Historical setting

Hilbert’s tenth problem, as stated in his 1900 list of mathematical problems, asks for a finite procedure deciding whether a polynomial equation with integer coefficients has a solution in the ordinary integers (Hilbert 1900, Problem 10). Its negative resolution grew out of a much stronger description of polynomial equations. A relation on the positive integers is recursively enumerable if an algorithm can list exactly its satisfying tuples. It is Diophantine if membership can be expressed by the existence of additional positive integers satisfying a polynomial equation with integer coefficients. Thus a Diophantine description expresses a condition entirely through polynomial solvability.

Davis, Putnam, and Robinson proved in 1961 that every recursively enumerable relation has such a representation if exponentiation is also allowed. They reduced the ordinary polynomial representation theorem to a growth criterion for a Diophantine relation (Davis et al. 1961, Theorem and Corollary 3). Matiyasevich supplied the required growth in 1970 through a Diophantine description of a Fibonacci relation, completing the theorem that every recursively enumerable relation on the positive integers is Diophantine (Matiyasevich 1970). The equivalent all-integer decision problem is therefore undecidable. This representation theorem explains the strength of the result: integer polynomial equations can encode arbitrary recursively enumerable conditions.

Letting the unknowns range over \(\mathbb Q\) changes the decision problem. Enumerating rational tuples will eventually find any rational zero, but gives no answer for an equation without one. A possible transfer of the integer theorem would be an existential definition of membership in \(\mathbb Z\) using only the ring operations of \(\mathbb Q\). One could then replace each integer unknown by a rational unknown subject to that membership condition, obtaining an equivalent rational existential question. More generally, a Diophantine model of integer arithmetic represents the integers by a Diophantine subset of some \(\mathbb Q^k\), with Diophantine relations for addition and multiplication. Such a model also translates integer equations into rational existential questions (Cornelissen and Zahidi 2000, Definition 3.1 and Observation 3.3). Here the defining polynomial equations and all their witnesses are interpreted over \(\mathbb Q\). These constructions are sufficient routes to undecidability; the decision problem does not require its solution to produce either one.

Julia Robinson showed in 1949 that the integers are first-order definable in the rationals using addition and multiplication. Her formula uses both universal and existential quantifiers, and her work also proves undecidability of the full first-order theory of rational arithmetic (Robinson 1949, Theorems 3.1 and 4.2). Later work reduced the quantifier complexity. Poonen defined \(\mathbb Z\) in \(\mathbb Q\) by a positive formula with two universal quantifiers followed by seven existential quantifiers (Poonen 2009, Theorem 4.1). Koenigsmann then gave a universal definition of \(\mathbb Z\); equivalently, its complement in \(\mathbb Q\) is Diophantine. His result also implies undecidability of the \(\forall\exists\) theory (Koenigsmann 2016, Theorem 1 and Corollaries 2–3). These results give substantial control of definitions inside \(\mathbb Q\), but do not supply the existential membership condition for the direct transfer above.

The distinction between a Diophantine model and undecidability also has a geometric setting. Mazur conjectured that, for a variety over \(\mathbb Q\), the closure of its rational points in its real points has only finitely many connected components (Mazur 1995, Conjecture 3). He observed that this conjecture would rule out a Diophantine definition of \(\mathbb Z\) in \(\mathbb Q\). Cornelissen and Zahidi showed that it would also rule out a Diophantine model of integer arithmetic in a finite Cartesian power of \(\mathbb Q\) (Cornelissen and Zahidi 2000, Theorem 3.6). These obstructions to definitions and models do not by themselves rule out a negative answer to the rational decision problem.

One approach studies the subrings \(\mathbb Z[S^{-1}]\subseteq\mathbb Q\), whose denominators use only primes in \(S\). Poonen constructed disjoint recursive sets of primes \(T_1,T_2\), each of natural density zero, such that positive-integer arithmetic has a Diophantine model in \(\mathbb Z[S^{-1}]\) whenever \(S\supseteq T_1\) and \(S\cap T_2=\varnothing\). Hilbert’s tenth problem is undecidable for every such ring. Taking \(S\) to be the complement of \(T_2\) gives a recursive set of density one and a proper subring of \(\mathbb Q\) (Poonen 2003, Theorem 1.3). The construction uses selected multiples of a point on a rank-one elliptic curve whose real coordinates approximate integers.

Eisenträger, Miller, Park, and Shlapentokh later constructed computably presentable subrings \(\mathbb Z[S^{-1}]\), including examples in which \(S\) has density one, whose Hilbert’s tenth problem is Turing equivalent to that over \(\mathbb Q\) (Eisenträger et al. 2017, Theorem 3.17). For these examples, a Turing reduction from integer solvability would already give a Turing reduction from integer solvability to the rational field problem. The density of the inverted primes alone therefore does not settle the relation to the field problem.

Elliptic curves also give a transfer between rings of integers of number fields. Poonen proved that, for \(F\subseteq K\), an elliptic curve \(E/F\) with \[\operatorname{rank}E(F)=\operatorname{rank}E(K)=1\] makes \(\mathcal O_F\) Diophantine in \(\mathcal O_K\) (Poonen 2002, Theorem 1). If \(\mathbb Z\) is already Diophantine in \(\mathcal O_F\), undecidability transfers to \(\mathcal O_K\). Mazur and Rubin pursued this route through quadratic twists and Selmer groups, conditional on the conjectured evenness of \(\dim_{\mathbb F_2}\operatorname{Sha}(E/L)[2]\). Assuming this for every elliptic curve over every number field, they obtained undecidability for every infinite finitely generated \(\mathbb Z\)-algebra (Mazur and Rubin 2010, Theorem 1.13, Corollary 1.14, and Theorem 8.1).

This number-field program has recently been completed unconditionally. Koymans and Pagano combined two-descent with additive combinatorics to prove that \(\mathbb Z\) is Diophantine in the ring of integers of every number field, and that Hilbert’s tenth problem is undecidable for every infinite finitely generated \(\mathbb Z\)-algebra (Koymans and Pagano 2026). Alpöge, Bhargava, Ho, and Shnidman gave another route: for every quadratic extension \(K/F\) of number fields, they construct an abelian variety over \(F\) with positive rank unchanged over \(K\), and deduce that \(\mathbb Z\) is Diophantine in every number-field ring of integers (Alpöge et al. 2026, Theorem 1.1 and Corollary 1.2). The field \(\mathbb Q\) is not a finitely generated \(\mathbb Z\)-algebra, so these results concern a different class of rings from the field in Theorem 1.

The elliptic approach also illuminates nearby rational decision problems. In a rank-one group, a multiple \(nP\) of a nontorsion point can represent the integer \(n\), and local coordinates or heights can reveal arithmetic information about that index. Cornelissen and Zahidi used elliptic divisibility sequences and existential valuation predicates in this direction. Under a conjectural odd primitive-divisor property for one such sequence, they obtained a Diophantine model of \((\mathbb Z,+,\mid)\) and proved undecidability of a positive \(\Pi_2\) fragment with one universal quantifier (Cornelissen and Zahidi 2007, Conjecture 3.16 and Theorems 3.17 and 5.3). The assertion with one universal quantifier uses the correction by Cornelissen and Shlapentokh (Cornelissen and Shlapentokh 2009, Remark 2.3 in the author version).

Garcia-Fritz, Pasten, and Vidaux instead adjoined predicates comparing the logarithmic heights of rational tuples to the rational ring language. In this expanded language they gave a positive existential interpretation of \(\mathbb N\), and hence proved undecidability; comparisons between tuples of length at most three suffice (Garcia-Fritz et al. 2025, Theorems 1.1–1.2). Their construction uses elliptic canonical heights to encode consecutive squares. The present paper concerns ordinary rational polynomial solvability, and its height estimates enter the proof of that ordinary statement.

Proof strategy

Constant polynomials can be decided directly. For each nonconstant integral polynomial \(f\), we construct an effectively generated sequence of finite tests. Each test can be answered by finitely many rational-solvability queries, and the sequence satisfies \[ f\text{ has an integer zero} \quad\Longleftrightarrow\quad \text{every test succeeds}. \tag{1}\] An integer zero supplies witnesses for all the tests. If a rational-solvability algorithm existed, we could run two searches in parallel: enumeration of integer zeros, and enumeration of tests until one fails. The arithmetic work is to prove that a failed test exists whenever \(f\) has no integer zero.

Suppose all the finite tests succeed. Compactness then gives a ring \(R\) containing a zero \(\mathbf a=(a_1,\ldots,a_n)\) of \(f\), embedded in an elementary extension of the rational field; we use a tilde for embedded values and points. The ambient extension also carries the integer and valuation data, height functions, elliptic multiple maps, and functions for finite products used in the proof. The tests attach an elliptic point \(T_a\) to each \(a\in R\), compatibly with addition, and require \(T_a\) to be nonidentity when \(a\ne0\). The embedded points lie in the transferred copy of a fixed infinite cyclic subgroup \(\mathbb ZP\) of finite index in a rank-one elliptic group, with \(T_1=P\). This defines unique indices \[\widetilde T_a=\eta(a)P, \qquad \eta(a)\in{}^*\mathbb Z.\] Here \({}^*\mathbb Z\) is the ambient integer structure, whose elements may be nonordinary. The point axioms make \(\eta:R\to{}^*\mathbb Z\) injective and additive, with \(\eta(1)=1\), but \(\eta\) need not respect multiplication. Thus the modeled equation does not yet give an equation satisfied by the indices.

Two local comparisons address this obstruction. The primary test compares an embedded ring value with a quotient of truncated elliptic logarithms. The formal logarithm can recover the corresponding quotient of point indices once that quotient is integral at the prime. A second comparison, on the conjugate elliptic curve, supplies this control using only additivity of \(\eta\). Where the comparisons apply, they show that every \(\widetilde a\) is integral at the prime and agrees with \(\eta(a)\) to the prescribed precision. Each pair uses auxiliary choices that work for every ring element, which is essential for the later uniform bound.

The proof arranges these comparisons at primes detected by a height estimate. Fix the five rational points of \(\mathbf P^1\) supplied by that estimate and primitive integral linear forms vanishing at them; call these the contact forms. For a rational point \(s=(u:v)\) outside this set, written in coprime integer coordinates, let \(M(s)\) be the product, without repetition, of the nonexceptional primes at which one of these forms has odd valuation. We call the primes counted by \(M(s)\) its contact primes. The estimate bounds the absolute logarithmic Weil height by \[h(s)\le H M(s)^c\] for fixed constants \(H,c\). This definition and estimate transfer to the ambient extension. At each prescribed precision, the theory represents every ring element as a sum of three fractions of ring elements, called slopes, and equips factors of their contact forms with the local conditions. A prime-pattern theorem supplies the required primes for these representations in the ordinary integer model. A separate existential condition controls the parity of possible denominator valuations outside a fixed finite set. In a model satisfying the tests, consider the three slopes supplied for any ring element at that precision. After the further fixed exclusions, the parity condition lets the proof apply the local comparisons at every contact prime of those slopes. Enlarging the exceptional set changes the height constant but preserves the exponent \(c\).

Set \[\delta=f\bigl(\eta(a_1),\ldots,\eta(a_n)\bigr), \qquad B=\max\bigl(1,|\eta(a_1)|,\ldots,|\eta(a_n)|\bigr).\] If \(\delta=0\), elementarity gives an ordinary integer zero of \(f\). Otherwise \(\delta\) is a nonzero ambient integer whose size is bounded by a fixed polynomial in \(B\). Choose one ordinary precision \(K\) large enough in terms of the degree of \(f\) and the exponent \(c\). For any \(A\in R\), choose a tested representation at precision \(K\). Every contact prime of its slopes divides this same \(\delta\) to order at least \(K\), because all modeled root coordinates agree locally with their indices. The height estimate then bounds each slope height by a constant times \(B\). The logarithmic height inequality for their sum gives \[h(\widetilde A)\ll_f B \qquad\text{for every }A\in R,\] with one ordinary constant.

For each nonzero root coordinate \(a_j\), consider its attached point \(T_{a_j}\). Applying the uniform bound to the ring elements representing \(x(T_{a_j})\) gives \(h(x(\eta(a_j)P))\ll_f B\). The transferred canonical-height comparison for \(P\) makes this height grow quadratically in \(|\eta(a_j)|\). If \(B>1\), choose \(j\) with \(|\eta(a_j)|=B\); the two bounds give \(B^2\ll_f B\). If \(B=1\), the required bound is immediate. Thus all root indices lie in an ordinary finite interval. Additivity and injectivity identify the modeled root coordinates with those ordinary integers, contradicting \(\delta\ne0\). This proves the finite-test equivalence and hence the undecidability theorem.

