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HILBERT'S TENTH PROBLEM OVER THE RATIONAL NUMBERS
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A complete proof, written to be readable with high-school algebra.
CONTENTS
--------
0. The question and the answer
1. Background: the integer case
2. The overall strategy in one page
3. The elliptic curve and its integer indices
4. Contact forms, five points, and the ring reinterpretation
5. The theory T, its finite tests, and compactness
6. Pole parity: an existential formula with even-pole restriction
7. A height bound from odd contact with five points
8. The formal logarithm and its local control
9. Primes with useful reduction orders
10. Turing degree and quartic normal form
11. Closing remarks and how to read this paper
Extended notes; index of symbols
0. THE QUESTION AND THE ANSWER
------------------------------
Hilbert's tenth problem asks: is there a mechanical procedure that,
given a polynomial with integer coefficients, decides whether it has an
integer solution?
In 1970 Matiyasevich, building on work of Davis, Putnam and Robinson,
proved the answer is NO. The decision problem for integer solutions is
undecidable.
The question of this paper replaces "integer" by "rational". Does a
mechanical procedure exist that decides whether a polynomial with
integer coefficients has a solution in rational numbers? The number of
variables is part of the input.
THEOREM 0.1. No. There is no algorithm which, given
f in Z[X_1, ..., X_n]
with n part of the input, decides whether f has a zero in Q^n.
Here is a first reason why the rational case is genuinely different
from the integer one. For integer solutions, "search all integer
tuples" covers every candidate: if a solution exists, another step
will find it. For rational solutions, a search over rational tuples
still finds every solution that exists, but it says nothing when no
solution exists. An undecidability proof therefore cannot be a
"search this space" argument. It has to build, from f alone, a finite
object that fails exactly when f has no integer solution.
The heart of the paper is exactly that object. For each nonconstant f
we build an effectively generated sequence of FINITE tests, each of
which is a rational solvability question of a specific shape. The
sequence satisfies
f has an integer zero <=> every test succeeds. (0.1)
If f has an integer zero, that zero gives witnesses for every test,
so the forward direction is pure bookkeeping (Section 5). The hard
direction is: if every test succeeds, then f has an integer zero.
That is a compactness-plus-arithmetic argument (Sections 5-10), and
every bit of arithmetic in the paper exists to make that direction
work.
One clue about what the tests look like. Each test asks for rational
witnesses u_1, u_2, v, a "prime-like" ring element r, and two
auxiliary ring elements h, j_0 such that certain valuation congruences
hold at each of the two residue branches of r. The witnesses describe
the modeled integer A in two different ways: additively, through an
elliptic multiple A P, and multiplicatively, through truncated formal
logarithms at carefully chosen rational primes. When all tests
succeed in a compactness limit, these two descriptions are forced to
agree. A height argument then shows that the modelled "integer"
coordinates are actually ordinary integers.
Section 2 restates this whole scheme in one page before any of its
parts are proved. Readers who want an outline should read Section 2
first.
1. BACKGROUND: THE INTEGER CASE
-------------------------------
Fix the smallest amount of language that gets the job done. A relation
R on the positive integers is
- COMPUTABLY ENUMERABLE (c.e.) if some algorithm lists exactly
its satisfying tuples;
- DIOPHANTINE if membership can be witnessed by another tuple
of positive integers y_1, ..., y_k satisfying a single integer
polynomial equation with integer coefficients.
So a Diophantine relation is one describable purely by polynomial
solvability. For example "x is prime" is Diophantine: witness y and
z, and require x = (y+2)(z+2) + 1 (with the cases x = 2 handled by a
small disjunction); "x is a square" is Diophantine: witness y and
require x = y^2.
The Matiyasevich-Davis-Putnam-Robinson theorem is:
THEOREM 1.1 (DPRM). Every c.e. relation on the positive integers is
Diophantine.
Two immediate corollaries make this theorem usable:
COROLLARY 1.2. Hilbert's tenth problem over Z is undecidable.
Proof: take a c.e. set known to be undecidable (the halting set, say),
encode it as a Diophantine relation, and a general decision procedure
would decide halting.
COROLLARY 1.3. For every c.e. set S of positive integers there is a
single polynomial F(x, y_1, ..., y_k) with integer coefficients such
that x is in S iff F(x, y) = 0 for some positive integers y.
We do not re-prove Theorem 1.1. What we need from it is only this
direction: certain explicitly given sets of positive integers are
Diophantine, and their Diophantine descriptions are effective. That
is the content of Corollary 1.3, applied to a few sets constructed
in Section 9.
There is one more background ingredient, essentially Julia Robinson's
1949 observation.
THEOREM 1.4 (Robinson). The set Z is first-order definable inside Q
using only + and *.
Robinson's formula uses both universal and existential quantifiers.
Two refinements: Poonen (2009) gave a formula with two universal plus
seven existential quantifiers, and Koenigsmann (2016) gave a universal
definition of Z in Q (equivalently, a Diophantine description of its
complement).
Definitions of Z inside Q do NOT immediately settle the rational
problem. To transfer the integer case, one would want to say "every
integer variable is a particular rational" using only +, * and
existential quantifiers. Robinson's formula is not of that form.
Koenigsmann's is not either. Section 2 explains how we sidestep this
gap.
One more background notion, so that later statements read correctly.
For a variety V over Q, Mazur's conjecture on the topology of
rational points asserts that the closure of V(Q) in V(R) has only
finitely many connected components. Mazur observed that his
conjecture would rule out a Diophantine definition of Z in Q, and
Cornelissen-Zahidi extended this: the conjecture would also rule out
a Diophantine MODEL of integer arithmetic inside Q. Both are
obstructions to the "clean" route. Neither rules out an
undecidability proof like the one below, which avoids Diophantine
definitions and Diophantine models entirely.
