. Lift each projective rational representative to \(\mathop{\mathrm{GSp}}_4(\mathbb Q)\), using \(H^1(\mathbb Q,\mathbb G_m)=0\), and clear denominators on the integral homology lattices. Its multiplier becomes a positive integer \(m\). This gives an isogeny of multiplier \(m\) on the local correspondence. To retain that multiplier after taking closure, work over the fixed fine moduli space in characteristic zero. Subgroups \(K\subset A[m]\) that are maximal isotropic for the Weil pairing and have order \(m^2\) form a finite étale scheme over this space: étale-locally the torsion is a constant finite group. Each such subgroup gives the quotient \(B=A/K\) with its principal polarization characterized by \(q^*\lambda_B=m\lambda_A\). The resulting map to \(\mathcal A_2\times\mathcal A_2\) is finite. Indeed it is finite over the first coarse factor, hence proper into the product because the second factor is separated, and its fibres are finite. Its image is therefore closed, and every geometric point of that image admits an isogeny of multiplier \(m\). The closure of the local correspondence lies in this image, including at singular coarse points and points with extra automorphisms.

Taking all representatives and the finitely many curve components of their intersections with the pairwise products that dominate both factors gives the required finite union. Additional level structures change only the finite covers of these moduli problems, not the underlying isogenies or their multipliers. ◻

Proof. We first preserve the branch order on a curve of controlled degree, then use the ordinary differential equation to bound that order independently of coefficient heights.

Reduction to a curve. Write \(k=\dim W\), and let \(\widetilde D\) be the normalization divisor defining the branch. At a general point \(p\) of \(\widetilde D\), both the normalization and \(\widetilde D\) are smooth, and the finite map from \(\widetilde D\) to its image \(D_0\) is étale at \(p\) after restricting to dense open subsets, by separability in characteristic zero. Choose \(k-1\) general affine linear functions whose differentials restrict to a basis of the cotangent space of \(D_0\) at the image of \(p\). Their pullbacks restrict to a basis on \(\widetilde D\). Their common level set through \(p\) is consequently a smooth curve germ transverse to \(\widetilde D\) in the normalization. The downstairs linear section is proper and has total projective degree at most \(D\). The transverse germ leaves the nonnormal locus, so its map to its image curve is generically the normalization isomorphism. It is thus the corresponding smooth germ on the normalization of an irreducible component of that linear section.

This slice preserves formal orders as well. If \(\operatorname{ord}_{\widetilde D}a=e>0\) and \(\operatorname{ord}_{\widetilde D}E=N\), only the terms with \(j\leq N/e\) contribute to the leading coefficient. After deleting a proper closed subset of \(\widetilde D\), write \(a=t^ev\) and that finite truncation as \(t^Nu\), where \(t\) is a local equation of \(\widetilde D\) and \(u,v\) are units. On the transverse curve \(t\) is a uniformizer. Hence the orders are exactly \(e\) and \(N\) after restriction. All these conditions hold at a general slice point.

Let \(T_0\) be the smooth projective normalization of this sliced curve. Its genus is polynomial in \(D\), by a general birational plane projection and the arithmetic genus of a plane curve. Every ambient coordinate on \(T_0\) is a quotient \(X_i/X_0\), so its polar divisor has degree at most \(D\). In particular the maps defined by \(a\) and \(aR\) have degrees at most \(D\) and \(2D\), respectively.

Adjoin the points on the fixed curves specified by the algebraic chart germs. Constant argument maps contribute constant chart factors and require no cover; in what follows index only the nonconstant argument maps. If the corresponding fixed maps to \(\mathbb P^1\) have degrees \(d_i\) and branch sets \(B_i\), including infinity when it is a branch value, the chosen normalized fibre-product component \(T\to T_0\) has degree at most \(N_0=\prod_i d_i\). It is étale outside \(\bigcup_i\phi_i^{-1}(B_i)\), where \(\phi_i\) is \(a\) or \(aR\). This set has at most \(\sum_i\#B_i\deg\phi_i=O(D)\) points, and the ramification contribution above each is at most \(N_0\). Thus Riemann–Hurwitz gives \[2g(T)-2\leq N_0\max\{0,2g(T_0)-2\}+O(D).\] The degrees of the maps from \(T\) to the fixed curves are bounded by \(N_0\deg\phi_i\), so all pulled-back fixed divisors have polynomial degree. Poles of \(aR\) are included in this calculation through the fibre above infinity. All ramification indices are bounded as well; any further ramification multiplies the two orders in (11) by the same factor.

The ordinary differential system. Let \(p\in T\) be the point of the specified branch. In direct sums and tensor products of the fixed pulled-back systems the expression takes the form \[E=r_0y,\] where \(y\) is a horizontal column of a fixed rank \(N\) and \(r_0\) is a rational row. Polynomial algebraic factors are absorbed in \(r_0\). A constant map to one of the fixed curves contributes a constant horizontal factor. The rational frames here are the pullbacks of the fixed frames; they are regular and invertible at \(p\). In fact \(a=0\) and \(R\) is regular at the chosen affine divisor, so both arguments \(a\) and \(aR\) are at their ordinary centres. The connection and \(r_0\) are therefore regular at \(p\). Their global polar divisors have degrees polynomial in \(D\), by the bounds on the degrees of the pullback maps and of the coordinate functions.

Choose a rational function \(z\) on \(T\) which is a uniformizer at \(p\) and whose polar divisor has degree at most \(2g(T)+1\). Such a function follows from Riemann–Roch by prescribing its first two jets at \(p\) and allowing a pole at another point. For \(g(T)=0\) a degree-one function suffices. Put \(\delta=d/dz\), and write the horizontal equation as \(\delta y=Ay\). The poles of the rational vector field \(\delta\) are bounded in degree by the zero divisor of \(\mathrm dz\), hence by \(2g(T)-2+2\deg z\). Differentiating a rational function adds at most its reduced polar divisor and the polar divisor of this vector field. It follows that \(A\), \(r_0\), and any fixed number of their derivatives have polar degrees polynomial in \(D\). They remain regular at \(p\); using a uniformizer rather than dividing by \(\mathrm da\) accommodates arbitrary ramification of \(a\).

Set \[r_{i+1}=\delta r_i+r_iA.\] Then \(\delta^iE=r_iy\). The successive rows become dependent after at most \(N\) steps. If \(r_\rho\) is the first row dependent on its predecessors, their span \(M=\langle r_0,\ldots,r_{\rho-1}\rangle_{\mathbb C(T)}\) is stable under the dual connection: differentiating the dependence shows that every later row belongs to it. Here \(\rho\leq N\).

Let \(\mathcal O=\mathcal O_{T,p}\) and saturate the common kernel of these rows in \(\mathcal O^N\). Saturation makes it a subbundle locally. It is stable under the ordinary connection, since its derivative is regular and belongs to the kernel generically. The quotient has an ordinary connection as well. Explicitly, if \(v\) is a local section annihilated by every row in \(M\), stability of \(M\) under the dual connection gives \(r(\nabla_\delta v)=0\) for every \(r\in M\). The ordinary connection makes \(\nabla_\delta v\) regular, so it belongs to the saturated kernel. The kernel and its quotient are free over the local discrete valuation ring, and the induced quotient connection has no pole. The image \(\bar y\) of \(y\) in this quotient is a nonzero horizontal section: a zero image would give \(r_0y=E=0\). Conversely, if \(E\) were identically zero, all its derivatives \(r_i y\) would be zero and \(y\) would lie in the kernel. The nonzero horizontal quotient section has nonzero reduction at \(p\): the coefficient recurrence of an ordinary linear differential equation forces a horizontal section with zero initial value to be zero.

The integral dual of this free quotient is exactly the saturated row lattice \(\mathcal L=M\cap\mathcal O^N\): its rows are the integral functionals annihilating the common kernel. Since \(\bar y\) has nonzero reduction, one such functional \(\lambda\in\mathcal L\) has \(\lambda y\) a unit. Choose a nonzero \(\rho\times\rho\) minor \(\Delta\) of the matrix with rows \(r_0,\ldots,r_{\rho-1}\). All these rows are regular at \(p\). Cramer’s rule on the selected columns gives \[\Delta\lambda=\sum_{i=0}^{\rho-1}c_ir_i, \qquad c_i\in\mathcal O.\] Hence at least one \(i<\rho\) satisfies \(\operatorname{ord}_p(r_iy)\leq\operatorname{ord}_p\Delta\). This is the elementary-divisor bound for the row lattice, expressed without choosing a new global frame. A nonzero rational function on \(T\) has equal total zero and pole degrees, so \(\operatorname{ord}_p\Delta\) is bounded by its polar degree, which is polynomial in \(D\). Finally, since \(z\) is a uniformizer, \[\operatorname{ord}_p E \leq\operatorname{ord}_p(\delta^iE)+i \leq\operatorname{ord}_p\Delta+N-1.\] Increasing the resulting polynomial by \(1\) proves (11), since \(\operatorname{ord}_p a\geq1\). This construction used only degrees, genera, ramification and fixed ranks; it used no arithmetic height. ◻

Proof. Write \(A=\overline{\mathbb Q}[W]\), \(u=f/a^m\), and let \(B\) be the finite normalization of \(A\). The graph has coordinate ring \(A[u]\), and \(B[u]\) is finite over it. Suppose there were a point of the graph above a generic point \(\mathfrak p\) of a component of \(V_W(a)\). Lying over in \(B[u]\) and then contraction to \(B\) give a prime \(\mathfrak r\) above \(\mathfrak p\). The local ring \(B_{\mathfrak r}\) is a discrete valuation ring. The hypothesis says \(u=v\pi^{-s}\) there, for a unit \(v\), a uniformizer \(\pi\), and \(s>0\). Its map into the local ring of the lifted graph point would satisfy \(\pi^su=v\) with \(\pi\) in the maximal ideal and \(v\) invertible, a contradiction.