The detailed reduction in Section 2 makes these steps precise. Section 3 proves the pole-parity condition; its passage from two-descent data to rational points uses the pointwise \(2\)-converse (OpenAI 2026a, Theorem 1.1) and the Cassels–Tate pairing (Cassels 1962). Section 4 proves the five-point height estimate. Its modularity step uses the odd regular Fontaine–Mazur theorem at \(2\) (OpenAI 2026b, Theorem 1.1), whose statement has no hypothesis on the residual image. Section 5 proves the rank-one and local elliptic results, and Section 6 supplies the prime patterns and simultaneous local witnesses for the ordinary integer model. Section 7 gives Turing degree \(0'\) and the quartic normal form.

At rational primes, valuations are normalized by \(v_q(q)=1\), with \(v_q(0)=+\infty\). The support of a nonzero rational number is the set of primes dividing its numerator or denominator in lowest terms. The notation \(X\ll Y\) means \(X\le CY\) for a positive constant depending only on the fixed data in the statement; additional dependencies are indicated when they occur.

Elliptic indices and the local comparison

Fix the number field, involution, and elliptic curve \[ F=\mathbb Q(\sqrt2),\qquad d=75-53\sqrt2,\qquad E:\ y^2=x^3-d^2x, \tag{2}\] where \(\sigma\) is the nontrivial automorphism of \(F/\mathbb Q\). The following result, proved in Section 5, supplies the integer indices.

We now describe the point data that the recursive theory will impose. Let \(R\) be an integral domain of characteristic zero for which \(\sqrt2\notin\mathop{\mathrm{Frac}}(R)\), and write \[R_F=R[\sqrt2],\qquad F_R=\mathop{\mathrm{Frac}}(R)(\sqrt2).\] For each \(a\in R\), consider a point \(T_a\in E(F_R)\) subject to \[ \begin{gathered} T_0=O,\qquad T_1=P,\qquad T_{a+a'}=T_a+T_{a'}\quad(a,a'\in R),\\ T_a\in mE(F_R)\quad(a\in R),\\ a\ne0\Longrightarrow T_a\text{ is finite and }x(T_a)y(T_a)\ne0. \end{gathered} \tag{3}\] The theory will supply six unary ring-valued functions \(x_0,x_1,x_2,y_0,y_1,y_2\). For \(a\ne0\), they are required to satisfy \(x_2(a)y_2(a)\ne0\) and \[ x(T_a)=\frac{x_0(a)+x_1(a)\sqrt2}{x_2(a)},\qquad y(T_a)=\frac{y_0(a)+y_1(a)\sqrt2}{y_2(a)}. \tag{4}\] Their values at \(a=0\) are unrestricted, since \(T_0=O\). In the integer model we will take \(T_a=aP\). Representing the coordinates by ring elements will also let a uniform height bound on \(R\) control their heights.

For the arithmetic analysis, consider an elementary extension of the ordinary many-sorted structure containing \(\mathbb Q,\mathbb Z,\mathbb R,F\), the prime and valuation relations, the height functions, the elliptic multiple maps, and the finite-product functions. We call elements of the original structure ordinary; starred sorts satisfy the same first-order statements with ordinary parameters. For the moment, suppose that \(R\) embeds in the rational sort \({}^*\mathbb Q\) of such an extension. Its fraction-field interpretation embeds \(F_R\) in \({}^*F\). A tilde denotes the image of a ring element, field element, or point. Later compactness will supply an extension and an embedded ring with these properties.

The identity \(mE(F)=\mathbb ZP\) transfers to this extension: every point of \(mE({}^*F)\) has a unique expression \(nP\), with \(n\in{}^*\mathbb Z\). Define \[ \widetilde T_a=\eta(a)P. \tag{5}\] The group-law axioms give \[ \eta(a+a')=\eta(a)+\eta(a'),\qquad \eta(1)=1. \tag{6}\] If \(\eta(a)=0\), then \(T_a=O\), and the finite-point axiom forces \(a=0\). Thus \(\eta:R\to{}^*\mathbb Z\) is injective and additive. Conjugating (5) gives \(\widetilde T_a^\sigma=\eta(a)P^\sigma\). We make no assumption about \(\eta(aa')\).

For nonzero \(h\in R\), the index ratio of \(T_{ha}\) and \(T_h\) is \(\eta(ha)/\eta(h)\), which need not equal \(\eta(a)\). We will recover this ratio locally from point parameters, and use a second comparison to identify it with \(\eta(a)\) to the required precision.

For a finite point with \(y\ne0\), put \(z=-x/y\), and set \(z(O)=0\). Every nonzero multiple of \(P\) avoids the two-torsion points, so \(z(nP)\) is defined for every integer \(n\). Write \[\ell(Z)=Z+\sum_{t\ge2}c_tZ^t\in F[[Z]],\qquad \ell_K(Z)=Z+\sum_{2\le t\le K}c_tZ^t\] for the formal logarithm in this parameter and its truncation through degree \(K\). The truncations are effectively computable from the fixed equation. Section 5 proves the following uniform statements about integer multiples of the fixed \(F\)-rational points \(P,P^\sigma\).

The two estimates provide different kinds of local information. Equation (7) determines the valuation of an index ratio from the point parameters without assuming that the ratio is integral. Once an index ratio is known to be integral, (8) approximates it by a ratio of finite polynomial evaluations. The next lemma combines both forms of control.

Reduction to \(O\) has its usual valuation meaning, transferred to the elementary extension. Only the stated assertions about integer multiples of the fixed \(F\)-rational points \(P,P^\sigma\) and evaluations of the finite polynomials \(\ell_K,\ell_K^\sigma\) are transferred; no infinite formal series is evaluated in the extension.

The same \(h\) must serve every \(a\), because the calculation uses both \(a\) and \(1\). We will impose ring-language versions of these comparisons and obtain their valuation hypotheses at primes detected by a height estimate.

Contact primes and ordinary integer witnesses

Throughout the paper, \(h\) denotes the absolute logarithmic Weil height. The next input identifies primes whose product controls the height of a rational point. It is proved in Section 4.

Multiplying \((u,v)\) by a nonzero rational scalar adds the same valuation to both terms in (11), so each contact order is independent of the chosen homogeneous coordinates. It is nonnegative after scaling the coordinates to be integral and primitive at \(q\). With fixed primitive integral coordinates, the nonzero integers \(L_b(u,v)\) have only finitely many prime divisors, so the product defining \(M_S(s)\) is finite.

For a positive integer \(K\) and a nonzero integer \(e\) satisfying \(M_S(s)^K\mid e\), the estimate gives \[h(s)\le H|e|^{c/K}.\] In the reduction, the same polynomial error will play the role of \(e\) for representations of every ring element. This is why the exponent \(c\) must be fixed, although its precise value is not needed. The following immediate consequence permits the further fixed exclusions needed by the local tests.

To apply this estimate uniformly, the theory will write each ring element as a sum of three fractions and impose local conditions on factors of their contact forms. The following ordinary integer result supplies witnesses for those axioms. It is a direct consequence of the prime-pattern lemma and the Chinese remainder calculation proved in Section 6.

The two places in part (2) are the two residue branches above a split prime. At either one, conditions (i) and (ii) use the two conjugate curves at that same place. If \(\mathfrak p\) is the prime ideal of \(\mathbb Z[\sqrt2]\) corresponding to \(v_r\), its valuation ring is \((\mathbb Z[\sqrt2])_{\mathfrak p}\), with uniformizer \(r\). Thus the two valuation inequalities in part (2) are exactly membership in \(r^K(\mathbb Z[\sqrt2])_{\mathfrak p}\); for a finite indicated point, reduction to \(O\) is equivalent to \(1/x\in r(\mathbb Z[\sqrt2])_{\mathfrak p}\). We now encode these ordinary witnesses by formulas that also make sense for a general ring.

A recursive theory with an ordinary integer model

Let \(\mathcal T\) be a theory in a countable recursive expansion of the language of rings. Its ring \(R\) is required to be an integral domain of characteristic zero with \(\sqrt2\notin\mathop{\mathrm{Frac}}(R)\). Add the six unary ring-valued functions above. Interpret \(T_0\) as \(O\) and, for \(a\ne0\), require the two fractions in (4) to define the coordinates of \(T_a\in E(F_R)\), with both denominators nonzero. Impose the point conditions (3).

Use the five points \(\mathcal C\), primitive contact forms \(L_b\), and finite multiplier set \(\mathcal K\) of the preceding inputs. For each ordinary integer \(K\ge1\), add the following representation axiom. For every \(A\in R\) there exist \(u_1,u_2,v\in R\), with \(v\ne0\), such that, on setting \[u_3=Av-u_1-u_2,\] each member of \[ v,\qquad L_b(u_i,v)\quad (1\le i\le3,\ b\in\mathcal C\setminus\{\infty\}) \tag{14}\] equals \(\varepsilon k r\), where \(\varepsilon\in\{1,-1\}\), \(k\in\mathcal K\), and \(r\in R\) has the following properties. The choices of \(r\) and its auxiliary data are made separately for the different members of the list.

First require \(r\ne0\), \(rR\) proper, and \(R/rR\) a field. Require \(B_r\in R\) satisfying \[B_r^2\equiv2\pmod{rR},\qquad 2B_r\text{ invertible in }R/rR.\] These conditions define two maximal ideals of \(R_F\): \[ \mathfrak p_\pm =\ker\bigl(R_F\longrightarrow R/rR,\ \sqrt2\longmapsto\pm B_r\bigr). \tag{15}\] They are the two residue branches to be tested. In the integer model they correspond to the two places of \(F\) above a supplied split prime \(r\). In a general model they are defined by the quotient field, independently of an ambient valuation.

For each \(\mathfrak p\in\{\mathfrak p_+,\mathfrak p_-\}\), require nonzero \(h,j_0\in R\), chosen at that branch and independently of \(a\), such that the following hold for every \(a\in R\). A point is called formal for this localization if it is \(O\), or if \(1/x\in r(R_F)_{\mathfrak p}\).

1. The points \(T_h,T_{ha}\) are formal for this localization, \(\ell_K(z(T_h))\ne0\), and \[ \frac{\ell_K(z(T_{ha}))}{\ell_K(z(T_h))}-a \in r^K(R_F)_{\mathfrak p}. \tag{16}\]

2. The points \(T_{j_0}^{\sigma},T_{(h-4)a}^{\sigma}\) are formal for the conjugate curve at the same localization, and \[ \frac{z(T_{(h-4)a}^{\sigma})}{z(T_{j_0}^{\sigma})} \in r^K(R_F)_{\mathfrak p}. \tag{17}\]

Here \(z(O)=0\), and the denominator in (17) is nonzero by \(j_0\ne0\) and (3). The pairs \(h,j_0\) may differ between the two branches. The scheme contains every ordinary \(K\); the termination proof will later select a single precision after \(f\) is fixed.

All these conditions have recursive first-order expressions in \(R\). The absence of \(\sqrt2\) from \(\mathop{\mathrm{Frac}}(R)\) is expressed by \[\forall a,b\in R\quad b\ne0\Longrightarrow a^2\ne2b^2.\] Elements of \(R_F\) are pairs of ring elements, and elements of \(F_R\) are fractions \((A+B\sqrt2)/D\), with \(A,B,D\in R\) and \(D\ne0\). Equations with fixed coefficients in \(F\) can therefore be separated into two coordinates and cleared of denominators. The elliptic group law, including the identity and exceptional addition cases, is given by finitely many field equations and inequations. Membership in \(mE(F_R)\) is expressed by existential coordinates for a point whose \(m\)-multiple is the prescribed point. These conditions are ring formulas under the same interpretation.

Saying that \(R/rR\) is a field means that \(1\notin rR\) and every element outside \(rR\) has an inverse modulo \(rR\). For \(D'\in F_R\), the assertion \[ D'\in r^i(R_F)_{\mathfrak p_\pm} \tag{18}\] means that \(D'=r^iX/Y\) for some \(X,Y\in R_F\) such that the evaluation of \(Y\) under \(\sqrt2\mapsto\pm B_r\) is nonzero modulo \(rR\), hence a unit in that field. These representations and inverse conditions are expressed by ring quantifiers. The exponent \(i\) is fixed in each axiom; neither variable exponentiation nor a valuation function is part of \(\mathcal T\).