The recent history is worth recording for orientation. 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 Z[S^{-1}] whenever S contains T_1 and is
disjoint from T_2, giving undecidability for those rings. Taking S to
be the complement of T_2 gives a recursive set of density one and a
proper subring of Q. Eisentrager, Miller, Park and Shlapentokh later
produced computably presentable subrings of Q, some of density one,
whose Hilbert's tenth problem is Turing equivalent to that over Q.
Koymans-Pagano (2026) and Alpoge-Bhargava-Ho-Shnidman (2026) then
settled Hilbert's tenth problem for all infinite finitely generated
Z-algebras and for all rings of integers of number fields. None of
these results includes Q itself: Q is not a finitely generated
Z-algebra and not the ring of integers of any number field. The
proof below is the first to reach Q.
2. THE OVERALL STRATEGY IN ONE PAGE
-----------------------------------
Fix nonconstant f in Z[X_1, ..., X_n]. We want to decide f's integer
solvability using ONLY rational solvability queries of a specific
finite shape. The strategy has four layers.
LAYER 1. A first-order theory T with Z as a model.
We specify a language and a theory T and prove:
(a) T is recursive (axioms listed by a computable procedure),
(b) T has Z as a model,
(c) Z is the "integer sort" of any model of T,
(d) the ring of integer-sort elements is an integral domain,
(e) the ring does not contain sqrt(2) in its fraction field.
The axioms come in two kinds: standard ring axioms (cheap), and
"elliptic axioms" that remember each element a through a point T_a = aP
on a fixed rank-one elliptic curve, together with local valuation
conditions.
LAYER 2. Finite tests for f.
Add the named constants a = (a_1, ..., a_n) and the equation
f(a) = 0 to T, and Skolemise the resulting theory. A single finite
test is: assign a rational number to each ground term in a chosen
finite set, and require every ground instance of every universal axiom
to hold under that assignment, together with Phi on every ground-term
value. Here Phi is the existential formula of Section 6.
Each test is a rational solvability question of the shape
"do there exist rationals (u_1, u_2, v, r, B_r, h, j_0, ...)
satisfying a finite system of polynomial equations and
inequations?"
where a conjunction g_1 = ... = g_s = 0 is encoded as sum g_i^2 = 0,
and an inequation g != 0 is encoded as g y = 1 with a new variable y.
So a single finite test reduces to finitely many integer polynomial
solvability queries over Q.
Soundness (Section 5.5). If f has an integer zero, that zero satisfies
every test.
Completeness (Sections 5-10). If every finite test succeeds, then f
has an integer zero.
Then the two searches
(S1) enumerate integer tuples; check f(tuple) = 0;
(S2) enumerate finite tests; check rational solvability;
run in parallel decide f's integer solvability in finite time, given a
rational oracle.
LAYER 3. Completeness via compactness.
Suppose every finite test succeeds. By the compactness theorem of
first-order logic, all tests hold simultaneously in some model. The
ambient structure is an elementary extension of an ordinary many-sorted
structure containing Q, Z, R, F = Q(sqrt 2), the prime relations, the
valuation functions, the height functions, the elliptic multiple maps
a -> aP, and the finite-product maps used below. Elementarity gives
transferred versions of every first-order statement about Q, Z, R, F.
Inside that model, the ring R of "integer" values embeds into *Q. The
elliptic axioms are satisfied, so the transfer of "m E(F) = Z P" gives
an additive map
eta : R -> *Z, T_a = eta(a) P, eta(1) = 1,
which is injective by the (a != 0 => T_a finite with x y != 0) axiom.
But nothing forces eta to respect multiplication. Closing this gap is
the whole problem.
LAYER 4. Four arithmetic ingredients close the gap.
(i) POLE PARITY (Section 6). A positive existential formula Phi in
the language of rings, with fixed rational coefficients, such
that
- Phi(b) holds for every integer b;
- if Phi(b) holds at a rational b, then for all but finitely
many primes q, v_q(b) < 0 implies v_q(b) is even.
In one word: "Phi forces every pole to have even order".
(ii) LOCAL COMPARISON AT SPLIT PRIMES (Sections 3-4). At a prime r
splitting in F and satisfying certain reductions, on E and its
conjugate E^sigma simultaneously, truncated formal logarithms
recover the elliptic index of T_a from a ring element a. The
comparison is
v_r ( ell_K(z(T_{h a})) / ell_K(z(T_h)) - a ) >= K,
v_r ( z(T_{(h-4)a}^sigma) / z(T_{j_0}^sigma) ) >= K,
with h, j_0 chosen once and for all at r, independent of a.
(iii) PRIME PATTERNS (Section 9). For every integer A and every real
threshold Y_0 there exist integers u_1, u_2, v with v != 0 and
u_3 = A v - u_1 - u_2, so that thirteen fixed linear forms
evaluated at (u_i, v) are each of the form
(sign) * k * r,
for a fixed finite set of multipliers k and a rational prime r >
Y_0 that splits in F, is of good reduction for both E and
E^sigma, and satisfies gcd(d_1, d_2) | 4 at its two places,
where d_1 and d_2 are the local reduction orders. The finite
set of multipliers is fixed and independent of A and Y_0.
(iv) FIVE-POINT HEIGHT BOUND (Section 7). Five fixed rational points
C on P^1, five fixed contact forms L_b, and fixed constants
H, c > 0, so that every rational s = (u:v) outside C satisfies
h(s) <= H * M_S(s)^c,
where M_S(s) is the squarefree product of primes q outside a
fixed finite set S at which some normalized contact order
v_q(L_b(u, v)) - min{v_q(u), v_q(v)}
is odd.