The constructible image of \(V_{W'}(a)\) therefore contains none of the generic points of \(V_W(a)\). Each component of the latter has dimension \(k-1\), so the closure of this image has dimension at most \(k-2\). On imposing \(q=-g\), projection from the graph to \(W\) is a closed immersion: in its coordinate ring the extra coordinate is already specified by \(g\). Its image lies in the preceding constructible image, which proves (12). ◻

Proof. Fix a degree-compatible monomial order and a Gröbner basis of \(\mathfrak j=(\mathfrak b,a,F_0)\), together with representations of its elements in these displayed generators. Their degrees and heights have the bounds recorded above. For each reducible monomial, fix a basis element to divide by. Division in degree at most \(T\) can then be performed by a fixed decreasing sweep through all monomials of that degree range, allowing zero coefficients. Thus, for \(\deg H\leq T\), it gives linearly in \(H\) \[ H=\operatorname{rem}_{\mathfrak j}H +\sum_\lambda C_\lambda G_\lambda+aC+F_0D. \tag{14}\] There is a fixed bound \(\Delta\), polynomial in the complexities at this stage, such that every term on the right has degree at most \(T+\Delta\). Indeed, degree-compatible division produces quotient multipliers of degree at most \(T-\deg H_\alpha\) for a basis element \(H_\alpha\). Substituting its fixed generator representation adds its excess degree, at most \(\Delta\); it does not multiply \(T\).

Work modulo \((\mathfrak b,F_n,a^n)\). In (14), the \(aC\) term advances one layer, while \[F_0D\equiv-\sum_{j=1}^{n-1}a^jg_jD\] advances at least one layer. Starting with a polynomial of degree at most \(L\), apply this division to each successive coefficient of \(1,a,\ldots,a^{n-1}\). The coefficient being divided at layer \(i\) has degree at most \(L+B_0(i+1)\), for a polynomially bounded \(B_0\). To verify this induction, a jump by \(j\geq1\) adds at most \(\Delta+C(m+j+1)\leq B_0j\) to the degree. The sum of all jumps to layer \(i\) is \(i\), and there is no substitution at the same layer. We retain every remainder and require it to be zero.

When \(k=1\), the dimension hypothesis makes \(\mathfrak j=(1)\), so there are no remainder conditions. If \(\mathfrak j\ne(1)\), then \(k\ge2\), and the number of standard monomials of degree at most \(T\) is at most \[B_1(1+T)^{k-2},\] with \(B_1\) polynomially bounded. Here is an elementary uniform bound. If the monomial initial ideal has generators of degree at most \(D_1\), in a standard monomial at most \(\dim V(\mathfrak j)\) exponents can be at least \(D_1\). Otherwise its variables of large exponent would contain the support of a monomial generator, by the coordinate-prime description of the radical of a monomial ideal. Choose those variables and bound every other exponent by \(D_1-1\). This gives the asserted bound with a constant depending polynomially on \(D_1\), since \(M\) is fixed. If \(\mathfrak j=(1)\) there are no standard monomials and no conditions. Thus at most \[ B_1n(1+L+B_0n)^{k-2} \tag{15}\] linear conditions suffice when the ideal is proper.

Choose \(k\) algebraically independent functions among the ambient coordinate functions on \(W'\). Such a subset exists because these functions generate its function field. Let \(V_L\) be the span of their monomials of total degree at most \(L\), regarded as actual polynomials in the ambient coordinates. Then \[\dim V_L=\binom{L+k}{k},\qquad V_L\cap I(W')=0.\] This chooses independent polynomial representatives; no remainder map is presumed to descend canonically to classes modulo \(I(W')\). One can equally extend \(V_L\) to a linear complement of \(I(W')\cap\overline{\mathbb Q}[a,U,q]_{\leq L}\).

Set \(n=rL\). For a proper \(\mathfrak j\), the bound (15) is at most \[B_2r(1+B_0r)^{k-2}L^{k-1}\] for a polynomially bounded \(B_2\). A sufficiently large polynomial choice of \(L\) makes it smaller than \(\binom{L+k}{k}\). In the unit-ideal case the conclusion holds already for any nonzero source vector. There is consequently a nonzero element \(P\in V_L\) for which all remainders vanish. This proves both membership and \(P\notin I(W')\).

For completeness, the construction also gives the asserted affine height bounds. Division in bounded degree is a linear operation on a coefficient space of polynomial dimension. During division the leading monomial strictly decreases among the monomials of degree at most \(T\); thus its coefficient transformation is a product of polynomially many matrices with controlled affine heights. For composable matrices with inner dimension \(b\), the coefficient-tuple convention gives \[\mathrm h\!\left(AB\right)\leq\mathrm h\!\left(A\right)+\mathrm h\!\left(B\right)+\log b.\] Indeed, locally every entry is a sum of \(b\) products, and only infinite places contribute the factor \(b\). The \(n\) layer operations therefore have polynomial affine heights. Their source basis consists of monomials, so has height zero. A kernel vector obtained by minors and normalized to have a nonzero coordinate equal to \(1\) has affine height \(Q(d,n,L)\Psi(h)\).

The tracked divisions give a membership representation of bounded degree. Alternatively, apply the arbitrary-membership bound to \((\mathfrak b,F_n,a^n)\) and solve its coefficient equations with the chosen \(P\): the degrees and number of unknowns are polynomial in \(d,n,L\), and minors give the same affine height estimate. This bounds the scale of \(P,A_\lambda,A,B\) together. Since \(r,L,n\) are polynomial in \(d\), (13) follows with the stated bounds. ◻

Proof of 6. We descend through irreducible subvarieties \(W\subseteq X\) containing \(z_*\). At every step their degrees and degrees of chosen set-theoretic equations are polynomial in \(d\), and the affine heights of those equations are at most \(Q(d)\Psi(h)\). Initially take a component of \(X\) containing the point. The algebraic bounds above preserve these conditions under each operation below.

If \(a\) is constant on \(W\), its constant value has height at most \(Q(d)\Psi(h)\), by projection to the \(a\)-line and the component height bound. On the other hand the product formula and (8) give \[ \mathrm h\!\left(a_*\right)=\sum_vw_v\log\max(1,\left\lvert a_*\right\rvert_v^{-1}) \geq S_{\rm small}\geq h/Q_1(d). \tag{16}\] Hence \(h\leq Q(d)\Psi(h)\), which implies a polynomial bound in \(d\): for fixed \(c\), \(1+(\log h)^c\leq C_c\sqrt h\) for \(h\geq2\).

Suppose now that \(a\) is nonconstant and \(k=\dim W\geq1\). If \(a\) is nonconstant but \(V_W(a)\) is empty, this case can be settled without an analytic equation. For the controlled ideal \(\mathfrak b_W=(G_1,\ldots,G_t)\), the ideal \((\mathfrak b_W,a)\) is the unit ideal. The membership and coefficient bounds provide \[1=\sum_i A_iG_i+aB\] with polynomial degrees in \(d\) and joint affine height \(Q(d)\Psi(h)\). At the target \(1=a_*B(z_*)\). Evaluating \(B\) on the selected embeddings costs only its coefficient height and the small local sizes of \(U_*\), since \(\left\lvert a_*\right\rvert_v<1\). It follows that \[S_{\rm small}=\sum_{v\in\mathcal V}w_v\log\left\lvert B(z_*)\right\rvert_v \leq Q(d)\Psi(h).\] Again \(h\leq Q(d)\Psi(h)\) gives the required polynomial bound. We may therefore suppose that \(V_W(a)\) is nonempty.

The total number of generic normalization branches above \(V_W(a)\) is at most \(\deg W\). Indeed take one general transverse linear slice meeting every component \(D_j\) of this fibre at general affine points. The curve components have total degree at most \(\deg W\). Their normalizations contain the transverse germs above all these points. If a normalization divisor above \(D_j\) has residue degree \(f_i\) and \(e_i=\operatorname{ord}_i(a)\), it contributes \(\deg(D_j)f_ie_i\) to the zero divisor of \(a\) on the normalized slice. That divisor has degree equal to the polar degree of \(a\), at most \(\deg W\). In particular the number of branches is bounded. This counts residue degrees and ramification multiplicities, even though the normalization itself is birational.

At each such branch choose the lowest nonzero order among the \(E_\nu\). To avoid cancellation of these leading terms in a combination, set \[E=\sum_{\nu=1}^s t^{\nu-1}E_\nu\] for a positive integer \(t\). Each branch excludes at most \(s-1\) values of \(t\), since its leading-coefficient polynomial is nonzero. There is therefore a permissible \(t\) of polynomial size in \(d\). This combination preserves the coefficient and tail bounds: at finite places its coefficients are integral, and the extra triangle-inequality factor at infinite places has total logarithm polynomial in \(d\). Absorb it in the \(b_v\). Write \(E=\sum a^j e_j(U)\).

By 7, an integer \(m\) polynomially bounded in \(d\) can be chosen so that, for \[f=\sum_{j=0}^{m-1}a^je_j(U),\] one has \(\operatorname{ord}_{\mathfrak p}f <m\operatorname{ord}_{\mathfrak p}a\) on every relevant branch. Indeed the omitted terms have order at least \(m\operatorname{ord}_{\mathfrak p}a\), since all their coefficients are regular on the affine variety. Form the graph \(W'\) of \(q=f/a^m\). Its equations with controlled support are obtained by saturating \((\mathfrak b_W,a^mq-f)\) by \(a\); the closure of its support is exactly the graph. This takes place in one more variable and has polynomial degree and affine height bounds.