Finite rational tests and compactness

The finite tests will also impose a positive existential condition on every ring value. This input holds on all integers and controls the parity of possible poles; Section 3 constructs it.

The second property does not rule out even pole orders. Its later use rests on a simple product observation. If \(q\notin S_1\) and nonzero \(a,c\in\mathbb Q\) satisfy \(\Phi(a)\) and \(\Phi(ac)\), with both \(v_q(a)\) and \(v_q(ac)\) negative, then \[v_q(c)=v_q(ac)-v_q(a)\in2\mathbb Z.\] Section 2.5 applies this observation when \(\Phi\) holds on every element of the ring supplied by the finite tests.

An integer zero can therefore supply rational witnesses for all the conditions imposed below. In the converse direction, pole parity will connect a contact prime in the ambient integer structure to one of the tested localizations of the embedded ring.

Assume that an algorithm decides rational solvability of integral polynomials. Given \(f\in\mathbb Z[X_1,\ldots,X_n]\), the constant case can be settled immediately, so assume \(f\) is nonconstant. Add constants \(\mathbf a=(a_1,\ldots,a_n)\) and the equation \(f(\mathbf a)=0\) to \(\mathcal T\), and Skolemize the resulting theory. This is effective for a recursive first-order theory: after prenex conversion, each existential quantifier is replaced by a function of the preceding universal variables. All resulting function symbols are ring-valued. The Skolemized axioms are universal sentences with quantifier-free matrices.

A ground term is a term with no variables, built from the named constants using the ring operations and added functions in this language. A compatible assignment of values to all ground terms will interpret the added functions on the ring of term values. Enumerate these terms, associate a rational variable to each, and enumerate the following constraints:

1. the constants \(0,1\) and the ring operations have their usual values;

2. equal input tuples have equal outputs for each function symbol;

3. every ground instance of every universal axiom holds;

4. the value of every ground term satisfies the positive existential formula \(\Phi\) from Proposition 9.

The witnesses for \(\Phi\) in (d) are auxiliary rational variables. They are not values of additional functions of the ring and do not create new ground terms subject to (d). The same distinction applies to the inverse variables introduced only when translating a finite test to polynomial equations; they are not added to the term-generated ring.

Every finite initial part of this list is decidable by the assumed rational-solvability algorithm. Indeed, after introducing the witnesses for \(\Phi\), the finite collection is an existential Boolean combination of polynomial equalities and inequations. An inequation \(g\ne0\) becomes \(gy=1\) with a new variable; a finite Boolean formula can be put into disjunctive normal form; and a conjunction \(g_1=\cdots=g_s=0\) is equivalent over \(\mathbb Q\) to \(\sum_i g_i^2=0\). Clearing fixed rational denominators gives finitely many integral polynomial queries.

If any test fails, report that \(f\) has no integer zero. This answer is sound: an integer zero extends the model of Lemma 8 and admits Skolem functions; all its ground-term values are integers and hence satisfy \(\Phi\).

We now analyze any model and embedding supplied by Lemma 10. All starred objects refer to its one ambient extension. The point conditions give the additive injection \(\eta\) of (5)–(6).

Odd valuations select a tested branch

The height radical supplies primes of the ambient integer sort, whereas the tests are imposed at ideals above an element \(r\in R\). We now connect these two kinds of data. Fix an ordinary precision \(K\ge1\). Enlarge the finite exceptional set \(S\) in the height estimate to a finite set \(S_K\) containing: the exceptional set for \(\Phi\), the supports of all multipliers in \(\mathcal K\), the primes needed to keep the contact forms distinct with their required unit determinants, the bad or ramified primes of the elliptic data and \(F\), and the finitely many primes excluded by Proposition 3 at precision \(K\).

Write \(M_K(s)=M_{S_K}(s)\) for the radical with these larger exceptions. Corollary 6 keeps the exponent unchanged: if \(M(s)=M_S(s)\) is the original radical, then \[M(s)\le\left(\prod_{q\in S_K\setminus S}q\right)M_K(s), \qquad h(s)\le H_K M_K(s)^c\] with an ordinary constant \(H_K\). Enlarge \(c\) to a positive integer once and for all if necessary.

Choose the representation axiom at precision \(K\) for any \(A\in R\), and put \(\widetilde s_i=(\widetilde u_i:\widetilde v)\). All its contact forms are nonzero, so \(\widetilde s_i\notin\mathcal C\). Consider any ambient prime \(q\) counted in some \(M_K(\widetilde s_i)\). Here and below a radical, prime, or valuation on starred sorts is the transfer of its ordinary definition.

Let \(t=\min\{v_q(\widetilde u_i),v_q(\widetilde v)\}\). At \(q\notin S_K\), the minimum of the valuations of the five contact forms equals \(t\): all their coefficients are integral and a pair has unit determinant. Since some contact depth is odd, these five valuations cannot all have the same parity. In particular one of them is odd. For its expression \(\varepsilon k r\) in (14), \(k\) is a \(q\)-unit, and hence \[ v_q(\widetilde r)\text{ is odd}. \tag{19}\]

The odd valuation links this ring element \(r\) to the ambient prime \(q\), but it does not make the tested localization a subring of the ambient valuation ring: \(R\) may have denominators at \(q\). We will quotient the value group by the convex subgroup generated by the negative valuations of nonzero ring elements. Pole parity ensures that the odd value of \(r\) survives positively in this quotient. The center of the resulting valuation will then select one of the two tested branches.

Index congruences at every contact prime

The height contradiction

The formula and the pole restriction

For now, let \(\mathcal M\subset\mathbb Q^\times\) be a finite nonempty list, and for each \(m\in\mathcal M\) let \(\mathcal D_m\subset\mathbb Q^\times\) be a finite nonempty list. The construction below associates a formula \(\Phi\) to these lists. Let \(S_1\) be a finite set containing \(2\), \(3\), and the supports of every member of all the lists. We will make one fixed choice of the lists in the next subsection.

For \(l\in\mathbb Q^\times\), write \[ E_l:\quad y^2=x(x-l)(x+3l). \tag{30}\] The three roots are distinct, so this is an elliptic curve with its point at infinity \(O\) as origin. Define \(\Phi(b)\) to hold if \(b=0\), or if \(b\ne0\) and, for some \(m\in\mathcal M\), \(m_d\in\mathcal D_m\), and \(t,e\in\mathbb Q\), the following conditions hold: \[ d=m_db,\qquad b'=mb,\qquad U=b\frac{1-dt^2}{1+dt^2},\qquad H'=b'(b'-U^2), \tag{31}\] \[1+dt^2\ne0,\qquad eH'\ne0,\] and each of \(E_e\) and \(E_{eH'}\) has a rational point whose \(x\)-coordinate belongs to \(b'\mathbb Q^{\times2}\).

For each fixed choice of the lists, this is a positive existential formula. The choices of multipliers are finite disjunctions. A nonzero condition \(a\ne0\) is the existential equation \(az=1\); the condition on \(x\) is expressed by \(x=b'z^2\) with \(z\ne0\). Introduce \(U\) as a variable and multiply its defining equality by \(1+dt^2\). All remaining conditions are polynomial equalities with fixed rational coefficients, which can be cleared to integer coefficients.

Preparing integer witnesses

We now choose the multiplier lists so that the formula also holds on every integer. These lists will be fixed independently of the input \(b\). Choose a finite list \(\mathcal M\subset\mathbb Q^\times\) with the following property: for every \(b\in\mathbb Q^\times\), some \(m\in\mathcal M\) makes \[ b'=mb>0,\qquad b'\in\mathbb Q_2^{\times2},\qquad b'\in\text{the unit nonsquare class of }\mathbb Q_3^\times. \tag{32}\] Here the last condition means that \(b'\) has even \(3\)-adic valuation and nonsquare unit part. Such a list exists because \(\mathbb R^\times/\mathbb R^{\times2}\), \(\mathbb Q_2^\times/\mathbb Q_2^{\times2}\), and \(\mathbb Q_3^\times/\mathbb Q_3^{\times2}\) are finite, their classes are open, and weak approximation supplies a rational representative for each specified finite tuple of classes. For each \(m\in\mathcal M\), set \[S(m)=\{2,3\}\cup\operatorname{Supp}(m).\] The same argument gives a finite list \(\mathcal D_m\subset\mathbb Q^\times\) such that, for any \(b\ne0\), some \(m_d\in\mathcal D_m\) makes \[ d=m_db>0,\qquad d\in\mathbb Q_q^{\times2}\quad(q\in S(m)). \tag{33}\] Use these fixed lists in \(\Phi\), and fix \(S_1\) containing \(2\), \(3\), and all their supports as above. The normalizations (32)–(33) are used only to select entries \(m,m_d\) for an integer input; they are not additional conjuncts of \(\Phi\). Lemma 14 already proves the pole restriction for this choice.

Fix \(b\in\mathbb Z\setminus\{0\}\) and choose \(m,m_d\) so that (32) and (33) hold. We choose \(t\) to serve two purposes. The value \(U\) should be small at the places where \(d\) was made a local square, while at primes outside \(S(m)\) where \(v_q(b')\) is odd it should retain the valuation of \(b\). These properties will make \(H'\) a local square at every prime where \(v_q(b')\) is odd, and \(b'\) a local square at every prime where \(v_q(H')\) is odd.

At the real place and at every \(q\in S(m)\), choose a local solution of \(dt^2=1\). As \(t\) tends to this solution, the rational function \(U\) tends to zero: its denominator tends to the nonzero number \(2\). Thus, in sufficiently small neighborhoods, \(H'>0\) at the real place, and \[H'=b'^2\left(1-\frac{U^2}{b'}\right)\in\mathbb Q_q^{\times2} \quad(q\in S(m)),\] by openness of the local square subgroup. At every \(q\in\operatorname{Supp}(m_d)\setminus S(m)\), require \(t\) to be so small that \(v_q(dt^2)>0\). Weak approximation satisfies all these conditions simultaneously with a rational \(t\). In particular \(1+dt^2\ne0\) and \(H'>0\).

Let \(\beta,\theta\) be the unique positive squarefree integers representing the square classes of \(b',H'\), and let \[B=\operatorname{Supp}(\beta),\qquad J=\operatorname{Supp}(\theta).\]

Write \(H'=\theta c^2\) with \(c\in\mathbb Q^\times\). For every \(n\in\mathbb Q^\times\) the isomorphism \[ E_{n\theta}\longrightarrow E_{nH'},\qquad (x,y)\longmapsto(c^2x,c^3y) \tag{35}\] preserves \(x\)-coordinate square classes. Thus, taking \(e=n\) in the formula, it remains to find one \(n\) for which both \(E_n\) and \(E_{n\theta}\) have a rational point with nonzero \(x\)-coordinate of class \(\beta\), the class of \(b'\). We first give a criterion for one parameter \(l\), and then construct a single \(n\) for which it applies to both twists.

A criterion for one twist

We recall the exact part of full two-descent that we use; see (Silverman 2009, X, Proposition 1.4, p. 315). For a split separable cubic \(y^2=\prod_{i=1}^3(x-e_i)\) over a field \(k\) of characteristic zero, choosing two roots identifies \(H^1(k,E[2])\) with \((k^\times/k^{\times2})^2\). In the coordinate corresponding to \(e_i\), the Kummer map takes a point to the class of \(x-e_i\); at \((e_i,0)\) the value is replaced by \(\prod_{j\ne i}(e_i-e_j)\), and at \(O\) it is \(1\). The Kummer map is an injection of \(E(k)/2E(k)\) into this group. The group \(\mathop{\mathrm{Sel}}_2(E/\mathbb Q)\) is the subgroup of rational pairs lying in the Kummer image at every completion. The standard Selmer exact sequence (Silverman 2009, X, Theorem 4.2, p. 333) is \[ 0\longrightarrow E(\mathbb Q)/2E(\mathbb Q) \longrightarrow\mathop{\mathrm{Sel}}_2(E/\mathbb Q) \longrightarrow\mathop{\mathrm{Sha}}(E/\mathbb Q)[2]\longrightarrow0. \tag{36}\]

We use the coordinates \((x+3l,x-l)\) for \(E_l\). For a finite rational point \(T\) with \(x(T)\notin\{0,l,-3l\}\), the curve equation gives \[[x(T)]=[x(T)+3l]\,[x(T)-l].\] Consequently, if the class \((\beta,1)\) is represented by a rational point and is distinct from the classes of all four rational two-torsion points, that point has the required nonzero \(x\)-coordinate class \(\beta\). The next calculation identifies conditions that make \((\beta,1)\) the one additional Selmer direction beyond rational two-torsion.