THE FINAL ARGUMENT.
Suppose every finite test succeeds. Then:
(a) Compactness gives a model R -> *Q of T, with root tuple a in R^n
satisfying f(a) = 0 and Phi(a) for every a in R.
(b) Define
delta = f(eta(a_1), ..., eta(a_n)) in *Z,
B = max(1, |eta(a_1)|, ..., |eta(a_n)|).
Polynomial growth gives |delta| <= C_f B^D with D = deg f.
(c) If delta = 0, the ambient integer sort satisfies "f has an integer
zero", hence by elementarity so does ordinary Z.
(d) Suppose delta != 0. Fix an ordinary K large enough (K >= c D).
For any A in R, the model supplies a tested representation
A = (u_1 + u_2 + u_3)/v, v != 0,
whose contact forms are all tested primes.
(e) For each contact prime q of u_i/v, the odd-contact argument
selects a specific tested prime r whose q-valuation is odd.
Lemma 5.4 (coarsening, driven by pole parity) then produces a
coarsening of the q-adic valuation whose centre is one of the two
residue branches of r. Local comparison at r applies. Result:
v_q(A - eta(A)) >= K for every A in R.
(f) Apply (e) to A = a_j. For every q dividing M_K(u_i : v),
v_q(delta) >= K. Since M_K(u_i : v) is squarefree,
M_K(u_i : v)^K divides delta.
(g) Combine with (b) and (iv): h(u_i : v) <<_f B for each i, hence
h(A) <<_f B for every A in R.
(h) Apply this to the ring elements x_0(a_j), x_1(a_j), x_2(a_j)
witnessing x(T_{a_j}). Then h(x(eta(a_j) P)) <<_f B. Canonical
height on a fixed elliptic curve gives
h(x(nP)) = 2 h_hat(P) n^2 + O(1), h_hat(P) > 0.
If B > 1, pick j maximising |eta(a_j)| = B. Then
2 h_hat(P) B^2 <= C B + C',
forcing B bounded by an ordinary constant.
(i) An ambient integer lying in an ordinary finite interval is
ordinary. So eta(a_j) = n_j in Z. Additivity and eta(1) = 1
give eta(n_j) = n_j. Injectivity of eta gives a_j = n_j. Hence
f has an ordinary integer zero, contradicting delta != 0.
Sections 3-10 prove every line of this argument. Section 10 extracts
the Turing degree of the rational problem and the restricted-input
normal forms.
3. THE ELLIPTIC CURVE AND ITS INTEGER INDICES
---------------------------------------------
The proof runs on a single elliptic curve. Its job is to represent
"the integer a" by an elliptic point T_a = a P on a rank-one group.
3.1. The data.
F = Q(sqrt(2)), sigma = nontrivial automorphism of F/Q,
d = 75 - 53 sqrt(2),
E: y^2 = x^3 - d^2 x over F.
The integer d is positive at both real embeddings of F and has norm
d * d^sigma = 7. So the ideal (d) is a prime ideal of O_F = Z[sqrt 2]
lying over the rational prime 7, with residue field F_7.
At any prime r at which 7 is a nonsquare mod r, the reductions of E and
E^sigma = y^2 = x^3 - (d^sigma)^2 x are quadratic twists of one another
by opposite classes, so their Frobenius traces are opposite. Section 9
uses this.
3.2. Rank one.
PROPOSITION 3.1. E(F) has rank one. Consequently there is a positive
integer m and a non-torsion P in E(F) with
m E(F) = Z P.
Proof. O_F = Z[sqrt 2] is a PID. Minkowski's bound is sqrt(8)/2 =
sqrt(2) < 2, so every ideal class contains an ideal of norm one, and
every ideal is principal. The unit group satisfies
|O_F^* / (O_F^*)^2| = 4, and the units -1 and 1 + sqrt 2 realise all
four sign patterns at the two real embeddings, so every totally
positive unit is a square.
Consider the 2-isogenous curve E': y^2 = x^3 + 4 d^2 x with isogenies
phi: E -> E', hat phi: E' -> E, hat phi o phi = [2].
For a point with nonzero x-coordinate, define
alpha : E(F) -> F^*/(F^*)^2, alpha(T) = class of x(T),
with the convention that the origin goes to the trivial class and the
2-torsion point (0, 0) goes to the class of the linear coefficient.
Similarly define alpha' on E'(F). The kernels of alpha and alpha' are
hat phi E'(F) and phi E(F) respectively. The standard 2-isogeny
descent (Silverman, Arithmetic of Elliptic Curves, X.4.9) gives the
index identity
|im alpha| * |im alpha'| = 4 * 2^{rank E(F)}. (3.1)
For completeness, E has full rational 2-torsion, and either 2-torsion
point (±d, 0) maps under phi to the nonidentity point of ker hat phi,
so the kernel in the index count is trivial. Taking indices in
2 E(F) ⊂ hat phi E'(F) ⊂ E(F)
gives |E(F) / 2 E(F)| = 2^{rank E(F) + 2} = |im alpha| |im alpha'|.
Unit-coefficient lemma. If the linear coefficient of y^2 = x^3 + c x
is a unit at a finite place, then every nonzero x-coordinate has even
valuation there. Proof: if v(x) < 0 then v(x^3) < v(c x); if v(x) > 0
then v(c x) < v(x^3). Either way v(y^2) has the same parity as v(x),
and v(y^2) is even.