The target lifts to \(z_*'=(a_*,U_*,q_*)\). Its global height is bounded using the rational expression for \(q_*\): \[ \mathrm h\!\left(z_*'\right)\leq Q_5(d)h+Q_6(d)\Psi(h). \tag{17}\] Here \(Q_5\) is fixed at this stage, before the interpolation degree and order are chosen. Its selected local size must instead be estimated using the exact equation \(E(z_*)=0\). The tail bound gives \[ \left\lvert q_*\right\rvert_v =\left|a_*^{-m}\sum_{j\geq m}a_*^je_j(U_*)\right|_v \leq\exp((m+1)b_v). \tag{18}\] In particular the sum of the positive local logarithmic sizes of \((U_*,q_*)\) is at most \(Q_7(d)\Psi(h)\). The new equation and its finite truncations are \[F=q+\sum_{j\geq0}a^je_{m+j}(U),\qquad F_n=q+\sum_{j=0}^{n-1}a^je_{m+j}(U).\] At the target, \[ \left\lvert F_n(z_*')\right\rvert_v \leq\left\lvert a_*\right\rvert_v^n\exp((m+n+1)b_v). \tag{19}\]

By 8, \(V(I(W'),a,q+e_m)\) has dimension at most \(k-2\). For a controlled ideal \(\mathfrak b\) defining \(W'\) set-theoretically, \(V(\mathfrak b,a,q+e_m)\) is the same set and has the same dimension. Apply 9, with the ratio \(r=n/L\) to be chosen shortly. At the lifted target its identity becomes \[P(z_*')=A(z_*')F_n(z_*')+a_*^nB(z_*').\] Equations (7), (18), and (19), together with the coefficient and membership degree bounds, imply \[ \sum_{v\in\mathcal V}w_v\log\left\lvert P(z_*')\right\rvert_v \leq -n S_{\rm small}+Q_8(d,n,L)\Psi(h). \tag{20}\] All positive contributions here are sums of affine coefficient heights, positive local coordinate sizes, the \(b_v\), and infinite-place triangle-inequality terms. There is no error proportional to the number of finite places. At the remaining places evaluate \(P\) itself, rather than its membership representation. Since \(\deg P\leq L\), (17) gives \[ \sum_{v\notin\mathcal V}w_v\log\left\lvert P(z_*')\right\rvert_v \leq LQ_5(d)h+Q_9(d,n,L)\Psi(h). \tag{21}\]

The choices can now be made in the required order. The graph and \(m\) have already fixed \(Q_5\). Choose an integer \(r>2Q_1(d)(Q_5(d)+1)+1\), polynomially bounded in \(d\). Next choose the polynomially large \(L\) supplied by 9, and set \(n=rL\). If \(P(z_*')\ne0\), its product formula and (20)–(21) give \[0\leq \left(-\frac n{Q_1(d)}+LQ_5(d)\right)h +Q_{10}(d,n,L)\Psi(h).\] The coefficient of \(h\) is at most \(-L\). As \(n,L\) are now fixed polynomials in \(d\), this inequality again implies \(h\leq Q(d)\Psi(h)\) and hence a polynomial height bound. Thus, beyond a polynomial threshold chosen after \(m,r,L,n\), the value \(P(z_*')\) must be zero.

Since \(P\notin I(W')\), the intersection \(W'\cap V(P)\) has dimension at most \(k-1\). Project it to the original \((a,U)\)-space, take its closure, and choose an irreducible component containing \(z_*\). Such a component exists because \(a_*\ne0\) and the graph projection is an isomorphism over that open set. Projection cannot increase dimension. The degree and arithmetic bounds above preserve the induction hypotheses. Discard the graph coordinate \(q\); at the next step construct a new graph from the original common equations on this smaller variety. Consequently the ambient number of variables stays fixed.

The dimension drops at every continuing step, so there are at most \(\dim X\) such steps. A zero-dimensional final variety has constant \(a\) and is covered by (16). Taking the maximum of the finitely many polynomial thresholds proves the theorem. All component and coefficient field extensions made during the proof extend the selected local data by all embeddings, as specified above; they do not alter any smallness sum or introduce a field-degree factor. ◻

Proof. Fix a finite place. The semistable reduction theorem (Bosch et al. 1990, sec. 7.4, Theorem 1), followed by a further finite extension, gives semistable reduction and defines all geometric endomorphisms. They extend to the Néron model and preserve the torus in its connected special fiber. If that torus has positive rank \(r\), its rational character space is a nonzero unital module, on one side or the other, for the division algebra \(D=\mathop{\mathrm{End}}^0(A_{\overline K})\). Every finite dimensional module over a division algebra is free over it, so its rational dimension is a positive multiple of \(\dim_\mathbb QD=4\). This contradicts \(r\le2\). The toric rank is therefore zero, and the semistable model is an abelian scheme. The extension used here is local; no uniform bound on its degree is needed for an absolute height computation. ◻

Proof. The Galois action on the geometric endomorphism lattice has finite image in \(\mathrm{GL}_r(\mathbb Z)\), with \(r\le16\). Indeed a finite set of lattice generators is defined over a finite extension. A finite subgroup of \(\mathrm{GL}_r(\mathbb Z)\) injects into \(\mathrm{GL}_r(\mathbf F_3)\), because the kernel of reduction modulo \(3\) is torsion free. Its order is consequently bounded in terms of \(r\) alone. The fixed field of the kernel gives the asserted \(L\).

For a principally polarized abelian variety of fixed dimension, the endomorphism estimate of Masser–Wüstholz gives a rational spanning set of integral endomorphisms whose Rosati lengths are bounded by a fixed power of \[\max\{2,[L:\mathbb Q],h_F(A_t)\}.\] Here polynomial dependence on the field degree is essential. It is explicit in the Proposition of (Masser and Wüstholz 1994, sec. 4), with its parameter \(m\) equal to the dimension of the abelian variety; the alternative involving a smaller image is then zero. Its lattice Lemma 2.1 also gives a basis with the same kind of bound. See also (Daw and Orr 2022, Theorem 6.6 and Section 6.7) for the polynomial dependence in this application. The theta-height comparison (Pazuki 2012, Theorem 1.1 and Corollary 1.3), on one fixed theta level, and functoriality of heights on a fixed curve give \[h_F(A_t)\le C h.\] The degree of passage to that fixed level is bounded.

For clarity, the projection to the required plane can be made without target-dependent denominators. The map \[u\longmapsto 4(u+u^\dagger)-\mathop{\mathrm{Tr}}(u+u^\dagger)1\] preserves integrality, has symmetric trace-free image, and increases the Rosati norm by at most a fixed factor. By 3 its rational image has dimension two. Apply this map to the small spanning set and select two independent images. This proves (22). The norm and trace assertions are unchanged under field conjugacy: they are traces of actual endomorphisms and their Rosati adjoints. ◻

Proof. Global finite-place bounds. We first bound the denominators of integral endomorphisms in the rational frame. Potential good reduction supplies an integral de Rham lattice at each target; a fixed ideal measures the denominators of the frame and its dual relative to that lattice. Pass, for this argument only, to a fixed common cover carrying two coprime fine levels \(N_1,N_2\ge3\). The integral fine moduli scheme \(\mathcal A_2(N_i)\) is quasi-projective over \(\mathcal O_K[1/N_i]\) (Chai 1985, I, Theorem 1.4 and Section 1.7). Embed it in projective space over this ring, take its scheme-theoretic closure, and then take the closure of the graph of the curve map in the product with the fixed projective curve model. Denote this proper graph model by \(X_i\). Take its projective closure over \(\mathcal O_K\) when using global model heights. The inverse image \(U_i\) of the abelian interior carries the integral vector bundle \(\mathcal H_i\) of first de Rham cohomology and its dual. At a prime \(p\nmid N_i\), a target acquires good reduction by 10; its prime-to-\(p\) level extends after a finite local extension. Properness extends the point to \(X_i\). The good-reduction abelian scheme with prime-to-\(p\) level also supplies a valuation-ring point of the abelian interior. Its map to the projective closure agrees with the graph map on the generic fiber, hence everywhere by separatedness. The moduli image of the extended graph point therefore lies in the abelian interior. Consequently the extended point belongs to \(U_i\).

This last assertion concerns the moduli image. The point is allowed to reduce above a boundary point of the original curve. Thus we use \(U_i\), rather than a preassigned integral open of the curve, to compare lattices. The two choices of \(i\) cover every residue characteristic.

Choose a rational frame \(e_1,\ldots,e_4\) of \(\mathcal H\) such that it and its dual \(e^1,\ldots,e^4\) are regular at \(t_0\). Form on \(U_i\) the coherent denominator ideals \[\mathcal I_+=\{f:f e_j\in\mathcal H_i\text{ for all }j\}, \qquad \mathcal I_-=\{f:f e^j\in\mathcal H_i^\vee\text{ for all }j\}.\] These are coherent colon ideals, and extend to coherent ideals on the noetherian projective model \(X_i\) (The Stacks Project Authors 2026, Lemma 28.23.1(2), Tag 01PE). Their product, further multiplied by a denominator ideal for the fixed rational conversion from operator entries to our frame of \(\mathcal V\), gives a fixed ideal \(\mathcal J_i\). Multiplication by \(\mathcal J_i\) makes all the specified sections integral on \(U_i\). Remove its finitely many generic zeros from the curve. No duality assertion about arbitrary coherent extensions across the boundary is required. For a lifted target over a valuation ring \(R\), write \(\mathcal J_i(t)R=(\eta)\) and \(\Lambda=H^1_{\mathrm{dR}}(\mathcal A_t/R)\). The frame vectors belong to \(\eta^{-1}\Lambda\) and their duals belong to \(\eta^{-1}\Lambda^\vee\). An actual integral endomorphism preserves \(\Lambda\): it extends to the abelian scheme (Bosch et al. 1990, sec. 1.2, Proposition 8), and de Rham cohomology is functorial. Each of its matrix entries is therefore in \(\eta^{-2}R\). The extra fixed conversion of coordinates gives, for some fixed integer \(b\), \[ \log\max(1,\left\lVert P\right\rVert_v) \le b\lambda_{\mathcal J_i,v}(t)+c_v \quad(v\nmid N_i), \tag{25}\] where \(c_v=0\) outside finitely many fixed primes.

Here is explicitly why the right side has the claimed global cost. For a sufficiently ample fixed line bundle \(\mathcal L\) on \(X_i\), \(\mathcal J_i\mathcal L^n\) is generated by finitely many fixed sections \(s_0,\ldots,s_r\) (The Stacks Project Authors 2026, Proposition 28.27.13(7), Tag 01Q3). Choose \(s_k\) with \(s_k(t)\ne0\). With the model norm at a finite place, \(\lambda_{\mathcal J_i,v}(t)\le-\log\left\lVert s_k(t)\right\rVert_v\). The product formula for this nonzero element of the metrized line \(\mathcal L_t^n\) gives \[\sum_{v\nmid\infty}w_v\lambda_{\mathcal J_i,v}(t) \le n h_{\mathcal L}(t) +\sum_{v\mid\infty}w_v\log\left\lVert s_k(t)\right\rVert_v \le n h_{\mathcal L}(t)+C.\] The last bound uses the bounded sup norms of the finitely many fixed sections on the fixed projective complex model. The finite ideal heights are nonnegative, so this also bounds any restricted finite sum, such as the sum over \(p\nmid N_i\). Every generic projective curve and map used here is fixed, so \(h_{\mathcal L}(t)=O(h)\). Equation (25), summed using the two models, bounds all finite denominator contributions by \(O(h)\), independently of the local extension used for good reduction.