For the Selmer graph-matrix viewpoint underlying this calculation, see Monsky’s appendix to Heath-Brown (Heath-Brown 1994, Appendix, pp. 365–370). The matrix and simultaneous rank calculations below adapt that viewpoint to the present curve family.

At the primes dividing \(l\), the local square conditions will be recorded by binary residue symbols. For an odd prime \(q\) and a \(q\)-adic unit \(a\), write \([a/q]\in\mathbb F_2\) for the nonsquare bit: it is \(0\) for a square residue and \(1\) for a nonsquare residue. For a finite set \(Q\) of distinct primes different from \(2,3\), indexed by \(i\), and a vector \(w\in\mathbb F_2^Q\), write \(D_w\) for the diagonal matrix with diagonal \(w\). Put \[ s_i=[-1/q_i],\qquad \lambda_i=[3/q_i],\qquad (Ku)_i=\sum_{j\ne i}[q_j/q_i](u_j+u_i),\qquad N=K+D_s. \tag{37}\] All matrices and vectors in this calculation are over \(\mathbb F_2\). Write \(\mathbf1\) for the all-ones vector and \(\mathbf1_B\) for the indicator of \(B\) in \(Q\). Quadratic reciprocity gives \[ K+K^{\mathsf t}=ss^{\mathsf t}+D_s. \tag{38}\] In particular, if \(s^{\mathsf t}\mathbf1=1\), then \[ K\mathbf1=K^{\mathsf t}\mathbf1=0, \qquad N=K^{\mathsf t}+ss^{\mathsf t}. \tag{39}\]

The family \(y^2=x(x-n)(x+3n)\) also occurs in the study of \(\pi/3\)- and \(2\pi/3\)-congruent numbers. Mokrani adapted Monsky matrices to these families (Mokrani 2020). For this family, Wei and Guo give a two-Selmer matrix formulation and write the same quadratic-reciprocity identity for the associated Rédei matrix as (38) (Wei and Guo 2022, sec. 4 and Theorem 6.2). The simultaneous kernel conditions for the two twists required here are constructed below.

It remains to realize this particular Selmer class by a rational point. The consequence we use from Theorem 1.1 of (OpenAI 2026a) is the following: an elliptic curve \(C/\mathbb Q\) with \(C(\mathbb Q)[2]\ne0\) and \(\mathop{\mathrm{corank}}_{\mathbb Z_2}\mathop{\mathrm{Sel}}_{2^\infty}(C/\mathbb Q)\in\{0,1\}\) has finite \(\mathop{\mathrm{Sha}}(C/\mathbb Q)\). The next proof verifies the full Selmer-corank hypothesis before applying it.

One parameter for both twists

The criterion is now a condition on the prime support of one twist. We construct a single positive squarefree \(n\) for which it holds both on the support of \(n\) and on that of \(n\theta\). The mutual local-square relations from Lemma 15 are what allow the second support to be added without disturbing the first construction.

The five-point quaternionic quotient

We first construct the cover whose ramification will detect the odd contact orders in Theorem 5. Let \(B/\mathbb Q\) be the indefinite quaternion algebra of discriminant \(210=2\cdot3\cdot5\cdot7\), let \(\mathcal O\) be a maximal order, and put \(G=B^\times\). Fix \(B\otimes_\mathbb Q\mathbf R\simeq M_2(\mathbf R)\). The quaternionic Shimura datum is \((G,\mathfrak H^\pm)\), where the two half planes parametrize the conjugates of the elliptic-curve Hodge homomorphism. Its reflex field is \(\mathbb Q\): over \(\mathbf C\) its cocharacter has the usual \(\mathrm{GL}_2\) conjugacy class, and this class is invariant under the Galois action for the inner form \(G\).

We use canonical models and functoriality for Shimura varieties (Deligne 1971b, 1979). In the form needed here, they give the model over the reflex field with complex points \[G(\mathbb Q)\backslash \bigl(\mathfrak H^\pm\times G(\mathbf A_f)/K\bigr)\] for a compact open subgroup \(K\), and define level maps and finite-adelic right actions over that field. We use the full Shimura varieties over \(\mathbb Q\), including all their geometric components.

At maximal level, put \[X=\operatorname{Sh}_{\widehat{\mathcal O}^{\,\times}} (G,\mathfrak H^\pm).\] This curve is proper because \(B\) is a division algebra. Strong approximation for \(B^1\), the reduced-norm theorem, and \(\operatorname{Nrd}(\mathcal O_p^\times)=\mathbb Z_p^\times\) give \[G(\mathbf A_f)=G(\mathbb Q)^+\widehat{\mathcal O}^{\,\times}.\] Indeed, match the finite-adelic norm modulo \(\widehat\mathbb Z^\times\) by a positive rational reduced norm and then apply strong approximation to the norm-one part. There are also rational elements of negative norm, since no real place is ramified. Consequently \(X_{\mathbf C}\) is the connected compact curve \[(\mathcal O^1/\{\pm1\})\backslash\mathfrak H.\]

The uniformizing group has no nontrivial elliptic stabilizers. A noncentral norm-one unit fixing a point has integral trace of absolute value less than two, hence trace \(0\) or \(\pm1\). It would generate \(\mathbb Q(i)\) or \(\mathbb Q(\sqrt{-3})\). These fields cannot embed in \(B\), since they split at the ramified primes \(5\) and \(7\), respectively. The Eichler area formula, in the normalization of (Voight 2021, Theorem 39.1.2), therefore gives \[\frac{\operatorname{area}(X(\mathbf C))}{2\pi} =\frac16\prod_{p\mid210}(p-1)=8.\] There are no cusps or elliptic corrections, so \(2g(X)-2=8\) and \(g(X)=5\).

The finite-adelic normalizer of \(\widehat{\mathcal O}\) induces the Atkin–Lehner group \[\mathcal W\simeq(\mathbb Z/2\mathbb Z)^4\] over \(\mathbb Q\). At a split prime, the normalizer is scalars times order units. At a division prime, its quotient by scalars and order units has order two, detected by parity of reduced-norm valuation. Finite adelic scalars act trivially, since \(\mathbf A_f^\times=\mathbb Q^\times\widehat\mathbb Z^\times\). The resulting action is faithful: a generic domain point has rational stabilizer equal to the center, which accounts precisely for the scalars already removed. Let \(X^*=X/\mathcal W\).

The geometry of this discriminant-\(210\) quotient has been tabulated before. Long, Maclachlan, and Reid list the signature \((0;2^5;0)\), recording a genus-zero quotient with five elliptic cycles of order two and no cusps (Long et al. 2006, Table 3). Nualart Riera gives genus \(5\) for the original curve and the five involution labels \(30,42,70,105,210\), and identifies the full quotient over \(\mathbb Q\) with \(\mathbf P^1\) (Nualart Riera 2015, Propositions 4.1–4.2). The calculation here recovers this fixed configuration and establishes the rational branch values used in the reduction.

A finite stabilizer of a point on a smooth characteristic-zero curve acts faithfully on its tangent line and is cyclic. Since every nonidentity element of \(\mathcal W\) has order two, each nontrivial stabilizer has order two. If \(r\) is the number of geometric branch values of \(X\to X^*\), Riemann–Hurwitz gives \[ 2g(X^*)-2+\frac r2=\frac{2g(X)-2}{16}=\frac12. \tag{53}\] Thus \(r=5-4g(X^*)\le5\).

We exhibit five different stabilizer labels. Take \[m=30,42,70,105,210.\] The field \(\mathbb Q(\sqrt{-m})\) is nonsplit at every prime dividing \(210\). For the first three values, the only prime of \(210\) not dividing \(m\) is respectively \(7,5,3\), and the residues of \(-m\) there are the nonsquares \(5,3,2\). For \(m=105\), the quadratic discriminant is \(-420\), so \(2\) is ramified too. For \(m=210\), all four primes are ramified. The quaternion embedding criterion from Albert–Brauer–Hasse–Noether, as in (Voight 2021), gives an embedding of each field in \(B\).

An embedded \(\sqrt{-m}\) lies in some maximal order: at split local places an integral element stabilizes a lattice, and at division places it lies in the unique maximal order. Intersecting these local orders gives a global maximal order containing the embedded element. Strong approximation implies that all maximal orders in this indefinite rational quaternion algebra are conjugate by \(B^\times\) (Voight 2021, Theorems 28.2.10 and 28.2.11(b)), so a conjugate of the embedded element lies in our fixed \(\mathcal O\). It normalizes \(\mathcal O\) because it is a unit at every prime outside \(210\), and at a division prime every element normalizes the unique maximal order. Its norm is \(m\) and it fixes a point of \(\mathfrak H\), so it gives a fixed point of the involution \(w_m\) with the corresponding norm-parity label.

The five labels are different. A fiber of the quotient is one \(\mathcal W\)-orbit, and the stabilizers along this orbit agree because \(\mathcal W\) is abelian. Distinct labels thus yield distinct branch values, proving \(r\ge5\). Equation (53) now forces \(r=5\) and \(g(X^*)=0\). There is exactly one branch value for each of the five labels. Every \(w_m\) is defined over \(\mathbb Q\), so Galois preserves its uniquely labelled branch value. Each branch value is therefore individually rational. In particular \(X^*\) is a genus-zero curve with a rational point, hence is isomorphic over \(\mathbb Q\) to \(\mathbf P^1\). Fix an isomorphism sending one branch value to \(\infty\), and denote the five branch values by \(\mathcal C\).

The abelian surface family

We next pass to a fine level so that the cover carries abelian surfaces. The construction will provide a finite level map \(Y\to X\), unramified over \(X\), and a principally polarized abelian surface scheme \(\mathcal A\to Y\). Write \(\pi\) for the composite \[Y\longrightarrow X\longrightarrow X^*\simeq\mathbf P^1, \qquad \pi:Y\longrightarrow\mathbf P^1.\] Every fiber \(\mathcal A_y\) with \(y\in Y(\overline\mathbb Q)\) will have an embedding \(B\hookrightarrow\operatorname{End}^0_{\overline\mathbb Q}(\mathcal A_y)\) given by left multiplication. Fibers with the same image under \(\pi\) will be geometrically isogenous, and the polarized moduli map will be nonconstant on every geometric component. We now construct the family and prove these properties.

Canonical-model functoriality associates algebraic maps to morphisms of Shimura data at compatible levels (Deligne 1971b, 1979). A Siegel variety at principal level at least three carries the universal principally polarized abelian scheme. The required morphism to a Siegel datum comes from the following symplectic representation.

On the four-dimensional rational space \(V=B\), put \[\rho(g)(x)=x\bar g,\qquad \psi(x,y)=\operatorname{Trd}(\Delta x\bar y),\] where the bar is the standard quaternion involution and \(\Delta\) is a pure quaternion with \(\Delta^2<0\). Such a rational \(\Delta\) exists by real approximation in the trace-zero subspace. The identities \(\bar\Delta=-\Delta\) and cyclicity of reduced trace show that \(\psi\) is alternating. It is nondegenerate because the reduced-trace pairing is nondegenerate and \(\Delta\) is invertible. Moreover \[\psi(x\bar g,y\bar g)=\operatorname{Nrd}(g)\psi(x,y).\] Quaternion conjugation reverses products, as does composition of right multiplications; together these identities give \(\rho(gh)=\rho(g)\rho(h)\). Thus \(\rho\) is a faithful symplectic-similitude representation.

The Hodge homomorphism makes \(V_{\mathbf R}\) a complex vector space by right multiplication. At a base point represented by \(J=\left(\begin{smallmatrix}0&-1\\1&0\end{smallmatrix}\right)\), the associated real quadratic form is \[\psi(x,x\bar J)=\operatorname{tr}(J\Delta xx^{\mathrm t}).\] Since \(\Delta\) has trace zero, \(J\Delta\) is symmetric, and its determinant is positive; it is therefore definite. Choose its sign on one half plane. Conjugation by \(G(\mathbf R)\) gives all the other complex structures, and the displayed similitude identity gives the corresponding definite sign on the other half plane as well. This is the two-component Siegel datum. A real scalar acts on \(V\) with the homological weight of an elliptic curve, which is the required weight.