Applied to E, this lemma shows every class in im alpha is supported,
apart from its unit class, only at the prime ideal (d). Since O_F is a
PID, im alpha is bounded by 4 * 2 = 8, the first 4 being the four unit
classes and the second 2 being the possible inclusion of the class
of d.
Four classes are already supplied by 2-torsion:
1, -1, d, -d.
Let w = -5 + 4 sqrt 2 = d (1 + sqrt 2)^3. Direct calculation gives
w^3 - w = -8 d, and consequently
Q = (-d w, 2 sqrt 2 d^2) in E(F).
The element -d w has opposite signs at the two real embeddings, whereas
the four 2-torsion classes have equal signs at both embeddings. So
alpha(Q) lies outside the four-element subgroup, and |im alpha| = 8.
For alpha', the argument is parallel. On E' both real embeddings give
x^3 + 4 d^2 x the sign of x, so every nonzero x-coordinate is totally
positive. The unit-coefficient lemma handles all places except the
prime over 2 and the prime over d. At the prime over 2, normalise
v(2) = 1; then v(4 d^2) = 4. If v(x) is odd, the two terms 3 v(x) and
4 + v(x) have unequal valuations and their minimum is odd, impossible
for v(y^2). At the prime over d, v(4 d^2) = 2. If v(x) is odd, the
same argument is impossible unless v(x) = 1, in which case write
x = d u with u a local unit. The equation reads
y^2 = d^3 u (u^2 + 4).
Even valuation forces u^2 + 4 ≡ 0 mod (d), i.e. -4 ≡ 3 in F_7, but 3 is
not a square mod 7. So all finite valuations of x are even, the unit
part is a square (it is totally positive), and O_F is principal;
therefore x is a square in F^*. The 2-torsion point (0, 0) gives the
trivial class 4 d^2, and the origin does too. Hence |im alpha'| = 1.
By (3.1), rank E(F) = 1. Choose m divisible by the exponent of the
finite torsion subgroup of E(F). Then m E(F) is torsion-free of rank
one, generated by some P. Q.E.D.
For the rest of the paper, fix such m and P.
3.3. Points T_a and the index map eta.
Let R be an integral domain of characteristic zero with
sqrt(2) ∉ Frac(R). Write
R_F = R[sqrt 2], F_R = Frac(R)(sqrt 2).
Attach to every a in R a point T_a in E(F_R) subject to
T_0 = O,
T_1 = P,
T_{a + a'} = T_a + T_{a'} (additivity),
T_a in m E(F_R) (index integrality),
a != 0 => T_a finite and x(T_a) y(T_a) != 0 (injectivity).
Now work inside an elementary extension of an ordinary many-sorted
structure containing Q, Z, R, F, the prime and valuation relations,
height functions, the elliptic multiple maps, and finite-product maps.
The identity m E(F) = Z P transfers: every point of m E(*F) has a
unique expression n P, with n ∈ *Z. Define
T_a = eta(a) P, eta(a) ∈ *Z. (3.2)
The group axioms give
eta(a + a') = eta(a) + eta(a'), eta(1) = 1.
The injectivity axiom gives eta injective. So eta : R -> *Z is an
additive injection with eta(1) = 1.
Nothing so far constrains eta(a a'). That is the point: the elliptic
axioms only control the additive structure of R. Closing the additive-
multiplicative gap is the whole of Sections 4-10.
Conjugating (3.2) under sigma gives
T_a^sigma = eta(a) P^sigma.
3.4. Local coordinates and the formal logarithm.
For a finite point T in E with y != 0, put
z(T) = -x(T) / y(T), z(O) = 0.
Every nonzero multiple of P avoids the two 2-torsion points, so z(n P)
is defined for every integer n. Write
ell(Z) = Z + sum_{t >= 2} c_t Z^t in F[[Z]],
ell_K(Z) = Z + sum_{2 <= t <= K} c_t Z^t
for the formal logarithm in this local parameter and its truncation
through degree K. The coefficients c_t are effectively computable from
the fixed equation of E; Section 8 gives the derivation.
The next proposition is the technical engine of the reduction: it
recovers VALUATIONS of index ratios from point parameters, and QUOTIENTS
of index ratios once those quotients are known to be q-integral.
PROPOSITION 3.2 (Formal parameter valuation). There is a fixed finite
set of rational primes S_E with the following property. Let p ∉ S_E,
let v be a place of F above p, normalised by v(p) = 1, and let n, n' ∈ Z
with n' != 0. Suppose n P and n' P both reduce to O at v. Then
v( z(n P) / z(n' P) ) = v_p(n / n'). (3.3)
Further, for every integer K >= 1: if p > K + 1 and v_p(n/n') >= 0,
then ell_K(z(n' P)) != 0 and
v( ell_K(z(n P)) / ell_K(z(n' P)) - n / n' ) >= K. (3.4)
Both assertions also hold at the same place v after replacing
E, P, ell_K by E^sigma, P^sigma, ell_K^sigma and imposing the
corresponding reduction-to-O hypotheses on the conjugate points.
Proof in Section 8.
3.5. The local comparison lemma.
This is the single most-used arithmetic lemma in the paper. It
combines Proposition 3.2 with a compatibility statement about elliptic
multiples of P and P^sigma, and produces an additive congruence in the
ring from purely local point data.
LEMMA 3.3 (Local comparison). Let K >= 1 be an ordinary integer. Let
q be a prime of *Z (possibly nonordinary), with q ∉ S_E and q > K + 1.
Let v_* be a place of *F above q, normalised by v_*(q) = 1. Suppose
there are nonzero h, j_0 ∈ R, chosen independently of a ∈ R, such that
for every a ∈ R:
(i) The points T_h and T_{h a} reduce to O at v_*, the value
ell_K(z(T_h)) is nonzero, and
v_* ( ell_K(z(T_{h a})) / ell_K(z(T_h)) - a ) >= K.