Global archimedean bounds. At an archimedean place the Hodge norm of an actual endomorphism is \(q_{\lambda_t}^{1/2}\), up to a fixed convention. In a rational de Rham frame, the norms of the frame and dual have at most a fixed power of the inverse distance to the finitely many omitted points, times powers of its logarithm. The algebraic Gauss–Manin connection is regular singular by (Katz 1970, Theorem 14.1); this property passes to its duals, tensor products and subquotients (Katz 1970, Proposition 11.3 and Remark 11.3.4). After a fixed ramified cover at each puncture, compare its algebraic logarithmic lattice with the canonical extension of the variation. Both frames and their inverses differ by meromorphic matrices with finite pole orders. Schmid’s norm estimates for the variation and its dual (Schmid 1973, Theorems 4.9 and 6.6\('\)) then give the asserted power and logarithmic bounds for the rational frame and its dual. The sum of the positive logarithms of these factors is \(O(h)\) by the local height inequalities for the fixed omitted divisor. The contribution of (22) is \(O(\log h+\log d)\). Affine base coordinates and \(a\) are fixed rational functions and have height \(O(h)\). This proves (23).

Bounds on the center disks. For (24), outside finitely many primes the family, parameter chart, frame, and dual are integral near the center, with the frame invertible. All integral endomorphism coordinates then have norm at most one on that residue neighborhood. At a fixed exceptional prime \(p\), extend the local coefficient field once so that the fixed center has good reduction and a chosen full \(M\)-level structure, where \(M\ge3\) and \(p\nmid M\). Adding this level to the original characteristic-zero curve gives a finite étale cover; at the chosen lift of the center, the \(p\)-adic inverse function theorem gives an analytic section over a sufficiently small center disk. Every sufficiently close target on the original branch therefore has a lift on this section after coefficient extension, including when the original level is divisible by \(p\). Forgetting the original level maps the lifted disk to the prime-to-\(p\) fine moduli scheme. On a smaller closed disk its coordinates lie in an integral open about the fixed center and reduce to the same special point. An integral de Rham frame on this moduli open pulls back to a frame whose change-of-basis matrix with the original frame, and its inverse, are analytic and bounded on the disk. Thus both the finite-place comparison cost and the base-coordinate norms are bounded by fixed local constants.

At infinity choose compact disks within the regular charts; the frame and dual Hodge norms are bounded there. Only \(O(\log h+\log d)\) remains from the Rosati norm. The sum of weights above a fixed rational prime is one, as is the sum of archimedean weights, which proves the restricted sum.

Finally a polarized isogeny of fixed multiplier carries Rosati norms isometrically under conjugation on operators. Its inverse introduces only a bounded denominator into an actual endomorphism. This changes the finite estimates at finitely many fixed primes by fixed constants, and proves the last assertion. ◻

Proof. Work on a smooth integral prime-to-\(p\) fine moduli chart \(\mathfrak M\) over a Witt base, and let \(z\) be the common special point. The relative first crystalline cohomology of the universal abelian scheme defines a convergent isocrystal \(\mathcal E\) on the special fiber of this chart. Its value at \(z\) is the crystalline cohomology of the actual abelian variety \(\overline A\) over \(z\). The canonical realization of \(\mathcal E\) on any smooth proper lifting is its relative de Rham cohomology with Gauss–Manin connection; see (Maulik and Poonen 2012, Theorems 4.16 and 4.26), the latter using (Ogus 1984, Theorem 3.8.2).

Here is the common evaluation explicitly. Over the local field \(F\) in the statement, put \(\mathfrak T=\operatorname{Spf}\mathcal O_F\langle Z\rangle\) and substitute \(X=aZ\) into the chosen inverse branch. Shrinking the fixed center disks ensures that the resulting chart coordinates belong to \(\mathcal O_F\langle Z\rangle\) and differ from their centered values by elements of \(\mathfrak m_F\mathcal O_F\langle Z\rangle\), where \(\mathfrak m_F\) is the maximal ideal of \(\mathcal O_F\). Thus the map to \(\mathfrak M\) reduces constantly to \(z\) on \((\mathfrak T\bmod p)_{\mathrm{red}}=\operatorname{Spec}k[Z]\). This remains true when \(F\) is ramified and \(a\notin p\mathcal O_F\): the image of \(\mathfrak m_F\) in \(\mathcal O_F/p\) is nilpotent. Consequently \(\mathfrak T\) is an enlargement of \(z\), and the crystal transition for its structural map to the point enlargement gives a canonical horizontal identification \[\mathcal E_{\mathfrak T} \simeq D\otimes_{F_0}\mathcal O_{\mathfrak T}[1/p].\] This is the constant-reduction construction of (Maulik and Poonen 2012, Definitions 4.10–4.13 and Section 4.7).

At \(Z=0\) and \(Z=1\), the relative comparison gives precisely the two maps \(c_{t_0}\) and \(c_t\) of the statement. Their compatibility with restriction to a fiber identifies them with the functorial absolute crystalline–de Rham comparisons; an explicit absolute statement with arbitrary ramification is (Kubrak and Prikhodko 2022, Appendix B, Proposition B.4). Proper formal GAGA identifies completed and ordinary de Rham cohomology. An endomorphism of either individual fiber extends over its good-reduction abelian scheme and is carried to its actual specialization on \(\overline A\). This argument needs only that individual endomorphism, without an extension to neighboring fibers. The common fiber is \(\overline A\) itself, with the same specialization maps and prime-to-\(p\) markings as in smooth proper base change. The construction does not introduce an auxiliary isogeny between special fibers.

The crystal identification is horizontal and is the identity when the two evaluations coincide. Therefore \(c_{t_0}^{-1}c_t\) is the identity-normalized horizontal transport, and uniqueness for the ordinary differential equation identifies its operator realization with \(Y(a)\). The resulting relative matrix identity is also discussed in (Papas 2025a, Theorem 3.4 and Remark 3.6).

Outside finitely many fixed primes our curve chart and connection are integral; the Taylor comparison converges, including over ramified extensions, whenever \[ -\log|a|_v>\frac{2\log p}{p-1}. \tag{27}\] Indeed the divided-power terms have denominators dividing the corresponding factorial. At the finitely many other primes, 10 for the fixed center and a prime-to-characteristic moduli chart give the same assertion on a fixed smaller disk.

Functoriality identifies both operator realizations with the operators \(\beta_i,\alpha_j\) on the same crystalline vector space, proving (26). The characteristic polynomial of an endomorphism of an abelian variety has integral coefficients and is independent of the Weil cohomology: it is characterized by the degree polynomial \(\deg(n-u)\) (Milne 1986, sec. 12). For the crystalline realization used here, the cohomological assertion can be seen directly. Rational crystalline cohomology is a Weil cohomology (Maulik and Poonen 2012, Remark 4.15). Cup product identifies the cohomology of the abelian special fiber with the exterior algebra on \(H^1\): comparison with its good lift reduces this to the corresponding statement for a complex abelian variety. Thus \([n]-u\) acts on top cohomology by \(\det(n-u\mid H^1)\); the degree formula identifies that action with \(\deg([n]-u)\). Equality for all sufficiently large integers \(n\) identifies the characteristic polynomials, and hence their traces. This applies also to \(\beta_i\alpha_j\), so its trace is integral and the pairing in (26) belongs to \(\tfrac14\mathbb Z\). Specialization preserves the polarization, Rosati involution, and these trace polynomials. Positivity of the Rosati form in characteristic \(p\) (Milne 1986, Theorem 17.3) gives \[|\mathop{\mathrm{Tr}}(\beta_i\alpha_j)|^2 \le \mathop{\mathrm{Tr}}(\beta_i^2)\mathop{\mathrm{Tr}}(\alpha_j^2) \le C(dh)^c.\] At infinity parallel transport identifies integral first cohomology lattices, so the cross pairing again belongs to \(\tfrac14\mathbb Z\). On the compact center disk the Hodge norms at the two points are uniformly comparable under this transport. Cauchy–Schwarz for the resulting positive norm and (22) give the same size bound. Notice that the two transported operators need not both be Hodge endomorphisms at the center in this last argument. ◻

Proof. Use the effective divisor \(D_0=[t_0]\) on the smooth completion \(\overline S\). Since it has positive degree, some fixed multiple of \(D_0\) minus the ample divisor defining \(h\) is ample. The height of an ample divisor is bounded below, and hence \[h_{D_0}(t)\ge c h-C.\] Take local Weil functions for this divisor. On the actual branch through \(t_0\) they equal \(-\log|x(t)|_v\) up to fixed bounded errors. Outside the chosen branch neighborhoods their positive contributions are bounded; the error is zero at all but finitely many primes. At an unexceptional prime the integral parameter chart is étale at the center, so the branch is precisely the Henselian inverse branch with that reduction. Other zeros of the rational function \(x\) are not counted. Before the factorial cutoff, the weighted proximity sum is therefore at least \(ch-C\).