Choose a rational symplectic basis for \(V\). The standard lattice in that basis is self-dual for \(\psi\). Intersect a sufficiently small compact open subgroup of \(\widehat{\mathcal O}^{\,\times}\) with the inverse image of a principal Siegel level, and call the resulting subgroup \(K_Y\). Put \[Y=\operatorname{Sh}_{K_Y}(G,\mathfrak H^\pm).\] The induced morphism from \(Y\) to the fine Siegel moduli scheme pulls back its universal family to the principally polarized abelian surface scheme \(\mathcal A\to Y\). The self-dual lattice need not be preserved by the whole maximal order: the intersection of levels provides the integral moduli problem, while the rational Hodge structure retains the \(B\)-action. Fix this level and family once and for all.

There may be finitely many geometric components of the smooth projective curve \(Y\). The level map \(Y\to X\) is unramified by the uniformization and the absence of elliptic stabilizers on \(X\). Consequently its composite \(\pi\) has ramification index two at every point above \(\mathcal C\) and index one elsewhere.

Every geometric fiber \(A=\mathcal A_y\) has left \(B\)-multiplication in \(\operatorname{End}^0_{\overline\mathbb Q}(A)\). Left multiplication commutes with the right complex structures on \(V\); the equivalence between abelian varieties up to isogeny and their polarizable rational Hodge structures supplies the endomorphisms. Homomorphisms of abelian varieties defined over \(\overline\mathbb Q\) do not acquire new elements after extension to \(\mathbf C\) (Mumford 1970).

For non-CM points, the moduli construction of Guitart and Molina gives a related description by compatible isogenies (Guitart and Molina 2009, Lemma 1, Corollary 2, and Section 3). For this fine-level family, the following Hodge argument gives the underlying geometric isogenies for all fibers needed here; compatible choices are made in the descent argument. Changing level or applying the normalizer changes the finite-adelic coordinate, while domain coordinates of fibers above a common point of \(X^*\) can be identified up to \(G(\mathbb Q)\). Their rational Hodge structures are therefore isomorphic, so the fibers are geometrically isogenous. In particular, if \(\pi(y)=s\in\mathbf P^1(\mathbb Q)\), then \(\pi({}^g y)=g(s)=s\) and \(\mathcal A_y\) is isogenous to each of its Galois conjugates.

Finally, the polarized moduli map of the family is nonconstant on every geometric component. Its complex structures vary in a half plane, whereas a fixed rational Hodge structure has only countably many presentations on the fixed rational space \(V\). Thus a component cannot have a single geometric isomorphism class of fibers.

Rational fibers and odd-contact ramification

The ramification index two now explains the parity in the theorem. We make precise how even contact with a branch value gives unramified local lifts, and record the good reduction of the resulting surfaces.

Enlarge a fixed finite set \(S\) of rational primes whenever necessary in the following construction. All these enlargements depend only on the fixed curves, maps, and family, and occur before choosing \(s\). Include \(2\), primes at which the branch sections meet, and primes at which the chosen coordinates or the geometric models degenerate. Spread \(Y\), \(\mathbf P^1\), \(\pi\), and \(\mathcal A\) over \(\mathbb Z[S^{-1}]\) so that the curves are smooth and proper, \(\pi\) is finite and étale away from the branch sections, and \[\pi^*b=2D_b\] as relative Cartier divisors, with \(D_b\) the reduced inverse image, finite étale over the ground ring. The abelian scheme also spreads over this proper model of \(Y\).

This use of contact multiplicities has a classical specialization precedent. Darmon and Granville work with a fixed Galois cover of the projective line branched at \(0,1,\infty\). In that setting, their Proposition 3.2, credited there to Beckmann, relates divisibility of contact orders by branch indices to the absence of new ramification outside fixed bad primes in specialized fiber fields (Darmon and Granville 1995). For the cover used here, the local equation \(t=z^2\) makes the parity of the contact order relevant.

Comparison of base height and Faltings height

The fiber-field lemma identifies the primes that may ramify when a rational base point is lifted to the family. We also need to recover the height of that base point from the fiber. The next lemma depends only on the fixed family, and applies to every algebraic point of \(Y\).

A field of definition for homomorphisms

We now choose the fiber used in the arithmetic proof. Fix \(s\in\mathbf P^1(\mathbb Q)\setminus\mathcal C\), put \(M=M_S(s)\) and \(T=S\cup\{q:q\mid M\}\), and choose \(y\) above \(s\). Let \(A=\mathcal A_y\) and choose the splitting field \(K_0\) from Lemma 20. Since \(\pi\) is defined over \(\mathbb Q\) and \(s\) is rational, \(\pi({}^g y)=g(s)=s\) for every \(g\in G_\mathbb Q\). The common-base property of the family therefore makes the conjugates of \(A\) geometrically isogenous. The descent argument needs all their homomorphisms over one Galois field. We can define them without losing the bounded degree or introducing ramification at further primes.

All homomorphisms between conjugates, including the prescribed \(B\)-action, are now defined over \(L\). Its degree is uniformly bounded, and it is unramified outside \(T\). Extending \(K_0\) to \(L\) preserves good reduction of \(A\) outside the fixed set \(S\). We now bound \(h_{\mathrm F}(A)\) in terms of \(M\), beginning with its endomorphism algebra.

The endomorphism dichotomy and descent up to isogeny

The endomorphism algebra determines which comparison will give a height bound for \(A\). Put \(D=\operatorname{End}^0_{\overline\mathbb Q}(A)\), and let \(C_B\) be the centralizer of the prescribed left \(B\)-action in \(D\). On rational homology the centralizer of left \(B\) is right \(B\). A Hodge endomorphism in that centralizer must commute, over \(\mathbf R\), with the complex structure, whose centralizer in \(M_2(\mathbf R)\) is \(\mathbf C\). Thus \(C_B\) is either \(\mathbb Q\) or an imaginary quadratic field \(k\) contained in \(B\). More explicitly, a nonscalar rational element commuting with the complex structure generates such a quadratic field, and its real centralizer has dimension two, leaving no further possibilities.

Both cases use the following quantitative isogeny theorem to transfer a known height bound. If \(C_1,C_2\) are \(g\)-dimensional abelian varieties over a number field \(k_0\) of degree \(d\), and are isogenous over an extension \(K/k_0\), then there are \(K\)-isogenies in both directions with degree at most \[ \kappa(C_1)= \left((14g)^{64g^2}d \max\{h_{\mathrm F}(C_1),\log d,1\}^{2}\right)^{1024g^3}. \tag{57}\] This is (Gaudron and Rémond 2014a, Theorem 1.4, with the definition on p. 2058). It uses the original stable Faltings height and requires neither semistability over \(k_0\) nor a principal polarization. In our applications we take \(K=k_0\). The stable height is additive on products and satisfies \[ h_{\mathrm F}(C_2)\le h_{\mathrm F}(C_1) +\tfrac12\log\deg\phi \quad\text{for an isogeny }\phi:C_1\longrightarrow C_2 \tag{58}\] (Faltings 1983). We will always apply (57) to a source whose height has already been bounded.

If \(C_B=k\), the central-simple-algebra double-centralizer calculation gives \[D=B\otimes_\mathbb Qk\simeq M_2(k).\] The last isomorphism holds because an embedded quadratic field is a maximal subfield of \(B\) and splits it. Matrix idempotents, all defined over \(L\) by Lemma 23, give \(A\sim_L E^2\) for a CM elliptic curve \(E/L\): take the image of an integer multiple of a rank-one idempotent, and use the matrix units to identify the two elliptic factors up to isogeny. The degree of \(j(E)\) is at most \([L:\mathbb Q]\), hence is bounded. Complex multiplication theory identifies the degree of a singular modulus with the class number of its imaginary quadratic order. Class-number growth for imaginary quadratic fields and the formula for class numbers of orders imply that only finitely many such \(j\)-invariants have bounded degree (Cox 2013; Siegel 1935). Their stable elliptic heights, and therefore \(h_{\mathrm F}(E^2)\), are bounded. Apply (57) to the source \(E^2\), with dimension two, bounded height, and bounded \([L:\mathbb Q]\), and then apply (58). This bounds \(h_{\mathrm F}(A)\) uniformly. Lemma 21 already proves Theorem 5 for these fibers, since \(M\ge1\). The finiteness invoked here need not be effective.

We henceforth treat the other case, \(C_B=\mathbb Q\). The same double-centralizer calculation gives \(D=B\). One can check this calculation after an extension splitting \(B\): an algebra containing the full \(2\times2\) matrix units is a matrix algebra over their centralizer. Because \(B\) is division, \(A\) is absolutely simple; a proper isogeny factor would give a nontrivial idempotent in \(D\).

Our goal in this case is to find a primitive weight-two newform \(f\) of level \(N\), a Galois extension \(L'/\mathbb Q\) containing \(L\), and an abelian variety \(D_0/L'\) such that, for fixed constants \(C,a>0\), \[[L':\mathbb Q]\le CM,\qquad N\le CM^a,\qquad J_1(N)\sim_{L'}A\times D_0.\] Once the height of this modular Jacobian has been bounded in terms of \(N\), the factorization will let us transfer the bound to \(A\) over a field of controlled degree. We first use the conjugacy isogenies to construct a two-dimensional representation. Their discrepancy under composition will be reduced to a sign and then split with control on both the field degree and inertia.

Compatible-isogeny cocycles and their scalar splittings are part of the theory of \(\mathbb Q\)-curves and building blocks (Ribet 1992, sec. 6)(Pyle 1995, chaps. 4–5); compare (Quer 2001, sec. 4). The scalar cochain constructed below takes values among roots of unity of order \(O(M)\), and its inertia restrictions will control the modular level.

Put \(\Gamma=\operatorname{Gal}(L/\mathbb Q)\). For each \(g\in\Gamma\), choose a quasi-isogeny \(\mu_g:{}^gA\to A\), with \(\mu_1=1\), satisfying \[ \mu_g\,{}^g\alpha=\alpha\mu_g\qquad(\alpha\in B). \tag{59}\] To obtain it, start with an isogeny between the fibers above the common rational point. Transporting the \(B\)-action along this isogeny gives an automorphism of \(B=D\), which is inner by Skolem–Noether. Composing with the corresponding element of \(B^\times\) gives (59). All these maps are defined over \(L\) by Lemma 23.

The maps \(\mu_g\,{}^g\mu_h\) and \(\mu_{gh}\) have the same source and target and both intertwine the \(B\)-actions. Their quotient therefore lies in the centralizer \(\mathbb Q^\times\): \[\mu_g\,{}^g\mu_h=c(g,h)\mu_{gh}, \qquad c(g,h)\in\mathbb Q^\times.\] For a quasi-isogeny, use the degree obtained by extending the geometric degree multiplicatively from isogenies. A rational scalar \(c\) on a surface has degree \(c^4\), so the last identity gives \[|c(g,h)|= \left(\frac{\deg\mu_g\,\deg\mu_h}{\deg\mu_{gh}}\right)^{1/4}.\] Let \(a_g=(\deg\mu_g)^{-1/4}\) be the positive real algebraic fourth root. We extend the coefficient scalars in the isogeny category and put \[\widetilde\mu_g=a_g\mu_g \in\operatorname{Hom}_L^0({}^gA,A)\otimes_\mathbb Q\overline\mathbb Q.\] Here Galois transports the Hom factor and acts trivially on the coefficient factor \(\overline\mathbb Q\). The preceding degree identity now gives \[ \widetilde\mu_g\,{}^g\widetilde\mu_h =\epsilon(g,h)\widetilde\mu_{gh}, \qquad \epsilon(g,h)=\frac{c(g,h)}{|c(g,h)|}\in\{\pm1\}. \tag{60}\] Associativity makes \(\epsilon\) a normalized cocycle on \(\Gamma\); we inflate it to \(G_\mathbb Q\). This is the obstruction in the isogeny version of Weil descent. The normalized discrepancy depends only on the sign of \(c(g,h)\), so we will not need bounds for the individual degrees \(\deg\mu_g\).

A scalar cochain with coboundary \(\epsilon\) will cancel this sign. On \(G_L\) the inflated cocycle is trivial, so the restriction of such a cochain will be a character. We seek one whose restricted character is conjugacy invariant and has controlled order: its kernel will then define a Galois field over which the corrected action agrees with the Tate action of \(A\). Control on inertia will also be needed to bound the modular level. The following splitting supplies both properties.

Two-dimensional representations and modularity

We construct representations at coefficient primes \(2\) and \(3\). The \(2\)-adic one will yield a modular form. The \(3\)-adic one will measure the conductor at \(2\), where the conductor comparison requires a coefficient prime different from the residue prime. This requires identifying both representations with the same newform.