(ii) The points T_{j_0}^sigma and T_{(h-4) a}^sigma reduce to O on
the conjugate curve at the same place v_*, and
v_* ( z(T_{(h-4) a}^sigma) / z(T_{j_0}^sigma) ) >= K.
Then every a ∈ R is q-integral, and
v_q( a - eta(a) ) >= K for every a ∈ R.
Here z(O) = 0. The denominator in (ii) is nonzero because j_0 != 0
and nonidentity points have nonzero x- and y-coordinates.
Proof. Apply the formal-parameter valuation equality (3.3) of
Proposition 3.2 to the conjugate points at v_*. The comparison (ii)
gives
v_q( eta((h-4) a) / eta(j_0) ) >= K.
The denominator is a nonzero integer, so v_q(eta(j_0)) <= 0, and
v_q( eta((h-4) a) ) >= K. (3.5)
Write V_a = eta(h a). Additivity applied to (h-4) a = h a - 4 a, and
then to a = 1, gives
V_a ≡ 4 eta(a) (mod q^K),
V_1 ≡ 4 (mod q^K).
Since q ∤ 2 (because q ∉ S_E), V_1 is a q-unit. Both V_a and eta(a)
lie in *Z, hence are q-integral. So
V_a / V_1 is q-integral,
V_a / V_1 ≡ eta(a) (mod q^K). (3.6)
Apply the truncated-logarithm comparison (3.4). The points T_{h a} and
T_h have indices V_a and V_1, reduce to O at v_*, and the ratio V_a/V_1
is q-integral by (3.6). Therefore
v_* ( ell_K(z(T_{h a})) / ell_K(z(T_h)) - V_a / V_1 ) >= K. (3.7)
Comparison (i) relates the same logarithm ratio to a. Combining (i)
with (3.7) gives v_* ( a - V_a/V_1 ) >= K. Since v_* restricts to
v_q on *Q, this is
v_q( a - eta(a) ) >= K,
and it also shows a is q-integral. The case where eta(a) vanishes is
covered by the convention v_q(0) = +infty. Q.E.D.
Two remarks. First, the auxiliary integers h and j_0 depend on the
prime r and on the residue branch (Section 4), but not on the element
a being tested. The derivation of (3.6) uses additivity for both a and
1, so a single h must serve for both. Second, Lemma 3.3 is a finite
polynomial statement; it transfers verbatim from ordinary Z to any
elementary extension. No infinite formal series is ever evaluated,
only the fixed truncations ell_K and ell_K^sigma.
4. CONTACT FORMS, FIVE POINTS, AND THE RING INTERPRETATION
---------------------------------------------------------
The point of this section is to turn "the elliptic axioms hold" into a
statement built from ring operations on R. Every point T_a and
coordinate x(T_a) must become a ring-level inhabitant expressible by
quantifiers over R, so the axioms of T (Section 5) are ring formulas.
4.1. Coordinate functions.
Fix six unary ring-valued functions
x_0, x_1, x_2, y_0, y_1, y_2 : R -> R.
The intent is to represent the coordinates of T_a as
x(T_a) = (x_0(a) + x_1(a) sqrt 2) / x_2(a),
y(T_a) = (y_0(a) + y_1(a) sqrt 2) / y_2(a). (4.1)
Values at a = 0 are unconstrained (T_0 = O). For a != 0, the theory
requires x_2(a) y_2(a) != 0, that (4.1) both make sense as elements of
F_R, that they satisfy the Weierstrass equation of E, and that T_a lie
in m E(F_R), expressed by existential coordinates for a point whose
m-multiple is T_a.
Because T_a is written through (4.1), the elliptic group law, the
identity T_0 = O, the exceptional case T_a + T_b = O (i.e. T_a is the
negation of T_b in the sum), and the mE-membership condition all become
FINITE systems of polynomial equations and inequations in the ring
elements x_i(a), y_i(a), x_i(b), y_i(a+b), etc. Every field equation
with coefficients in F separates into the two coordinates of sqrt 2
and clears denominators from x_2 and y_2.
Consequence. In the language of rings augmented by these six
functions, the elliptic axioms of Section 3.3 have recursive
first-order expressions. Every axiom of T is a finite Boolean
combination of ring equations and inequations. That is what allows
T to exist (Section 5).
4.2. Contact forms and five points.
Throughout the paper, h(s) is the absolute logarithmic Weil height of a
rational point s in P^1(Q). Section 7 constructs five fixed points
C = { b_1, b_2, b_3, b_4, b_5 } ⊂ P^1(Q),
including the point at infinity, together with a fixed finite set S of
rational primes and fixed constants H, c > 0.
For every b in C\{infinity}, and only for those b, fix ONCE AND FOR
ALL a primitive integral linear form
L_b(U, V) = alpha_b U + gamma_b V, gcd(alpha_b, gamma_b) = 1,
alpha_b != 0,
L_b, L_{b'} have different zero sets for b != b',
L_b, L_{b'} have unit determinant on the reductions outside S,
and L_infinity(U, V) = V.
The forms L_b for b in C are called the CONTACT FORMS associated with
the five fixed points. Their coefficients and their determinants are
all FIXED data of the problem, independent of the input f and of every
object constructed below.