Here is a uniform bound for the loss caused by that cutoff. Put \(D=[K(a):\mathbb Q]\le Cd\) and \(H=\mathrm h\!\left(a\right)\le Ch\). If a rational prime \(p\) has any place where \(|a|_v<1\), it contributes at least \((\log p)/D\) to \(\mathrm h\!\left(a^{-1}\right)=H\): writing a local valuation in integral units cancels its ramification index against the absolute weight. For the set \(\mathcal P(a)\) of these primes, \[ \sum_{p\in\mathcal P(a)}\log p\le DH, \qquad \#\mathcal P(a)\le DH/\log2. \tag{29}\] Write \(\alpha_p=\log p/(p-1)\). At places over \(p\) discarded by (27), the total weighted loss is at most \(2\alpha_p\), since their weights sum to at most one. For any set of \(N\) primes, \[\sum\alpha_p \le C\sum_{k=1}^{N}\frac{\log(k+1)}k \le C\bigl(1+\log^2(N+1)\bigr).\] Here \(\log x/(x-1)\) decreases for \(x>1\), and the \(k\)th smallest prime is at least \(k+1\). Apply this with (29). Restricting the finitely many exceptional and archimedean disks costs only a fixed further constant. This proves (28); its last assertion follows by elementary absorption of the logarithmic term for \(h\ge Q_0(d)\). ◻

Proof. First choose a field \(L\) defining the complete labeled tuple: the point, its two endomorphisms, and the fixed curve, center, and frames. By 11, \([L:\mathbb Q]\le Cd\). Let \(M\) be a normal closure. We do not bound \([M:\mathbb Q]\). Compute weights by extending every selected embedding to all embeddings of \(M\); every old weighted sum is preserved.

At infinity or a fixed exceptional prime, the four entries of the cross trace are the rational constants of 13. At any one such place type, an embedding of the labeled field \(L\) into its local algebraic closure determines the Taylor evaluation and the constant. There are at most \([L:\mathbb Q]\) possibilities. Group by these embeddings, by the finitely many fixed charts, and then by the constant matrix. This gives \(O(d)\) groups of the form (31), with \(\mathrm h\!\left(T\right)\le C(1+\log d+\log h)\). We count the constants actually obtained from these embeddings; we do not count all rational matrices of that height.

At an unexceptional prime \(p\), for each embedding \(\rho:M\hookrightarrow\overline\mathbb Q_p\), choose \(\sigma\in\operatorname{Gal}(\overline\mathbb Q_p/\mathbb Q_p)\) inducing the \(p\)th-power automorphism of \(\overline{\mathbf F}_p\). Apply \(\sigma\) to (26). The trace on the right is rational, and is fixed by \(\sigma\). Thus \[\left\langle\rho B,Y^\rho(\rho a)\rho P\right\rangle =\left\langle\sigma\rho B,Y^{\sigma\rho}(\sigma\rho a)\sigma\rho P\right\rangle.\] On \(L\), the embedding \(\sigma\rho\) equals \(\rho\tau\) for one of the at most \([L:\mathbb Q]\) embeddings \(\tau:L\to M\). Partition by \(\tau\) and the fixed conjugate charts. Define \((B',P',t',b)=\tau(B,P,t,a)\) in this group. This gives the same analytic equation (32) at every embedding of the group. The compositum \(L\tau(L)\) has degree at most \([L:\mathbb Q]^2\), so the field defining the coordinates of this equation has polynomial degree even though \(M\) need not.

A local automorphism preserves the absolute value, whence \(|b|_v=|a|_v\) and \(|R|_v=1\). Globally \(\mathrm h\!\left(R\right)\le\mathrm h\!\left(a\right)+\mathrm h\!\left(b\right)\le Ch\). On the common special fiber, relative Frobenius \(F:\overline A\to \overline A^{(p)}\) satisfies \(F\alpha=\alpha^{(p)}F\) for every endomorphism \(\alpha\), and similarly for the centered endomorphisms. Thus conjugation by \(F\), which is the normalized isometry on the operator system, carries each labeled vector to its primed vector. This records an identity of actual specialized endomorphisms, in addition to the de Rham equation. No restriction on the Newton polygon has been used.

For the local sums retain the selected embeddings of \(M\) with their absolute weights. If a later algebraic operation introduces another coefficient field \(F'\), work over \(MF'\) and take all extensions of those selected embeddings. The common equation, the equality \(|R|_v=1\), and the local norm bounds are preserved by restriction of absolute values, and the new weights sum to the old ones. This step requires no new choice of a Frobenius lift. Thus the field indexing the selected embeddings can be large, while the field defining the algebraic tuple has polynomial degree. The degree of the normal closure never enters the coordinate-degree, height, or grouping estimates.

There are in total polynomially many groups. Apply 14 and select one to obtain (30). Put the ordinary coordinates \(z,P\) and, where present, \(z',P',R\) in \(U\). The constants \(B,B'\) belong to fixed data; \(T\) is a coefficient of the equation, not an extra variable. Equations (33) require the base coordinates to be those of the selected inverse branches. Their inclusion keeps the number of variables and equations fixed and prevents another sheet of the rational parameter from being treated as the chosen germ. Equation (34) follows from 12, its conjugate version, and the local and global bounds for \(R\).

Coefficient heights. It remains to bound the common series, uniformly across its selected places. Outside the fixed exceptional set, the algebraic chart series and the connection matrix in that chart have integral Taylor coefficients. If \(Z(X)=\sum Z_jX^j\) is an identity-normalized horizontal matrix with integral ordinary connection matrix, its recursion shows inductively that \[Z_j\in (j!)^{-1}\mathrm M_r(\mathcal O_v).\] For example \((j+1)Z_{j+1}\) is a sum of \(Z_k\) times integral connection coefficients for \(k\le j\), and \(j!/k!\) is integral. The same holds for inverse horizontal matrices and the fixed rank-five system. Hence \(\left\lVert Z_j\right\rVert_v\le\exp(j\alpha_p)\) at a good prime. Integral algebraic chart series satisfy the stronger bound \(\left\lVert z_j\right\rVert_v\le1\).

At every fixed exceptional finite place the implicit function theorem gives a positive radius of convergence for each algebraic chart series; its coefficients and those of the connection are bounded by \(c_v^{j+1}\) for a fixed \(c_v\ge1\). The same recursion then bounds a horizontal coefficient by \(c_v^j|1/j!|_v\), after increasing \(c_v\). At infinity the ordinary analytic solution and the chart series have fixed positive convergence radii, so their coefficients grow at most exponentially. Since only finitely many \(c_v\) differ from one, summing these bounds and using \(\sum_{p}v_p(j!)\log p=\log(j!)\) gives, for example, \[\mathrm h\!\left(Z_0,\ldots,Z_j,z_0,\ldots,z_j\right) \le C(j+1)\log(j+2).\] Only this polynomial estimate for logarithmic heights is needed. In the second series \(b=aR\), the coefficient of \(a^j\) has an additional factor \(R^j\), explaining the linear degree bound. The remaining coefficients are fixed algebraic coefficients, together with the four rational constants \(T\) in the constant case. Their affine heights, including scale, have just been bounded. Polynomial operations of fixed degree and concatenation through order \(l\) preserve a polynomial height bound. This proves (35) for the complete coefficient tuple.

Local tails. At an unexceptional finite place put \(M_v=\max(1,\left\lVert U\right\rVert_v)\). The ultrametric inequality and \(|R|_v=1\) give, for a fixed integer \(b\) and every \(l\), \[\left\|\sum_{j\ge l}a^j e_j(U)\right\|_v \le M_v^b |a|_v^l\exp(l\alpha_p).\] For \(l=0\) the tail is the entire common relation and vanishes at the target. For \(l\ge1\) the constant matrix \(T\) does not occur in the tail. The displayed inequality follows by taking the maximum over \(j\ge l\); (27) makes those bounds decrease. There is no multiplicative constant greater than one at every prime. Fixed products or tensor expressions satisfy the same estimate after increasing \(b\) and the fixed coefficient bounds, because the total Taylor index controls the factorial bound.

At the finitely many exceptional places and at infinity, choose the disks strictly inside a fixed convergence disk. Cauchy estimates, or the local differential recursion, then give the same tail estimate with \(\alpha_p\) replaced by a fixed constant; at infinity the geometric-series sum contributes one fixed further constant. Consequently one may take \[b_v\le C\log M_v+C\alpha_p\] at the unexceptional primes, with fixed additional constants at the other places. The contribution of \(\alpha_p\) over all selected primes is \(O(1+\log^2(dh))\) by (29) and the calculation in 14. Summing and using (34) proves (36)–(37). All operations used involve a fixed number of fixed ordinary systems and their fixed tensor constructions, as asserted. ◻

Proof. Write \(b(v,w)=q(v+w)-q(v)-q(w)\). The nondegenerate ternary space \(A^\perp\) is isotropic over \(k\), so it has a decomposition \[A^\perp=\langle e,f\rangle\perp\langle c\rangle, \qquad q(e)=q(f)=0,\quad b(e,f)=1,\quad q(c)=\delta\ne0.\] Here and below angle brackets around vectors mean their linear span. For any \(v,w\in A\), the plane \[T_* = \langle e+v,c+w\rangle\] is disjoint from \(A\), and \(A+T_*=e^\perp\). This hyperplane has radical \(ke\). We can arrange that \(T_*\) and \(T\) have isometric restricted forms. A nondegenerate binary form over \(k\) represents every nonzero scalar: in the split case use the form \(xy\), and in the anisotropic case use surjectivity of the norm \(k_2^\times\to k^\times\). The value zero is represented by the zero vector. Consequently the following choices are possible:

1. If \(q|_T=0\), take \(v=0\) and \(q(w)=-\delta\).

2. If \(q|_T\) has rank one, write it as \(\operatorname{diag}(0,\beta)\) with \(\beta\ne0\), take \(v=0\), and choose \(q(w)=\beta-\delta\).

3. If \(q|_T\) has rank two, choose a vector of norm \(\delta\) in \(T\) and an orthogonal complement of norm \(\alpha\ne0\). Take \(q(v)=\alpha\) and \(w=0\). Then \(q|_{T_*}=\operatorname{diag}(\alpha,\delta)\).

Witt extension, which applies also to isometries between degenerate subspaces of a nondegenerate space in characteristic different from two (Elman et al. 2008, Theorem 8.3), extends the chosen isometry \(T\to T_*\) to an element of \(\operatorname{O}(V)(k)\). If its determinant is \(-1\), compose it with reflection in an anisotropic vector of \(T_*^\perp\). Such a vector exists: a three-dimensional subspace of a nondegenerate five-dimensional space cannot be totally isotropic. The reflection fixes \(T_*\) pointwise. We have therefore obtained \(u_0\in\mathop{\mathrm{SO}}(V)(k)\) with \(u_0T=T_*\).