Fix coefficient embeddings \(\overline\mathbb Q\hookrightarrow \overline\mathbb Q_\ell\) for \(\ell=2,3\). On the covariant rational Tate module of \(A\) with these extended coefficients put \[ R_\ell(g)=\beta(g)\widetilde\mu_{\bar g}\circ g, \qquad g\in G_\mathbb Q,\quad \bar g=g|_L. \tag{62}\] Here \(g\) first transports torsion points to \({}^gA\), and the quasi-isogeny returns them to \(A\). In a composition the discrepancy of the maps is \(\epsilon(g,h)\), and the discrepancy of the scalar cochain is the same sign. Their product is one. Thus \(R_\ell\) is a genuine continuous representation. All algebraic coefficients and the splitting of \(B\) can be taken in one finite extension of \(\mathbb Q_\ell\) for each \(\ell\).

Equation (59) shows that \(R_\ell\) commutes with \(B\). After coefficient extension, \(B\otimes\overline\mathbb Q_\ell\simeq M_2(\overline\mathbb Q_\ell)\); the matrix units then give \[ R_\ell\simeq r_\ell\oplus r_\ell \tag{63}\] for a two-dimensional representation \(r_\ell\). On \(G_L\) the quasi-isogeny in (62) is \(\widetilde\mu_1=1\), so only the character \(\beta|_{G_L}\) separates \(R_\ell\) from the Tate action of \(A\). Define \[G_{L'}=\ker(\beta|_{G_L}).\] Conjugacy invariance of this character makes \(L'/\mathbb Q\) Galois, and its image has order at most \(2m\). Consequently \[ [L':\mathbb Q]\le 2m[L:\mathbb Q]\le C M. \tag{64}\] On \(G_{L'}\), the representation \(R_\ell\) is exactly the Tate-module representation of \(A\), with coefficients extended. This field will allow us to recover an abelian factor over \(L'\). Its degree may grow with \(M\); the local conductor argument will use \(L\), whose degree is uniformly bounded.

By Faltings’ semisimplicity and endomorphism theorems, \[\operatorname{End}_{G_{L'}}(V_\ell(A)) =\operatorname{End}_{L'}(A)\otimes\mathbb Q_\ell =B\otimes\mathbb Q_\ell\] (Faltings 1983). Therefore \(r_\ell|_{G_{L'}}\) is semisimple with scalar commutant after algebraic coefficient extension, and is absolutely irreducible. So is \(r_\ell\).

The de Rham comparison theorem for smooth proper varieties, together with its Hodge–Tate decomposition, applies to \(A\) over every completion of \(L'\) at its coefficient prime (Scholze 2013, Corollary 1.8); see also the corrected definitions in (Scholze 2016). Since the covariant Tate module is dual to first étale cohomology, \(V_\ell(A)\) is de Rham with cyclotomic exponents \(0,1\), each twice. The de Rham property descends through finite local extensions (Fontaine 1994, sec. 3.9(ii)). Thus \(R_\ell\) is de Rham over \(\mathbb Q_\ell\), and its decomposition (63) gives exponents \(0,1\), each once, for \(r_\ell\). Here the cyclotomic character has exponent \(1\) and Hodge–Tate weight \(-1\).

For complex conjugation \(j\), the involution \(r_\ell(j)\) cannot be scalar. If it were, (63) would make \(R_\ell(j)\) scalar too. Under comparison with Betti homology, \(\mu_{\bar j}\circ j\) is conjugation followed by a holomorphic quasi-isogeny. It exchanges the two nonzero Hodge subspaces, whereas a scalar preserves both. Multiplying by the nonzero algebraic coefficient in (62) does not change this fact. Hence the two eigenvalues of \(r_\ell(j)\) are \(1\) and \(-1\), and \(r_\ell\) is odd.

At a prime outside \(T\cup\{\ell\}\), inertia fixes \(L\), \(A\) has good reduction, and the corrected \(\beta\) is trivial. Thus \(r_\ell\) is unramified there. In particular it is ramified at only finitely many primes. All hypotheses of the stated modularity theorem now hold for \(r_2\). Applying it realizes \(r_2\) as a Tate twist of the representation associated with a classical cuspidal eigenform.

We use the arithmetic-Frobenius normalization in which a weight-\(k\) form \(f\) has good traces \(a_q(f)\) and determinant \(\psi_f\chi_{\mathrm{cyc}}^{k-1}\). Its cyclotomic exponents are \(0,k-1\); the companion fixes this convention in (OpenAI 2026b, sec. 2.1). The two exponents \(0,1\) of \(r_2\) therefore force weight two and Tate shift zero. Thus \(r_2\) is realized by the arithmetic \(2\)-adic representation of a normalized primitive weight-two newform.

For comparison with a dual presentation, if an intermediate realization is written as \(V_{f,2}^{\vee}\otimes\chi_{\mathrm{cyc}}^n\), then \[V_{f,2}^{\vee}\otimes\chi_{\mathrm{cyc}}^n \simeq V_{g,2}\otimes\chi_{\mathrm{cyc}}^{n+1-k}, \qquad g=(f\otimes\psi_f^{-1})_{\mathrm{new}}.\] Here \(g\) is the primitive newform of the indicated finite-character twist. The target exponents \(0,1\) give \(k=2\), \(n=1\) in this presentation, and hence arithmetic Tate exponent zero. Changing from arithmetic to geometric Frobenius inverts the evaluated eigenvalues; it does not dualize the representation. After this conversion when necessary, let \(f\) denote the normalized primitive newform whose arithmetic \(2\)-adic representation realizes \(r_2\), and let \(N\) be its level.

To use \(r_3\) for the dyadic conductor of this level, we must identify it with the same algebraic conjugate of \(f\). A single algebraic trace, obtained from the good special fiber, will compare the two coefficient primes.

A uniform bound for the modular level

The modular factor and the final height bound

For elliptic curves over \(\mathbb Q\), Murty and Pasten earlier obtained an effective \(O(N\log N)\) bound for Faltings height in terms of the conductor, with applications to \(S\)-unit equations (Murty and Pasten 2013, Theorems 1.1 and 7.1). The abelian-variety comparison used in this final step follows the modular height strategy of von Känel (Känel 2021, sec. 1.2 and 5); see also (Alpöge 2021). The preceding descent and ramification arguments provide the point-dependent fields and the level bound needed to apply that strategy here.

Let \(J=J_1(N)\). The weight-two Eichler–Shimura realization places \(r_2\) in \(V_2(J)\otimes\overline\mathbb Q_2\) (Ribet 1977, Theorem 2.1 and the following weight-two realization). We will turn this shared constituent into a nonzero homomorphism \(A\to J\) over \(L'\). The next argument descends a Hom from the extended coefficient field to \(\mathbb Q_2\), giving the abelian factor over \(L'\) itself.

The controlled field, level, and factorization sought in the non-CM case are now established. We turn to the height of the modular Jacobian. Write \(g_0=\dim J\). The modular group index bounds the degree of the map \(j:X_1(N)\to\mathbf P^1\) by \(C N^2\). It is branched only over \(0,1728,\infty\), so after rescaling the target it is a Belyi map. Riemann–Hurwitz, or the usual genus formula, also gives \(g_0\le C N^2\). Javanpeykar’s theorem (Javanpeykar 2014, Theorem 1.1.1 and Section 2.3) for a smooth projective connected curve \(X\) of genus \(g\ge1\) bounds its Jacobian’s stable height in his convention by \[13\cdot10^6 g\,\deg_B(X)^5.\] Javanpeykar’s height here equals the original stable Faltings height of the Jacobian used in (57). The addition of \((g/2)\log\pi\) in (Javanpeykar 2014, Lemma 2.4.4) converts instead to the Deligne normalization \(h_{\mathrm F}^{\mathrm D}\) of (Gaudron and Rémond 2014b, sec. 2.3). Thus no normalization correction is needed here; in either convention the difference is only \(O(g)\). Since Lemma 27 ensures \(g_0\ge2\), the genus hypothesis applies. We obtain \[ g_0\le C N^2, \qquad \max\{0,h_{\mathrm F}(J)\}\le C N^{12}. \tag{66}\]

For the final comparison we choose a principally polarized complement. An abelian variety over a number field has a polarization defined over that field and is consequently isogenous there to its dual. When \(D_0\ne0\), choose such a polarization over \(L'\). Applying Zarhin’s trick to it gives a principal polarization over \(L'\) on \[D_Z=(D_0\times D_0^\vee)^4\] (Zarhin 2023, Theorem 1.1(ii) and Remark 1.4); see also (Bost 1996, proof of Corollary 3.2). If \(D_0=0\), omit this factor and take all its height and dimension contributions below to be zero. In either case this choice of complement gives \[C_1=J^8\sim_{L'} C_2=A^8\times D_Z.\] Both sides have dimension \(g=8g_0\). Their common field has degree \(d=[L':\mathbb Q]\le C M\), and the source has known stable height \(h_{\mathrm F}(C_1)=8h_{\mathrm F}(J)\). From (65) and (66), \[g\le C M^{2a},\qquad \max\{0,h_{\mathrm F}(C_1)\}\le C M^{12a}.\] Applying (57) to this source gives an isogeny \(\phi:C_1\to C_2\) whose logarithmic degree is at most \[\begin{align*} \log\deg\phi &\le1024g^3\bigl(64g^2\log(14g)+\log d +2\log\max\{h_{\mathrm F}(C_1),\log d,1\}\bigr) \tag{67}\\ &\le C M^C. \end{align*}\] The last inequality follows by substituting the preceding bounds: every power of \(g\) is a fixed power of \(M\), while the remaining logarithms are bounded by \(C(1+\log M)\le C'M\) for \(M\ge1\). This is the point at which the explicit dependence on dimension in the isogeny theorem is needed. The theorem is applied with source \(J^8\), whose height is already bounded.

By additivity and (58), \[8h_{\mathrm F}(A)+h_{\mathrm F}(D_Z) \le h_{\mathrm F}(C_1)+\tfrac12\log\deg\phi \le C M^C.\] Bost’s lower bound applies to every abelian variety over a number field. For the complement \(D_Z\) chosen above, the form in (Gaudron and Rémond 2014b, Corollary 8.4) gives \[h_{\mathrm F}^{\mathrm D}(D_Z)\ge -\tfrac12(\dim D_Z)\log(2\pi).\] The normalization identity \(h_{\mathrm F}^{\mathrm D}=h_{\mathrm F} +\tfrac12(\dim D_Z)\log\pi\) therefore yields \[h_{\mathrm F}(D_Z)\ge -\tfrac12(\dim D_Z)\log(2\pi^2).\] Since \(\dim D_Z\le g\le C M^{2a}\), it follows that \[ h_{\mathrm F}(A)\le C M^C. \tag{68}\] Neither a degree bound for the original \(\mu_g\) nor a bound for the degree of a field of coefficients has entered this estimate.

Reduction orders and their possible common factors

Restrict attention to primes \[ r\equiv1\pmod8, \qquad r\equiv3\pmod7. \tag{74}\] Quadratic reciprocity gives \((2/r)=1\), so \(r\) splits in \(F\), and \((7/r)=(r/7)=-1\). At either chosen place above \(r\), the two nonzero reductions of \(d\) and \(d^\sigma\) have product \(7\). They therefore represent opposite square classes in \(\mathbb F_r^*\). The curves \(E\) and \(E^\sigma\) are the quadratic twists by these classes of \(E_0:y^2=x^3-x\), so their traces of Frobenius are opposite.

Here is the specific description of that trace which we need. Since \(r\equiv1\pmod4\), the automorphism \((x,y)\mapsto(-x,iy)\) of \(E_0\) is defined over \(\mathbb F_r\) and has square \([-1]\). Its centralizer in the rational geometric endomorphism algebra is \(\mathbb Q(i)\). Indeed, the faithful action of endomorphisms on an auxiliary two-dimensional rational Tate module bounds this centralizer by dimension two, while it already contains \(\mathbb Q(i)\). More explicitly one may use the \(3\)-adic Tate module for \(r>3\), on which the minimal polynomial \(T^2+1\) is irreducible and its matrix centralizer has dimension two. The faithful action remains injective after extending coefficients (Silverman 2009, Theorem III.7.4).