For a rational point s = (u : v) in P^1(Q), with u, v coprime integers
at each prime q (i.e. min{v_q(u), v_q(v)} = 0 at q), the CONTACT ORDER
of s at q with respect to b is
k_q(s, b) = v_q( L_b(u, v) ) - min{ v_q(u), v_q(v) }. (4.2)
The definition of k_q(s, b) does not depend on the choice of coprime
homogeneous coordinates: multiplying both u and v by a nonzero integer
c adds the same v_q(c) to both sides of (4.2), and the reader checks
that k_q(s, b) is unchanged. It is nonnegative after scaling.
For fixed coprime representatives (u, v) of s, the integers L_b(u, v)
for the five b's have only finitely many prime divisors, so the
following product is finite.
DEFINITION 4.1. Let M_S(s) be the product, taken WITHOUT repetition, of
the primes q outside the fixed finite set S at which at least one of the
five contact orders k_q(s, b) is odd. Call the primes counted by
M_S(s) the CONTACT PRIMES of s.
The height bound in Section 7 bounds h(s) in terms of M_S(s). The
five points C are chosen to make that bound possible; their explicit
construction uses a quaternionic Shimura quotient of genus zero with
exactly five ramification values.
4.3. From contact orders to a single tested prime.
The reduction hands us a representation
A = (u_1 + u_2 + u_3) / v, v != 0,
of every ring element A in terms of three "slopes" u_i / v.
Suppose there is a prime q counted by M_S(s_i) for one of the slopes
s_i = (u_i : v). Then at least one of the five contact orders is odd
at q. Because the coefficients alpha_b, gamma_b of the contact forms
are q-units and the pairs L_b, L_{b'} have unit determinant outside S,
the minimum of the five integer valuations v_q(L_b(u_i, v)) equals
t = min{ v_q(u_i), v_q(v) }.
So the five valuations v_q(L_b(u_i, v)) are all congruent to t modulo 2
iff all contact orders k_q(s_i, b) are even. Since one k is odd, at
least one of the five valuations v_q(L_b(u_i, v)) is odd, i.e. at least
one of the integers L_b(u_i, v) has odd q-valuation.
In the tested representation, this integer L_b(u_i, v) is required to
factor as (sign) * k * r for some multiplier k from a fixed finite set
K ⊂ Z_{> 0}
and some ring element r in R satisfying "rR is a proper ideal and R/rR
is a field". Since k is a q-unit (K is a fixed finite set of integers
outside the enlarged exceptional set S_K), the parity of v_q(L_b(u_i,v))
is the parity of v_q(r).
So the height at the slope s_i forces a SPECIFIC PRIME-LIKE ring
element r in R whose q-valuation is odd. Section 5.7 uses r to select
a residue branch of R at r; Section 5.8 uses pole parity to push that
branch back to an actual valuation of the ambient field *F at q.
4.4. Reformulation in ring language.
The axioms of T require, for every A in R and every fixed precision
K >= 1:
There exist u_1, u_2, v in R with v != 0 such that
A v = u_1 + u_2 + u_3,
each of the thirteen numbers
v, L_b(u_i, v) for 1 <= i <= 3, b in C\{infinity}
equals (sign) * k * r for some k in K and some r in R,
and at each such r, both residue branches of R at r satisfy the
local comparison conditions of Lemma 3.3 for every a in R.
The number thirteen is v twice plus four contact forms per slope for
three slopes: 1 + 4 * 3 = 13.
4.4.1. Why thirteen. Every slope u_i/v, i = 1, 2, 3, contributes one
copy of v and four contact forms L_b(u_i, v), b in C\{infinity}.
Requiring each of these thirteen integers to factor as (sign) * k * r
means each is a "prime value" up to a fixed multiplier from the fixed
finite set K. That is the content of the prime-pattern lemma,
Section 9. The ordinary integer model supplies the required thirteen
objects for every integer A, and the compactness model inherits the
same property by elementarity.
4.4.2. Encoding residue branches. Saying "R/rR is a field" is a
first-order statement over R: 1 ∉ rR, and every element outside rR has
an inverse mod rR. In the concrete model r is always a rational prime;
in an abstract model r is a ring element required to behave like a
prime generator. Splitness at r is encoded by requiring
There exists B_r in R with B_r^2 ≡ 2 (mod rR)
and 2 B_r invertible in R/rR.
This defines two maximal ideals of R_F = R[sqrt 2]:
p_+ = ker( R_F -> R/rR, sqrt 2 |-> +B_r ),
p_- = ker( R_F -> R/rR, sqrt 2 |-> -B_r ). (4.3)
These are the two "residue branches" at r. In the ordinary integer
model, if r is a rational prime that splits in F = Q(sqrt 2), its two
places correspond to the two residues of sqrt 2 mod r; these are +B_r
and -B_r. So p_+ and p_- correspond exactly to the two places of F
above r.
4.4.3. Encoding local valuation conditions without valuations. For
D' in F_R, the assertion "D' lies in r^K (R_F)_{p_±}" is written
D' = r^K X / Y with X, Y in R_F,
and the reduction of Y under sqrt 2 |-> ± B_r is nonzero mod rR
(hence a unit in the field R/rR).
Every quantifier here ranges over R (through the representation of
elements of R_F as pairs (A + B sqrt 2), A, B in R, and of elements of
F_R as quotients of such pairs by nonzero denominators). So the whole
assertion "D' lies in r^K (R_F)_{p_±}" is expressed by first-order ring
quantifiers, with fixed exponents and no external valuation function.
That is what allows the local comparison of Lemma 3.3 to become axioms
of T.
4.4.4. f never appears here. Nothing in this section depends on the
input polynomial f. The five-point set C, the contact forms L_b, the
multipliers in K, the auxiliary h, j_0 at each tested prime, and the
constants H, c in Section 7 are all fixed once and for all. The input
f enters Section 5 only through "add the constants a and the equation
f(a) = 0 and Skolemise". That is what lets a single oracle machine for
the rational problem exist (Section 10).