It remains to correct the spinor norm. The setwise stabilizer of every \(2\)-plane \(E\) in \(\mathop{\mathrm{SO}}(V)(k)\) has surjective spinor norm. To see this, choose a hyperbolic plane \(\langle r,s\rangle\), normalized by \(b(r,s)=1\), and consider \[h_z(r)=zr,\qquad h_z(s)=z^{-1}s,\qquad h_z|_{\langle r,s\rangle^\perp}=1\quad(z\in k^\times).\] If \(R_v\) denotes reflection in \(v\), then \(h_z=R_{r+s}R_{r+zs}\), so its spinor norm is \(z\) modulo squares. When \(E\) is nondegenerate, take this hyperbolic plane inside the nondegenerate ternary space \(E^\perp\); then \(h_z\) fixes \(E\) pointwise. When \(E\) is degenerate, write \(E=\langle r,a\rangle\) with \(0\ne r\in\operatorname{rad}(E)\). Choose \(s_0\) with \(b(r,s_0)=1\) and \(b(a,s_0)=0\), and put \(s=s_0-q(s_0)r\). Then \(q(s)=0\) and \(a\perp\langle r,s\rangle\), so \(h_z\) preserves \(E\) setwise. This also treats the totally isotropic case.

Apply the preceding construction to \(E=T_*\), choosing the spinor norm of \(h_z\) to be the inverse of that of \(u_0\). The element \(u=h_zu_0\) still maps \(T\) to \(T_*\) and has trivial spinor norm. To see the lifting assertion directly, write an orthogonal transformation of determinant one as an even product of reflections (Schulze-Pillot 2020, Theorem 1.32). The corresponding product in the Clifford algebra has Clifford norm equal to the product of their quadratic norms. If this product is a square \(c^2\), rescaling the Clifford element by \(c^{-1}\) gives a norm-one element, hence an element of \(\mathop{\mathrm{Spin}}(V)(k)\) with the same orthogonal action (Schulze-Pillot 2020, Corollary 9.6 and Definition 9.7). Conversely a spin lift has trivial spinor norm. Thus \(u\) has the required lift. In dimension five over \(k\), this spin group is the split group \(\mathop{\mathrm{Sp}}_4\). ◻

Proof. The geometric monodromy subgroup in \(\mathop{\mathrm{Sp}}_4(\mathbb Z)\) is finitely generated, because \(S_*^{\mathrm{an}}\) has finitely generated fundamental group, and is Zariski dense by 4. Strong approximation for a finitely generated Zariski-dense subgroup of a simply connected absolutely almost simple group gives surjectivity onto \(\mathop{\mathrm{Sp}}_4(\mathbb F_\ell)\) for all sufficiently large primes \(\ell\) (Matthews et al. 1984, Theorem, p. 515). Choose such an \(\ell\), avoiding \(2\), the discriminant of the ambient quadratic lattice, the discriminant of \(L_{t_{0,*}}\), and every multiplier in \(\mathcal M\). After a fixed extension containing the relevant root of unity, full symplectic markings with the prescribed pairing form a torsor under \(\mathop{\mathrm{Sp}}_4(\mathbb F_\ell)\). Surjective geometric monodromy says exactly that its pullback to \(S_*\) is geometrically connected.

Fix a marking above \(t_{0,*}\). Its plane is nondegenerate modulo \(\ell\). For any target, its saturated plane still has dimension two modulo \(\ell\), and 16 supplies a change of marking in the image of \(\mathop{\mathrm{Sp}}_4(\mathbb F_\ell)\) giving (38). All markings occur on the same cover \(S\). Deleting finitely many points to choose frames and affine charts excludes only finitely many points downstairs. In particular, \(\ell\) has not been chosen to avoid the discriminants of the varying target planes.

At a common marked reduction, smooth proper base change identifies each integral \(\ell\)-adic cohomology lattice with that of the common reduction. The mod-\(\ell\) identifications agree with the markings. Take \(\mathbb Z_\ell\)-bases of the two saturated planes. Their four reductions are independent by (38), so some \(4\)-row minor of their coordinate matrix is an \(\ell\)-adic unit. These four vectors extend to a basis of \(\Lambda_\ell\); their sum \(M\) is a direct summand, and therefore satisfies the saturation equality in (39). Its reduction is precisely the degenerate space in (38). This proves local saturation at \(\ell\); no global saturation assertion about the sum of integral lattices is needed. Simultaneous conjugation carries both markings, both lattices, and their common pairing to the conjugate data, preserving independence and degeneracy. ◻

Proof. It suffices to treat one pair \((A_0,\lambda_0)\), \((A'_0,\lambda'_0)\) and one \(m\). Choose models over a localization of the ring of integers, with good reduction and with \(m\) invertible. A polarized isogeny \(\varphi\) with \(\varphi^*\lambda'_0=m\lambda_0\) has degree \(m^2\) and kernel contained in \(A_0[m]\). After a fixed finite extension and localization, all subgroup schemes of this fixed order and the required isotropic type in \(A_0[m]\) are the spreads of the finitely many characteristic-zero ones. Indeed \(A_0[m]\) is finite étale, and after splitting it these are subgroups of one fixed finite group.

Each possible kernel gives a fixed polarized quotient \((D,\lambda_D)\), with the quotient map pulling \(\lambda_D\) back to \(m\lambda_0\). The remaining data are polarized isomorphisms \((D,\lambda_D)\simeq(A'_0,\lambda'_0)\). Here isomorphisms are group isomorphisms of abelian schemes, preserving the zero section and the polarizations. Their scheme is unramified by rigidity and is of finite type: for relatively ample line bundles representing the polarizations, the restriction of their product to such a graph is numerically twice the source line bundle, and hence has a fixed Hilbert polynomial. Thus these graphs lie in one relative Hilbert scheme. The fibers of the polarized-isomorphism scheme are torsors under the finite polarized group-automorphism groups (Milne 1986, Proposition 17.5), so the scheme is quasi-finite.

A quasi-finite scheme of finite type over a ring of integers, with finite generic fiber, becomes the spread of that generic fiber after deleting finitely many primes. For completeness, discard its finitely many vertical irreducible components; its finitely many generic points then spread to a finite scheme after localization, and deleting the support of the remaining complement gives the assertion. Further localization makes this scheme finite étale. After a fixed extension splitting its generic fiber, this finite étale scheme is a disjoint union of sections: each component has generic degree one over the normal base. Its geometric points are therefore exactly the specializations of the fixed characteristic-zero maps, including every polarized-automorphism choice. Applying this to the finite list of kernels proves the lemma. ◻

Proof. From a branch identity to monodromy. Suppose that every equation is an identity on such a branch. On the normalization near a general point of its divisor, \(a\) vanishes and all affine coordinates, including \(R\) when present, are regular. The series in \(a\) and \(aR\) therefore converge after a complex embedding in a neighborhood of that point. In a transverse parameter on the normalization, the formal identity makes every coefficient zero at the generic point of the divisor. Convergence therefore gives an identity on a nonempty analytic open. Any fixed denominator cleared in a chart is nonzero at the target, hence is a nonzero function on \(W\) and has finite order on each normalization branch. It cannot turn a nonzero series into an identity. If one of the chart equations is not an identity we already have the desired conclusion; otherwise they identify the series arguments with the actual projections to \(S\) and \(S'\). The first projection is nonconstant since \(a\) is nonconstant.

Pass to a smooth dense open of \(W\) on which these projections and the algebraic frames are defined. An irreducible complex algebraic variety has connected smooth locus, so the analytic identity continues on its universal cover. At a fixed point of this open the vector coordinates are single-valued, whereas continuation changes the transport matrices by the monodromy of the pulled-back systems. A dominant map to a curve has fundamental-group image of finite index after deleting suitable closed subsets and passing to a resolution; thus 4 applies on this open. The vector pairs are generically independent, because they are independent at the target. A pair minor nonzero at the target is not identically zero on \(W\), so its nonvanishing locus meets this analytic open even if the pair vanishes on the divisor itself. They need not remain independent at every boundary point of \(W\).

Constant relations and independent projections. For (40), fix a general point of this open. The orbit of the transport matrix under continuation has Zariski closure a translate of \(\mathop{\mathrm{SO}}_5\). For nonzero vectors \(b,v\) in its standard representation, the function \(g\mapsto\left\langle b,gv\right\rangle\) is not constant. Indeed, the span of the differences \(gv-v\) is invariant, and is the whole representation by irreducibility and the absence of invariant vectors. A constant matrix coefficient would annihilate this span. Applying this to \(B_1\) and \(P_1\) contradicts the continued identity.

For (41), if the second projection is constant, only the first system varies and the same argument applies. If it is nonconstant, 4 gives product monodromy unless the closure of the paired image is contained in a dominating isogeny correspondence. In the latter case 5 puts the image in the fixed finite list; since these correspondences are closed, the target also belongs to that list, contrary to this case’s defining assumption. For product monodromy, fix the second transport matrix and vary the first. The resulting constant matrix coefficient is again impossible.

The exceptional relation forces a common sign. Consider now (43). At a general point \(z\) of \(W\), write \(Y_z:\mathcal V_z\to\mathcal V_{t_0}\) for transport. The continued identity holds with transport \(gY_z\) for every \(g\in\mathop{\mathrm{SO}}(\mathcal V_{t_0})\). The standard representation is absolutely irreducible, so Burnside’s theorem says that these matrices span \(\mathop{\mathrm{Hom}}(\mathcal V_z,\mathcal V_{t_0})\). For each \(i,j\), the linear functional \[M\longmapsto\left\langle B_i,MP_j\right\rangle-\left\langle D_i,M\widehat P_j\right\rangle\] therefore vanishes on every linear map \(M\) between these two spaces. The corresponding rank-one tensors are equal. Identifying their first factors with covectors by the fixed center trace pairing, we obtain \[ B_i\otimes P_j=D_i\otimes\widehat P_j \qquad(1\le i,j\le2). \tag{44}\] These are polynomial equations in the algebraic vector coordinates. They hold on a dense open of \(W\), and therefore at the target itself. There, all four pairs of factors are nonzero. The \((1,1)\) equation gives \(D_1=c_0B_1\) and \(\widehat P_1=c_0^{-1}P_1\) for some nonzero scalar \(c_0\); the \((2,1)\) and \((1,2)\) equations give the same scalar for the other indices. The self-pairing of \(B_1\) is a nonzero rational number, and is unchanged by conjugation or the isometry \(I_0\). Thus \(c_0^2=1\), and \[ D_i=\varepsilon B_i,\qquad \widehat P_j=\varepsilon P_j, \qquad \varepsilon\in\{1,-1\}. \tag{45}\] The \((2,1)\) and \((1,2)\) equations are essential here: they couple the two labels and give a single sign for both pairs.