Frobenius commutes with this automorphism. Its characteristic polynomial is integral, and its determinant is \(r\); these are the usual degree and trace identities for elliptic endomorphisms (Silverman 2009, Proposition III.8.6 and Chapter V, §2). It is thus an algebraic integer in \(\mathbb Q(i)\), hence equals \(a+bi\) for \(a,b\in\mathbb Z\), with \[ a^2+b^2=r,\qquad a\ne0. \tag{75}\] The last condition follows because a rational prime is not a square. Consequently, in either order, \[ d_1,d_2=r+1\pm2a. \tag{76}\] Both orders are divisible by four, since both curves have full rational two-torsion over \(\mathbb F_r\). Their sum is \(2(r+1)\), whose \(2\)-adic valuation is exactly two by (74). Thus the \(2\)-part of their greatest common divisor is exactly four. For sufficiently large \(r\), the Hasse bound gives \(0<d_i<2r\). Since \(4\mid d_i\) and \(r\) is odd, \(d_i\ne r\), proving \(r\nmid d_i\).

If an odd prime \(l\) divides both orders, their sum and difference show that \(l\mid r+1\) and \(l\mid a\). Equation (75) then gives \[ a\equiv0\pmod l,\qquad b^2\equiv-1\pmod l. \tag{77}\] In particular such an \(l\) is \(1\) modulo \(4\). We shall first exclude small \(l\) by congruences on \(r+1\), then use an upper sieve to remove the remaining primes \(r\) satisfying these two local conditions.

A finite multiplier set and the initial residue conditions

Write \(L_b(U,V)=\alpha_bU+\gamma_bV\) for \(b\ne\infty\). Here \(\alpha_b\ne0\), \(\gcd(\alpha_b,\gamma_b)=1\), and \(\alpha_b\gamma_c-\alpha_c\gamma_b\ne0\) for \(b\ne c\). For an integer \(A\), put \(T=(u_1,u_2,v)\) and denote the thirteen homogeneous forms in (72) by \(F_\nu(T)\), \(1\le\nu\le t=13\). Their coefficient vectors, other than \((0,0,1)\), are \[(\alpha_b,0,\gamma_b),\quad (0,\alpha_b,\gamma_b),\quad (-\alpha_b,-\alpha_b,\alpha_bA+\gamma_b).\] Every vector is primitive, including the last one because \(\gcd(\alpha_b,\alpha_bA+\gamma_b)=1\). There is a fixed threshold \(p_0\), independent of \(A\), above which any two vectors are linearly independent over \(\mathbb F_p\). Within one of the three families this follows from the fixed nonzero determinants of the \(L_b\); between different families it follows from the three distinct directions \((1,0),(0,1),(-1,-1)\) in the first two coordinates. The form \(v\) is independent of each remaining form whenever \(p\nmid\alpha_b\).

Choose once and for all a finite set \(S_0\) consisting of all primes up to a sufficiently large threshold, including \(2,7\), the exceptions just described, and all fixed bad primes needed for the elliptic curves. Increase the threshold so that \(p>2t\) and the later local-density estimates hold for every \(p\notin S_0\). Choose a fixed integer \(H_0\) with \(t2^{-H_0}<1\). For any \(A\) and any \(p\in S_0\), each primitive \(F_\nu\) maps \((\mathbb Z/p^{H_0}\mathbb Z)^3\) onto \(\mathbb Z/p^{H_0}\mathbb Z\). The proportion of vectors for which \(p^{H_0}\mid F_\nu(T)\) is therefore \(p^{-H_0}\). Since \(tp^{-H_0}<1\), there is a vector class modulo \(p^{H_0}\) on which none of the \(F_\nu\) is divisible by \(p^{H_0}\). Every lift of this class modulo \(p^{H_0+3}\) has \[v_p(F_\nu(T))<H_0\qquad(1\le\nu\le t).\] This proves the existence of the small-prime classes that we will select for a particular \(A\).

We next choose a finite catalogue of multipliers, without selecting \(A\) or any of those classes. Put \(m_2=8\) and \(m_p=p\) for odd \(p\in S_0\). Define units \(c_2=1\pmod8\), \(c_7=3\pmod7\), and \(c_p=1\pmod p\) for the remaining \(p\in S_0\). These are the desired residues of the positive primes in Lemma 29. A catalogue datum consists of integers and unit residues \[e_{p\nu}\in\{0,\ldots,H_0-1\},\qquad \xi_{p\nu}\in(\mathbb Z/m_p\mathbb Z)^* \quad(p\in S_0,\ 1\le\nu\le t).\] For such a datum set \(k_{0\nu}=\prod_{p\in S_0}p^{e_{p\nu}}\). The Chinese Remainder Theorem and Dirichlet’s Theorem (Davenport 2000, sec. 4) allow us to choose distinct primes \(h_1,\ldots,h_t\notin S_0\), hence above the fixed threshold, with \[ h_\nu\equiv c_p^{-1}\xi_{p\nu}\pmod{m_p} \qquad(p\in S_0). \tag{78}\] When the datum later comes from a chosen vector class, \(\xi_{p\nu}\) will be the signed unit remaining after the factors from \(S_0\) have been removed. The congruence gives \(\xi_{p\nu}h_\nu^{-1}\equiv c_p\pmod{m_p}\), so division by the additional factor \(h_\nu\) gives the desired residue. There are only finitely many catalogue data. Fix one such tuple of primes for each datum and let \(\mathcal K\) be the union of the resulting integers \(k_{0\nu}h_\nu\). Thus \(\mathcal K\) is finite and is fixed before \(A\), the cutoff \(L\), and the dilation \(X\).

Now fix the integer \(A\) for which the lemma is to be proved. Choose a bounded open rectangular box \(\mathcal B\) of positive volume whose closure avoids the zero planes of all \(F_\nu\), and let \(\epsilon_\nu\in\{1,-1\}\) be the sign of \(F_\nu\) on this box. For each \(p\in S_0\), choose one of the vector classes modulo \(p^{H_0+3}\) supplied above and put \(e_{p\nu}=v_p(F_\nu(T))\) on that class. These values do not depend on its representative. They define \(k_{0\nu}\) as above. The class also determines the unit \[\xi_{p\nu}= \epsilon_\nu\frac{F_\nu(T)}{p^{e_{p\nu}}} \left(\frac{k_{0\nu}}{p^{e_{p\nu}}}\right)^{-1} \pmod{m_p}.\] Here the second factor is a unit modulo \(m_p\), and the first quotient is known modulo \(p^{H_0+3-e_{p\nu}}\), a modulus containing at least four powers of \(p\). In particular it determines the required residue even when \(p=2\) and \(m_p=8\). Select the tuple \(h_\nu\) already fixed for this datum, and put \(k_\nu=k_{0\nu}h_\nu\).

At \(p=h_\nu\), impose \(F_\nu(T)=0\pmod p\). On this plane, each other \(F_j\) is nonconstant. Each condition \(\epsilon_jF_j(T)=0\) or \(-k_j\) removes at most one line, and at most \(2(t-1)p<p^2\) points are removed. There is therefore a point satisfying \[\epsilon_jF_j(T)\ne0,-k_j\pmod{h_\nu} \quad(j\ne\nu).\] Lift that point modulo \(p^2\) so that \[\epsilon_\nu F_\nu(T)/p \ne0,-k_{0\nu}\pmod p.\] Such a lift exists because \(F_\nu\) is primitive, so varying the lift changes \(F_\nu(T)/p\) through all residues modulo \(p\). The Chinese Remainder Theorem combines these choices with the chosen classes at \(S_0\) into one vector class modulo \[ W_0=\prod_{p\in S_0}p^{H_0+3} \prod_{\nu=1}^t h_\nu^2. \tag{79}\] On this class all \[\Psi_\nu(T)=\epsilon_\nu F_\nu(T)/k_\nu\] are integers and are nonzero modulo every prime dividing \(W_0\). At a prime in \(S_0\), the definition of \(\xi_{p\nu}\) and (78) give \(\Psi_\nu\equiv c_p\pmod{m_p}\). The conditions at each \(h_j\) give the remaining exclusions. Thus the divided forms equal \(1\) modulo \(8\) and \(3\) modulo \(7\), and avoid \(-1\) modulo every odd prime dividing \(W_0\). Notice also that \(W_0/k_\nu\) is divisible by every prime dividing \(W_0\). This extra divisibility will control the local factors of the prime-counting theorem.

Fix a smaller rectangular box \(\mathcal B'\) of positive volume with \(\overline{\mathcal B'}\subset\mathcal B\). Since the signs agree with the forms on \(\mathcal B\) and its closure avoids their zero planes, there are constants \(0<c_A<C_A\) such that \[ c_AX\le\Psi_\nu(T)\le C_AX \quad(T\in X\mathcal B,\ 1\le\nu\le t). \tag{80}\] All choices for this \(A\), including \(W_0\), are now fixed before choosing \(L\). The catalogue was finite, so \(W_0\) ranges in a fixed finite set as \(A\) varies.

A lower bound for the prime patterns

Let \(L\) be a fixed cutoff larger than every prime dividing \(W_0\), and put \[W=W_0\prod_{\substack{p\le L\\p\nmid W_0}}p.\] At each new prime \(p\le L\) allow all residue classes of \(T\) satisfying \[ \epsilon_\nu F_\nu(T)\ne0,-k_\nu\pmod p \quad(1\le\nu\le t). \tag{81}\] At most \(2t\) affine planes, each of \(p^2\) points, are excluded. Thus if \(\mathcal R_W\) is the set of all allowed vector classes modulo \(W\), including the one fixed class modulo \(W_0\), then \[ \frac{|\mathcal R_W|}{W^3} \ge W_0^{-3}\prod_{\substack{p\le L\\p\nmid W_0}}(1-2t/p) \ge c_0(\log L)^{-C_0}. \tag{82}\] The last bound follows by taking logarithms and using Mertens’ estimate \(\sum_{p\le L}p^{-1}=\log\log L+O(1)\) (Williams 1974, equation (1.1) and Theorem 1). The constants can be fixed independently of \(L\); since \(W_0\) ranges in a fixed finite set they could also be made independent of \(A\). Whenever \(\Psi_\nu(T)=r\) is prime, these conditions and the initial conditions imply that no odd \(l\le L\) divides \(r+1\). Such an \(l\) cannot divide both orders in (76).

Let \(N_{A,L}(X)\) be the number of \(T\in X\mathcal B\cap\mathbb Z^3\) for which every \(\Psi_\nu(T)\) is prime and all imposed congruences hold. We claim that \[ N_{A,L}(X)\ge c_A(\log L)^{-C_0}\frac{X^3}{(\log X)^t} \tag{83}\] for sufficiently large \(X\) depending on \(A,L\), with \(c_A>0\) independent of \(L\).

We use the following precise form of the Green–Tao–Ziegler theorem. For a fixed finite collection of integer affine-linear forms \(\psi_1,\ldots,\psi_t\) on \(\mathbb Z^d\) whose nonzero linear parts are pairwise nonproportional, and a fixed dilating box where all values are positive, the von Mangoldt weighted count is \[ \sum_{n\in\mathcal D_X\cap\mathbb Z^d}\prod_{\nu=1}^t\Lambda(\psi_\nu(n)) =\operatorname{vol}(\mathcal D_X)\prod_p\beta_p+o(X^d), \tag{84}\] where \[\beta_p=(1-1/p)^{-t}p^{-d} \#\{n\in\mathbb F_p^d:\psi_\nu(n)\ne0\text{ for every }\nu\}.\] Here all coefficients and the box are fixed before \(X\to\infty\). This is the finite-complexity case of the Main Theorem of (Green and Tao 2010), combined with the Möbius–nilsequence theorem (Green and Tao 2012, Theorem 1.1) and the inverse theorem (Green et al. 2012, Theorem 1.3), with its correction (Green et al. 2024). Pairwise nonproportional linear parts give finite complexity: for any chosen form, the other forms can be partitioned into singletons, none spanning it. The 2024 erratum corrects intermediate factorization and filtration statements and leaves the stated inverse theorem, and hence (84), unchanged.

For each \(T_0\in\mathcal R_W\), use its representative in \([0,W)^3\) and write \(T=T_0+Wn\). The forms \[\psi_{\nu,T_0}(n) =\epsilon_\nu F_\nu(T_0+Wn)/k_\nu\] have integer coefficients and pairwise nonproportional linear parts. Use \(n\in(X/W)\mathcal B'\). For all sufficiently large \(X\), depending on \(A,L\), this entails \(T\in X\mathcal B\), uniformly over the finite set of representatives \(T_0\). The volume in (84) is then \(\operatorname{vol}(\mathcal B')X^3/W^3\).