5. THE THEORY T, ITS FINITE TESTS, AND COMPACTNESS
--------------------------------------------------
Fix f in Z[X_1, ..., X_n] with n part of the input. Constant f is
settled by inspection, so assume f is nonconstant. This section states
T, proves T has an ordinary integer model, describes the finite tests,
and proves compactness and the finite-test completeness argument.
Sections 6-9 prove the four arithmetic inputs that Section 5 uses.
5.1. The language.
A many-sorted language with these sorts:
Q-sort for rational elements,
Z-sort for integer elements,
R-sort for real elements,
F-sort for elements of Q(sqrt 2).
Relations: the prime relation on the Z-sort; ternary valuation
relations v_q(n) = e for each fixed rational prime q; the height
function on P^1(Q); the elliptic multiple map (n, T) |-> nT on
Z-sort × E-sort; and finite-product maps.
Constants: the rational numbers; the elements of F; the coordinates of
the point P; the coefficients c_t of ell_K for every K.
Functions of T, in addition to + and *: x_0, x_1, x_2, y_0, y_1, y_2
from R-sort to R-sort.
Ordinary elements are elements of the intended ordinary many-sorted
structure Q, Z, R, F with their usual operations, valuations and
heights. Starred sorts (*Q, *Z, *R, *F) will denote the corresponding
sorts in the elementary extension from the compactness theorem.
5.2. The axioms of T.
Ring axioms, plus
(rings) R is an integral domain of characteristic zero;
(no sqrt 2) a^2 != 2 b^2 for all a, b ∈ R with b != 0;
(elliptic) elliptic point axioms of Section 3.3, with T_a
represented through (4.1) and the six coordinate
functions;
(test) for each K >= 1, the tested representation axiom.
The tested representation axiom reads:
For every A ∈ R there exist u_1, u_2, v ∈ R with v != 0 such
that, putting u_3 = A v - u_1 - u_2, each of the thirteen numbers
in
v, L_b(u_i, v) (1 <= i <= 3, b ∈ C\{infinity})
equals (sign) * k * r, where k is one of the multipliers in the
fixed finite set K of Section 4, and r ∈ R satisfies:
r != 0, rR is a proper ideal, R/rR is a field;
there is B_r ∈ R with B_r^2 ≡ 2 mod rR and 2 B_r invertible
in R/rR;
for each of the two residue branches p_± of (4.3), and
some nonzero h, j_0 ∈ R chosen at that branch independently
of a ∈ R, the following hold for every a ∈ R:
(i) The points T_h, T_{h a} are formal at p_± in the
sense below, ell_K(z(T_h)) != 0, and
ell_K(z(T_{h a})) / ell_K(z(T_h)) - a
lies in r^K (R_F)_{p_±}.
(ii) The points T_{j_0}^sigma, T_{(h - 4) a}^sigma are
formal at the conjugate curve at p_±, and
z(T_{(h-4) a}^sigma) / z(T_{j_0}^sigma)
lies in r^K (R_F)_{p_±}.
"Formal at p_±" means: T = O, or else 1/x(T) lies in r (R_F)_{p_±}.
Note the quantifier structure. For each K >= 1 we add one axiom scheme
with A universally quantified; for each A there exist u_i, v; each of
the thirteen numbers factors as (sign) * k * r; each r supports two
branches p_+, p_-, each of which must satisfy (i) and (ii) universally
in a.
The conditions "rR proper" and "R/rR a field" express, in a way that
works in any model of T, that r behaves like a rational prime. In the
ordinary integer model, r is literally a rational prime, so all of
this is automatic. In an abstract model, "R/rR a field" forces the
ideal rR to behave under v_q the way a rational prime does in *Q, which
is all we use.
The first-order sentence "b != 0 => a^2 != 2 b^2" expresses "sqrt 2 not
in Frac(R)". Elements of R_F are pairs (A + B sqrt 2) with A, B ∈ R;
elements of F_R are fractions of such pairs by nonzero denominators.
Every field equation with coefficients in F separates into the two
sqrt 2 coordinates and clears denominators.
5.3. The standard integer model.
LEMMA 5.1. T is recursive, and has a model whose ring is Z. In this
model T_a = a P for every integer a.
Proof. Recursive: list the axioms by primitive recursion; schemes are
indexed by K >= 1 and are effective. Finite constants are symbolic
tags.
Ring axioms hold for Z, and sqrt 2 is not in Frac(Z) = Q. Set T_a = aP.
For nonzero a, the point aP lies in E(F) with x(aP) != 0 and y(aP) != 0,
since nonzero multiples of the non-torsion P avoid O and the two
2-torsion points. Take arbitrary integers for the six coordinate
functions at a = 0; the representation (4.1) satisfies the point
axioms.
Fix K >= 1 and A ∈ Z. The ordinary prime-pattern theorem (Corollary
9.2) supplies integers u_1, u_2, v with v != 0, so that, putting
u_3 = A v - u_1 - u_2, each of the thirteen numbers in the tested list
is (sign) * k * r for some k ∈ K and a positive rational prime r above
any prescribed real lower bound, with r ≡ 1 (mod 8) and r ≡ 3 (mod 7).
These residue conditions make 2 a square mod r and 7 a nonsquare mod r,
so r splits in F.
For each such r, Z/rZ is a field. Splitness supplies B_r ∈ Z with
B_r^2 ≡ 2 mod r and 2 B_r invertible mod r (automatic since r is odd).