Specialization to a Frobenius quasi-isogeny. We now use that the vectors in (45) come from endomorphisms. De Rham realization on the rational endomorphism algebra of a characteristic-zero abelian variety is faithful: base change to \(\mathbb C\), compare with Betti cohomology, and use the faithful action on its rational first homology. Conjugation by an isogeny preserves the Rosati self-adjoint subspace. Consequently (45) asserts equalities of actual rational endomorphisms in the center and target fibers. These equalities specialize under good reduction.

Choose any place in the nonempty selected group and let \(p\) be its residue characteristic. The unprimed center and target have a common marked reduction \(\overline A\), and the primed reductions identify with its \(p\)-twist. Write \(F:\overline A\to\overline A^{(p)}\) for relative Frobenius, and write \(\overline\varphi_0\) for the common specialization of \(\varphi_0\) and \(\varphi_s\). The map \[u=\overline\varphi_0^{-1}F\in\mathop{\mathrm{End}}^0(\overline A)\] is a self-quasi-isogeny with polarized multiplier \(p/m\). Relative Frobenius transports the unprimed labeled endomorphisms to their primed specializations. The specialized equalities (45) therefore say that the conjugation action \(g\) of \(u\) on the quadratic \(\mathbb Q_\ell\)-space acts by \(\varepsilon\) on both specialized endomorphism planes. The sign action extends to their saturated \(\mathbb Z_\ell\)-lattices; the chosen small pairs need not be bases of those lattices. By 3, \(g\) belongs to \(\mathop{\mathrm{SO}}_5\).

Excluding the two signs. Let \(T\) be the rational span of those two planes in \(\mathop{\mathrm{End}}^0(\overline A)\). Its \(\ell\)-adic realization has dimension four by 17. The trace form on its rational symmetric endomorphisms is positive definite by Rosati positivity in arbitrary characteristic (Milne 1986, Theorem 17.3); hence \(T\) is nondegenerate over \(\mathbb Q\), and its realization \(T_\ell\) is nondegenerate over \(\mathbb Q_\ell\). The one-dimensional perpendicular is consequently \(g\)-stable. Since \(\det g=1\) and \(\det(g|_{T_\ell})=\varepsilon^4=1\), \(g\) acts as \(+1\) on that perpendicular.

If \(\varepsilon=+1\), it follows that \(g=1\). The kernel of the conjugation representation of \(\mathop{\mathrm{GSp}}_4\) on this quadratic space is its scalar center by 3. Put \(\mu=p/m\). With the covariant convention of 2, the action on first cohomology is \[T_u=\mu(u^*)^{-1}=(u^\dagger)^*;\] the equality follows from \(u^\dagger u=[\mu]\). Thus \((u^\dagger)^*=c\mathrm{id}\) for some \(c\in\mathbb Q_\ell\). The ordinary cohomological trace of the rational quasi-endomorphism \(u^\dagger\) is rational (Milne 1986, Proposition 12.9), so \(4c=\mathop{\mathrm{Tr}}(u^\dagger)\in\mathbb Q\). Faithfulness gives \(u^\dagger=c\mathrm{id}\), and applying Rosati gives \(u=c\mathrm{id}\). Its multiplier would satisfy \(c^2=p/m\). This is impossible because \(p\nmid m\), so the \(p\)-adic valuation of \(p/m\) is odd.

If \(\varepsilon=-1\), \(g\) is an involution with negative eigenspace exactly \(T_\ell\). Both \(F\) and \(\overline\varphi_0\) have multipliers prime to \(\ell\), so their actions are invertible on the integral Tate modules; conjugation by \(u\) preserves the integral quadratic lattice \(\Lambda_\ell\). This lattice is unimodular by the choice of \(\ell\). Since \(\ell\) is odd, the integral projectors \((1+g)/2\) and \((1-g)/2\) give an orthogonal direct sum \[\Lambda_\ell=\Lambda_+\perp\Lambda_-, \qquad \Lambda_-=\Lambda_\ell\cap T_\ell.\] Each summand is unimodular, so \(\Lambda_-/\ell\Lambda_-\) is nondegenerate. But 17 identifies \(\Lambda_-\) with the sum \(M\) of the two marked saturated plane lattices, whose reduction is degenerate by (39). This is the required contradiction.

Both cases contradict the assumed identity. The use of Rosati positivity and rational endomorphism trace is valid for every reduction type; no classification or ordinarity assertion about Frobenius has been used. ◻

Proof. If the target locus is finite, enlarge the constants to cover it. Otherwise use the initial family and center, the finite multiplier list, and then the fixed prime and connected marking cover, in the order already specified. For each target choose one lift satisfying 17; this does not enlarge any fixed datum. Remove the finite exceptional targets of the affine charts, and suppose that its height \(h\) exceeds a sufficiently large fixed polynomial in \(d=\max\{2,[K(t):K]\}\). The fields generated by the labeled single or paired point tuples in 15 have degree polynomial in \(d\). A normal closure is used only to index selected embeddings. A chosen fixed-multiplier isogeny requires only a bounded relative extension over the field of its two point arguments, so adjoining it preserves this polynomial bound. Selected embeddings are extended by all extensions; the new weights above an old embedding sum to its old weight, preserving the normalized smallness sum.

Arithmetic inputs. Apply 15, with the fixed exceptional primes chosen above. Partition first into its constant and Frobenius groups, and, if needed, subdivide the latter by the finite center-isogeny choice. There are polynomially many groups, and one supplies a common list of equations and a selected weighted set \(\mathcal V\) with \[\sum_{v\in\mathcal V}w_v\log\frac1{\left\lvert a\right\rvert_v} \ge \frac{h}{Q_1(d)},\qquad \sum_{v\in\mathcal V}w_v\log^+\left\lVert U\right\rVert_v \le Q_2(d)(1+\log^c h).\] The tuple \((a,U)\) has global logarithmic height at most \(Q_3(d)h\). The coefficient degrees are \(O(j+1)\), and the complete affine coefficient tuple through order \(l\), including scale, has logarithmic height at most \(Q_4(l,d)(1+\log^c h)\). The tails satisfy the hypotheses of 6. These assertions also hold for the transformed target pair \(\widehat P\) in the exceptional case: its fixed denominator and preserved Rosati norms give exactly the size hypotheses of 12, and only the original ordinary connection in \(a\) is used. In the other paired case \(\left\lvert R\right\rvert_v=1\) at every selected place. Thus its two ordinary systems are evaluated at \(a\) and \(aR\), as required. The chart equations use fixed algebraic ordinary germs.

Nonidentity and interpolation. Start the interpolation argument inside the fixed affine algebraic conditions on the base coordinates described above. They have bounded complexity. No inverse Gram matrices or inverse minors of the target endomorphism pairs have been adjoined. Every irreducible subvariety encountered in the dimension descent contains the target. If \(a\) varies there, 19 supplies nonidentity on every relevant branch, including branches on which a chart equation already fails. If \(a\) is constant, the stopping case in 6 applies directly.

6 now gives \(h\le C_3d^{C_4}\) for the selected lifts. Its constants are uniform because the cover, centers, allowed ordinary systems, and isogeny multipliers are fixed.

Descent from the selected lifts. The degree of the marking cover is fixed. If \(t_*\) is the image of \(t\), then \[[K(t):K]\le (\deg(S\to S_*))[K(t_*):K].\] Choose the ample height upstairs as the height of the pullback of an ample divisor downstairs; it equals the downstairs height up to a bounded error, and arbitrary fixed ample heights on these curves are comparable up to fixed positive constants and bounded errors. Thus the bound for the selected lift descends to \(t_*\). The further normalization and fine-level maps descend in the same way by 2. For any other lift \(\widetilde t\) in a fixed cover, height comparison and \([K(t_*):K]\le[K(\widetilde t):K]\) give the same numerical bound. Thus only the selected lifts require (38). At singular coarse points there are finitely many normalization preimages, all included among the finite exceptions. Increasing the constants includes all finitely many omitted points. The absolute height bound for a selected lift also holds for all its conjugates over the fixed data field, even when those conjugates were not selected relative to the original marking of \(t_0\); no fresh incidence choice is needed to deduce that numerical bound. All fixed enlargements of the data field \(K\) change the degree bounds by at most fixed factors. ◻

Proof. We first describe the charts and their metric estimates, including the punctures of \(S\). A compact subset away from the finitely many points of \(\bar S\setminus S\) has a finite cover by relatively compact simply connected coordinate disks. Shrink these disks inside slightly larger disks on which the flat trivializations and the period maps are holomorphic. Their real and imaginary parts are restricted analytic functions, hence definable.

At a puncture choose a rational function \(u\) on \(\bar S\) which is a local parameter there. Enlarge \(K\) once to define these finitely many choices. The local monodromy is quasi-unipotent. On a fixed local cover \(u=w^e\) it is unipotent, say \(\exp N\), with \(N\) nilpotent. The one-variable nilpotent orbit theorem (Schmid 1973, Theorem 4.9) gives, in the compact dual, the expression \[ \Phi(u)= \exp\left(\frac{\log w}{2\pi i}N\right)\Psi(w), \tag{47}\] where \(\Psi\) extends holomorphically across \(w=0\). Cover a sufficiently small punctured \(u\)-disk by finitely many open angular sectors of width less than \(2\pi\), and choose a branch of \(w\) and of \(\log w\) on each. The exponential in (47) is a polynomial in \(\log w\). The branch of \(w\) is algebraic, its argument is bounded, and \(\log\left\lvert w\right\rvert\) is definable in \(\mathbb R_{\mathrm{an},\exp}\). After shrinking the radius, \(\Psi\) is restricted analytic in an affine chart about \(\Psi(0)\). Thus the projective map (47) is definable. These local covers are used only to describe functions on the sectors; no point-dependent global cover is introduced.