At \(p\mid W\), the coefficients of the variable parts of every divided form vanish modulo \(p\), whereas their constant values are nonzero. Consequently \(\beta_p=(1-1/p)^{-t}\ge1\). At \(p\nmid W\), multiplication by \(W\) and by \(k_\nu^{-1}\) is invertible. The zero sets of the forms are distinct affine hyperplanes, and each pair has an intersection of codimension two. Inclusion–exclusion, using only the first two terms for a uniform error, gives \[p^{-3}\#\{n:\psi_{\nu,T_0}(n)\ne0\ (\forall\nu)\} =1-t/p+O_t(p^{-2}).\] Thus \(\beta_p=1+O_t(p^{-2})\). Taking the original fixed threshold large enough, the product over any subset of these primes is bounded below by one fixed positive convergent product. This lower bound is independent of \(L\) and \(T_0\).

Prime powers in (84) are negligible. On a box of scale \(X\), there are \(O(X^{1/2}\log X)\) proper prime powers in the range of each nonconstant form, and at most \(O_{A,L}(X^2)\) lattice points for each prescribed value of that form, by solving for one coordinate. Their total weighted contribution is therefore \(O_{A,L}(X^{5/2}(\log X)^{t+1})=o_{A,L}(X^3)\). For genuine prime values the weight is at most \(C_A(\log X)^t\), by (80).

Sum (84) over the finitely many allowed progressions. For each fixed \(L\) their errors still sum to \(o_{A,L}(X^3)\), while the \(W^{-3}\) volume factor cancels the \(W^3\) in the number of progressions in (82). The lower bound for the local-factor product, the removal of prime powers, and the bound on the prime weights therefore give (83), with a positive coefficient independent of \(L\). We require no uniformity in the little-oh error as \(A\) or \(L\) varies.

We must now remove the patterns for which some pair of reduction orders has an odd common factor \(l>L\). For one form, we will bound the number of possible bad prime values by \(O_A(X/(L\log X))\) plus terms of smaller order as \(X\to\infty\) for fixed \(L\). For each fixed bad value, a second count will bound the points on its affine fiber for which the other \(t-1\) forms remain prime by \(O_A(X^2/(\log X)^{t-1})\). The contribution from the term \(X/(L\log X)\) is therefore \(O_A(X^3/(L(\log X)^t))\). This retains the factor \((\log X)^{-t}\) in (83) and introduces the saving \(1/L\). The first count sieves the pairs \((a,b)\) in (75); the second sieves the two free coordinates on a fiber. Both use the following two-dimensional estimate.

An elementary upper sieve with uniform errors

Let \(\mathcal P_Z\) be a set of primes at most \(Z\), with forbidden subsets \(\Omega_p\subset\mathbb F_p^2\) having densities \(g(p)=|\Omega_p|/p^2\) in \((0,1/2)\). Extend \(g\) multiplicatively to squarefree integers supported on \(\mathcal P_Z\). Let \(\mathbf x\) range over a square in \(\mathbb Z^2\) with \(n\) integer choices in each coordinate. The simultaneous forbidden conditions at the primes dividing a squarefree integer \(e\) hold at \[ n^2g(e)+O(ne+e^2) \tag{85}\] points, uniformly in the location of the square. Indeed the Chinese Remainder Theorem gives \(e^2g(e)\) residue classes, and the count in each differs from \(n^2/e^2\) by \(O(n/e+1)\).

The following estimate uses Selberg’s \(\lambda^2\) method (Selberg 1947); compare (Montgomery and Vaughan 2007, sec. 3.2). We give the weights and error term needed for these two-dimensional boxes.

We shall always set \(Z=X^{1/100}\). Two consequences for \(G(Z)\) will be useful. If \(\mathcal P_Z\) contains all primes above a fixed threshold up to \(Z\), and \(g(p)=s/p+O_s(p^{-2})\) for a fixed positive integer \(s\), then \[ G(Z)\gg(\log X)^s. \tag{89}\] If instead it contains the primes \(p\equiv1\pmod4\) above a fixed threshold up to \(Z\), with at most one additional prime omitted, and \(g(p)=2/p+O(p^{-2})\), then \[ G(Z)\gg\log X, \tag{90}\] uniformly in that omitted prime. In these two estimates the implicit constants may depend on the fixed threshold and the fixed density estimates, as well as on \(s\), but not on \(X\) or the omitted prime.

Here are details of the truncation behind both claims. For \(y=Z^\theta\), where \(\theta>0\) will be fixed sufficiently small, the Euler product \[D(y)=\prod_{\substack{p\le y\\p\in\mathcal P_Z}} (1+g_*(p)^{-1})\] is respectively \(\gg(\log y)^s\) or \(\gg\log y\), by the ordinary and progression Mertens estimates (Williams 1974, equation (1.1) and Theorem 1). In the latter case removing one prime loses at most a factor two because \(g(p)<1/2\). In the expansion of \(D(y)\) give a squarefree \(e\) weight \(1/g_*(e)\). The weighted mean of \(\log e\) equals \[\sum_{\substack{p\le y\\p\in\mathcal P_Z}} g(p)\log p\ll\log y.\] This uses \(\sum_{p\le y}(\log p)/p\ll\log y\), and the bound is uniform when a prime is omitted. For a small enough fixed \(\theta\), Markov’s inequality shows that at least half the weight has \(e\le Z\). Those terms occur in \(G(Z)\), proving both assertions.

Counting bad prime values

Fix one index \(\nu\). All its possible prime values lie in the interval (80). Let \(B_{A,L}(X)\) be the number of primes in that interval satisfying (74) for which the two orders have a common odd factor \(l>L\). We claim \[ B_{A,L}(X)\ll_A \frac{X}{L\log X}+X^{7/8}+\sqrt X\log X. \tag{91}\] It suffices to count pairs \((a,b)\in\mathbb Z^2\) with \(|a|,|b|\le C_A\sqrt X\), \(a\ne0\), for which \(a^2+b^2\) is a prime in the interval and (77) holds for some prime \(l>L\). Every bad prime supplies such a pair. Overcounting pairs, and counting a pair several times if necessary, gives an upper bound.

First consider \(L<l\le X^{1/8}\). If \(-1\) has no square root modulo \(l\), there are no pairs. Otherwise take each of its two roots \(b_0\) and write \[a=lU,\qquad b=b_0+lV.\] The relevant \(U,V\) lie in an encompassing square with \[ n\asymp_A\sqrt X/l\gg_A X^{3/8} \tag{92}\] integer choices in each coordinate, uniformly for these \(l\) and for least nonnegative representatives \(b_0\). Sieve by all sufficiently large primes \(p\equiv1\pmod4\), \(p\ne l\), up to \(Z=X^{1/100}\). Modulo \(p\) the substitution \((U,V)\mapsto(lU,b_0+lV)\) is an affine bijection, so the forbidden equation \(a^2+b^2=0\) has exactly \(2p-1\) solutions: it is the union of two distinct lines meeting at one point. Thus \[g(p)=2/p-1/p^2,\] independently of \(l,b_0\). Since the prime value \(a^2+b^2\) is \(\asymp_A X>Z\), it avoids all these forbidden sets. Lemma 31 and (90) give \(O(n^2/\log X+nZ^6+Z^8)\) pairs. At the shortest possible squares, \[\frac{nZ^6+Z^8}{n^2/\log X} \ll_A(\log X) \bigl(X^{6/100-3/8}+X^{8/100-3/4}\bigr)=o_A(1).\] The error is therefore absorbed uniformly before summing over \(l\). Each \(l\) contributes \(O_A(X/(l^2\log X))\), and summing over even all integers \(l>L\) gives \(O_A(X/(L\log X))\).

For \(l>X^{1/8}\) use a lattice count without a sieve. Since \(a\ne0\) and \(l\mid a\), necessarily \(l\le C_A\sqrt X\), and the number of possible \(a\) is \(O_A(\sqrt X/l)\), with no additive constant. There are at most two classes for \(b\), giving \(O_A(\sqrt X/l+1)\) possibilities. The count for one \(l\) is thus \[O_A(X/l^2+\sqrt X/l).\] To sum it explicitly, split \(X^{1/8}<l\le C_A\sqrt X\) into dyadic blocks \(D<l\le2D\). Even if every integer in a block were allowed, its contribution would be \(O_A(X/D+\sqrt X)\). The first terms form a geometric sum \(O_A(X^{7/8})\), while there are \(O_A(\log X)\) blocks. Their total is \(O_A(X^{7/8}+\sqrt X\log X)\). This proves (91) with constants independent of \(L\).

A uniform sieve on each affine fiber

For a fixed prime value \(r\asymp_A X\) of the \(\nu\)th form, consider \[ F_\nu(T)=\epsilon_\nu k_\nu r. \tag{93}\] We claim that the number of \(T\in X\mathcal B\cap\mathbb Z^3\) on this fiber for which all other \(\Psi_j(T)\) are primes is \[ \ll_A\frac{X^2}{(\log X)^{t-1}}, \tag{94}\] uniformly in \(r\) and independently of \(L\). In this upper bound we discard all the congruence conditions used for the lower bound.

Since \(F_\nu\) is primitive, an integral unimodular coordinate change makes it the first coordinate. The other two coordinates of every point of the fiber inside \(X\mathcal B\) lie in an encompassing square with \(n\asymp_A X\) integer choices in each direction. The coordinate change depends on \(A,\nu\) but not on \(r,L\). Sieve by all primes up to \(Z=X^{1/100}\) above a fixed sufficiently large threshold, excluding every prime dividing any multiplier. For all sufficiently large \(X\), every sieving prime differs from \(r\), so the fiber constant \(c=\epsilon_\nu k_\nu r\) is nonzero modulo each such prime \(p\).

On the affine plane \(F_\nu=c\) over \(\mathbb F_p\), every other \(F_j\) restricts to a nonconstant affine function, because \(F_j\) is not proportional to \(F_\nu\). Furthermore the \(t-1\) zero lines of these restrictions are distinct. If two coincided, their affine functions would be scalar multiples on that plane. For some \(\lambda\in\mathbb F_p^*\) and \(\mu\in\mathbb F_p\) this would give the identity \[F_j-\lambda F_k=\mu(F_\nu-c).\] The left side is homogeneous, so comparison of constant terms gives \(\mu c=0\). Since \(c\ne0\), one has \(\mu=0\), contradicting the pairwise independence of \(F_j,F_k\). This argument also covers the case that three ambient forms are jointly dependent: a nonzero constant shift then gives distinct parallel lines, rather than coincident ones.

The union of the \(t-1\) distinct zero lines has \((t-1)p+O_t(1)\) points. All must be avoided, since the other prime values have size \(\asymp_A X\) and exceed the sieving primes; the multiplier is invertible modulo every sieving prime. Thus \(g(p)=(t-1)/p+O_t(p^{-2})\), uniformly in the fiber. Lemma 31 and (89) give \[O_A\!\left(\frac{X^2}{(\log X)^{t-1}}+XZ^6+Z^8\right) =O_A\!\left(\frac{X^2}{(\log X)^{t-1}}\right),\] which proves (94). All excluded sieving primes were fixed in advance of \(r,L\), and residue-count errors are uniform in the position of the encompassing square. This proves the claimed uniformity.

Completing the prime-pattern construction

For each of the \(t\) forms, multiply the number of its possible bad prime values in (91) by the uniform fiber bound (94). A union bound shows that at most \[ \frac{C_AX^3}{L(\log X)^t} +o_A\!\left(\frac{X^3}{(\log X)^t}\right) \tag{95}\] of the prime patterns can have a common odd factor \(l>L\) in one of their pairs of reduction orders. More explicitly, the ratios of the two other terms to \(X^3/(\log X)^t\) are \(O_A(X^{-1/8}\log X)\) and \(O_A(X^{-1/2}(\log X)^2)\); both tend to zero. None of the constants in this upper estimate depends on \(L\).

Choose \(L\), after all data depending on \(A\) have been fixed, large enough that \[C_A/L<\tfrac12 c_A(\log L)^{-C_0}.\] This is possible because \((\log L)^{C_0}/L\to0\). Then choose \(X\) sufficiently large for (83), for all the upper estimates, and for the residual little-oh in (95) to be smaller than the remaining lower-bound coefficient. At least one prime pattern survives. Small common odd factors were already excluded by the CRT conditions, and all larger ones have just been removed. The \(2\)-part and the prime-to-\(r\) assertions proved from (76) now give (73). The primes have size at least \(c_AX\), so increasing \(X\) makes all of them exceed any prescribed \(B\). This proves Lemma 29.

Together with Corollary 30, the prime-pattern lemma proves Corollary 7, completing the ordinary integer witnesses used in Section 2. The analysis there combines the tested representations with the parity, height, and elliptic inputs to turn success of all finite tests into an ordinary integer zero.