Write \(\left\lVert\cdot\right\rVert_{H,u}\) for the Hodge norm. Schmid’s abstract one-variable norm estimate (Schmid 1973, Theorem 6.6\('\)), applied both to \(\mathbb V\) and to its dual, bounds the norms of each fixed flat basis and dual basis by a power of \(1+\log(1/\left\lvert u\right\rvert)\), uniformly on each sector. Indeed every fixed flat vector belongs to some step of the monodromy weight filtration, and that theorem supplies such a power; only finitely many basis vectors occur. Passing from \(w\) to \(u=w^e\) merely changes the constants. Consequently, with \(e_1,\ldots,e_5\) an integral flat basis and \(e_1^*,\ldots,e_5^*\) its dual, we have fixed \(c_0,r>0\) such that \[ \max_i\bigl\{\left\lVert e_i\right\rVert_{H,u}, \left\lVert e_i^*\right\rVert_{H,u}^{\vee}\bigr\} \le c_0\bigl(1+\log(1/\left\lvert u\right\rvert)\bigr)^r. \tag{48}\] On the compact charts the corresponding bound is constant. In particular (48) controls the metric and its inverse in flat coordinates.

For completeness, the logarithm in this bound can be controlled at an individual complex embedding, not merely on average. Let \(F=K(t)\) and let \(\sigma:F\hookrightarrow\mathbb C\) fix the chosen embedding of \(K\). Whenever \(\sigma(t)\) belongs to the sector, \(u(t)\ne0\), and the absolute height identity \(\mathrm h\!\left(u(t)^{-1}\right)=\mathrm h\!\left(u(t)\right)\) gives \[\begin{align*} \log^+\frac1{\left\lvert\sigma(u(t))\right\rvert} &\le [F:\mathbb Q]\mathrm h\!\left(u(t)\right)\\ &\le c_1d(t)h(t). \tag{49}\end{align*}\] The first inequality follows by retaining that archimedean summand in the height of \(u(t)^{-1}\), using the usual nonsquared absolute value. For the second, the fixed rational function \(u\) defines a morphism \(\bar S\to\mathbb P^1\), whose height is bounded above by a constant times \(1+h_{\bar S}\). This also explains why algebraic local parameters were chosen.

Let \(v\) be either of the two chosen endomorphism vectors, at any such embedding. If \(v\) realizes the symmetric endomorphism \(\alpha\), the Weil operator fixes its type \((0,0)\), and its rationality gives \[\left\lVert v\right\rVert_H^2=b(v,v)=\tfrac14\mathop{\mathrm{Tr}}(\alpha^2) =\tfrac14 q_{\lambda_t}(\alpha).\] The bound in 11 therefore gives \(\left\lVert v\right\rVert_{H}\le c_2(d(t)h(t))^{r_2}\). If \(v=\sum_i n_i e_i\), the dual norm inequality is \[\left\lvert n_i\right\rvert=\left\lvert e_i^*(v)\right\rvert \le\left\lVert e_i^*\right\rVert_{H}^{\vee}\left\lVert v\right\rVert_{H}.\] Combining this with (48) and (49) proves a polynomial bound on the numerical sizes of all ten integers. This is precisely (46).

The vectors are realizations of actual integral endomorphisms. Conjugation preserves integrality, independence, and the rational Rosati trace norms. Although labeling the endomorphisms may require an extension of \(K(t)\) of bounded relative degree, each embedding of \(K(t)\) extends to that field. Conjugating the labeled pair along any such extension supplies the asserted pair at the conjugate point. Neither the degree estimate for \(u(t)\) nor the number of conjugate points is diminished by this labeling extension. ◻

Proof. The set in question is stable under \(\mathop{\mathrm{Gal}}(\overline{\mathbb Q}/K)\). If its degrees were unbounded, take a point \(t\) of sufficiently large degree \(d=[K(t):K]\). Its full orbit has \(d\) distinct points on \(S(\mathbb C)\), all satisfying (50). Assign each conjugate to one chart of 21. If there are \(J\) charts, one chart contains at least \(d/J\) of these distinct points. For each of them choose a conjugate of the labeled pair at \(t\). Their integral coordinate tuples have \[ H_{\mathrm{rat}}(y)\le T=c_3d^B \tag{51}\] for fixed \(c_3,B>0\).

In the selected chart write \(z\in D\subset\mathbb R^2\) for the curve coordinate and \(\Phi(z)\in Q_V\) for the period line. Identify the flat real space with \(\mathbb R^5\). Consider the fixed definable set \[ Z=\left\{((v_1,v_2),z)\in(\mathbb R^5)^2\times D: \begin{array}{l} (b(v_i,v_j))_{i,j=1}^2\text{ is positive definite},\\ b(v_1,\Phi(z))=b(v_2,\Phi(z))=0 \end{array}\right\}. \tag{52}\] Orthogonality to a projective line means orthogonality to any nonzero representative; it is a well-defined algebraic incidence condition. The coordinate \(z\) is injective on the chart. The tuples just chosen thus form a set \(\Sigma\subset Z\) with at least \(d/J\) distinct second projections and rational first coordinates bounded as in (51).

Choose \(\eta>0\) with \(B\eta<1\). The finitely many chart incidences have a common counting constant, obtained by taking their maximum. For large enough \(d\) we have \[d/J>c_Z(\eta)(c_3d^B)^\eta.\] The quoted corollary therefore gives a path \(\beta(s)=(\gamma(s),z(s))\) in \(Z\), with \(\gamma\) semialgebraic and \(z\) nonconstant. It remains to show that this path would force the period image into a proper algebraic subset of \(Q_V\). The first projection supplies at most one algebraic parameter, and for each nondegenerate pair the orthogonal period lines form a conic.

Let \(\mathcal A\subset(\mathbb C^5)^2\) be the complex Zariski closure of \(\gamma((0,1))\). A semialgebraic set of real dimension at most one has real Zariski closure of dimension at most one, and complexification preserves that dimension. Hence \(\dim_{\mathbb C}\mathcal A\le1\). Equivalently, after a finite semialgebraic decomposition, the arc has Nash coordinate functions, which are algebraic over its real parameter; their algebraic closure has transcendence degree at most one.

Crucially, impose nondegeneracy before closing the orthogonality incidence. Let \[\mathcal U= \{(v_1,v_2):\det(b(v_i,v_j))_{i,j=1}^2\ne0\} \subset(\mathbb C^5)^2,\] and define the locally closed algebraic set \[\mathcal I= \{(v_1,v_2,[w])\in(\mathcal A\cap\mathcal U)\times Q_V: b(v_1,w)=b(v_2,w)=0\}.\] For each \((v_1,v_2)\in\mathcal A\cap\mathcal U\), their span is a nondegenerate two-space. Its perpendicular is a nondegenerate three-space, whose isotropic lines form a smooth projective conic. Every fiber of \(\mathcal I\to\mathcal A\cap\mathcal U\) therefore has dimension one, so \[\dim_{\mathbb C}\mathcal I\le2.\] Let \(X\subset Q_V\) be the Zariski closure of its image under the second projection. The dimension of an image and its closure cannot exceed the dimension of its source, whence \[ \dim_{\mathbb C}X\le2. \tag{53}\] This construction closes only the incidence over \(\mathcal A\cap\mathcal U\); it does not add arbitrary fibers over degenerate pairs. Since the pairs on \(\gamma\) have positive definite Gram matrix, they lie in \(\mathcal U\), and consequently \(\Phi(z(s))\in X\) for all \(s\in(0,1)\).

The continuous nonconstant path \(z(s)\) contains infinitely many distinct points accumulating inside the chart. Indeed one may restrict it to a closed interior interval on which it remains nonconstant. The inverse image \(\Phi^{-1}(X)\) is a complex analytic subset of a connected one-dimensional chart. The identity theorem therefore makes it the entire chart. Continuing the defining homogeneous polynomial identities along \(S\) gives \[\widetilde\Phi(\widetilde S)\subset X,\] where \(\widetilde S\) is the connected universal cover and the flat trivialization defining \(\widetilde\Phi\) extends the one in the chosen chart.

Fix a period line \(L\) in this image. Equivariance places its entire monodromy orbit \(\Gamma L\) in \(X\). By 4, \(\Gamma\) is Zariski dense in \(\mathop{\mathrm{SO}}(V_{\mathbb C},b)\), and this group acts transitively on the nonsingular isotropic quadric \(Q_V\). The orbit \(\Gamma L\) is therefore Zariski dense in \(Q_V\). This contradicts (53) and \(\dim Q_V=3\).

Degrees in (50) are thus bounded. That inequality also bounds their ample heights, and Northcott’s theorem on the fixed projective curve \(\bar S\) proves finiteness. ◻

Proof of 1. Suppose the stated locus on \(C\) were infinite. By 2, normalize \(C\) and take a connected component of a fixed fine-level cover containing infinitely many lifted target points. Removing the finitely many points needed for the smooth family and the algebraic charts leaves infinitely many. Choose an interior QM center and then the fixed auxiliary cover of 17. All these choices, including their fields of definition, are made once.

For every remaining target to which the construction is applied, 20 supplies a selected lift on this fixed cover with \(h(t)\le A d(t)^\kappa\), for fixed \(A,\kappa\). The degrees of a point and its lift are comparable by the fixed degree of the finite map, and the ample heights are comparable by the pullback of ample divisors, as in 2. Thus the constants are uniform as the quaternion algebra and the target vary.

Apply 22 on the fixed covering curve. Its counting argument uses the full orbit over the fixed field \(K\): the height inequality for a selected lift is invariant under that orbit, and conjugation preserves its two integral Hodge vectors and their Rosati bounds. A conjugate marking need not satisfy the incidence imposed relative to the chosen center. That incidence was used to prove the height inequality; the incidence (52) depends only on the target’s Hodge vectors. Hence all these selected lifts belong to the finite set of 22, whose final step uses Northcott’s theorem. Their images on \(C\) form a finite set. Adding back the finitely many removed points contradicts the assumed infinitude. This proves the asserted finiteness for full geometric endomorphism algebra an indefinite quaternion division algebra over \(\mathbb Q\), with no restriction on its discriminant or order. ◻