The two residue branches are
p_+ = ker(Z[sqrt 2] -> Z/rZ, sqrt 2 |-> +B_r),
p_- = ker(Z[sqrt 2] -> Z/rZ, sqrt 2 |-> -B_r),
corresponding to the two places of F above r.
For each branch, Corollary 9.3 supplies nonzero integers h, j_0 with
the two formal-kernel hypotheses, fixed for all a ∈ Z, satisfying
v_r( ell_K(z(h a P)) / ell_K(z(h P)) - a ) >= K, (5.1)
v_r( z((h-4) a P^sigma) / z(j_0 P^sigma) ) >= K. (5.2)
Equation (5.1) follows from Section 8 with indices h a and h; their
ratio is a. Equation (5.2) follows from (3.3) of Section 3 applied to
the conjugate curve with indices (h - 4) a and j_0. The conditions
"reduce to O at r" and "1/x lies in r (O_F)_{p_±}" translate directly
into the ring-language memberships of Section 4.4.3. So all axioms of
T hold in Z.
The coefficients c_t of ell_K are computable from E (Section 8). The
fixed sets, fixed integers, rational coefficients, and coordinates of
P are finitely many exact constants that can be hardcoded into T.
Their existence suffices; no effective search for them is required.
No list of all true arithmetic sentences enters the construction.
Q.E.D.
5.4. Finite tests.
Add the constants a = (a_1, ..., a_n) and the equation f(a) = 0 to T,
and Skolemise. "Skolemise" means: convert to prenex form and then
replace each existential quantifier with a function symbol applied to
preceding universal variables. The result is a universal theory with
quantifier-free matrices. Every added function symbol is ring-valued.
Define a GROUND TERM as a term without variables, built from named
constants using ring operations and added Skolem functions. A FINITE
TEST consists of: pick any finite set of ground terms; assign a
rational number to each such term; require
(a) the constants 0, 1 and the ring operations have their
usual values;
(b) equal input tuples give equal outputs for each function
symbol;
(c) every ground instance of every universal axiom holds;
(d) every ground-term value satisfies Phi.
Here Phi is the positive existential formula of Section 6. Its
witnesses are auxiliary rational variables, not values of additional
ring functions; they do not create new ground terms subject to (d).
Similarly, inverse variables introduced to test a polynomial
inequation are auxiliary witnesses, not new ring elements.
A single finite collection of (a)-(d) is decidable by rational
solvability alone. Introduce the Phi-witnesses; the finite collection
becomes an existential Boolean combination of polynomial equalities
and inequations. Encode each inequation g != 0 as g y = 1 with a new
variable y. Encode each conjunction g_1 = ... = g_s = 0 as
sum g_i^2 = 0. Clearing fixed rational denominators reduces the whole
test to finitely many integer polynomial solvability questions over Q.
5.5. Soundness.
Suppose f has an integer zero a ∈ Z^n. Extend the model of Lemma 5.1
by interpreting the constants a as this integer tuple, and the Skolem
functions by any chosen witnesses. All ground-term values are
integers, and Phi holds on every integer (Proposition 6.1, clause 1).
So every finite test succeeds.
5.6. Compactness.
LEMMA 5.2 (Compactness). If every finite rational test succeeds, then
there exist
(a) an elementary extension of the many-sorted structure specified
in Section 5.1,
(b) a model R of T, and
(c) a tuple a ∈ R^n with f(a) = 0,
together with an embedding R -> *Q into the rational sort such that
Phi holds at every element of the image.
Proof. Fix the ordinary many-sorted structure. Expand it by
constants naming all ordinary elements in their respective sorts. Its
ELEMENTARY DIAGRAM consists of all first-order sentences in that
language that are true in the expanded structure. Use this diagram,
and add:
- constants assigning a value of *Q to every ground term;
- the ring axioms and the functionality constraints;
- all ground universal instances of the Skolemised axioms;
- Phi for every term value;
- the axioms of T.
Every finite subset has a model. Its finitely many constraints are
among a successful rational test (Section 5.4), interpreted in the
ordinary structure. Finitely many sentences of the elementary diagram
are already true there.
Compactness (Marker, Model Theory, 2002) therefore produces a
simultaneous model in an elementary extension.
Let R be the set of values of the ground terms. Constraint (a) makes
R a subring of *Q. Interpret each added function by applying its
symbol to representatives of ground terms; (b) makes the result
independent of representatives. Every element of R is a ground-term
value, so (c) holds in R by definition. Its reduct is a model of T.
The embedding R -> *Q is injective by (a), because distinct ground-term
values are distinct rationals. The fraction-field interpretation
extends to F_R -> *F. Q.E.D.
Three remarks. (i) The elementary diagram is used only in the
existence argument; it need not be recursive and is never queried by
the finite rational tests. (ii) The point axioms of T hold in R, so
the additive injection eta : R -> *Z of Section 3.3 is available in
this model with T_a = eta(a) P. (iii) The condition "Phi holds on
every element of the image" makes pole parity available for every
a ∈ R and preserved under negation, by transfer of the first-order
statement "Phi(a) holds" in the ambient structure.
5.7. Odd valuations pick a tested branch.
We now have a model R and an embedding R -> *Q satisfying every axiom
of T.
Fix an ordinary K >= 1. Enlarge the fixed finite set S of Section 7
to a finite set S_K containing:
- the exceptional set for Phi;
- the supports of every multiplier in K;
- the primes needed to keep the contact forms distinct with unit
determinants;
- the bad or ramified primes of E, E^sigma, F;
- the finitely many primes excluded by Section 8 at precision K.
Write M_K(s) = M_{S_K}(s) for the contact radical with these larger
exceptions. Corollary 7.3 gives M_S(s) <= (prod_{q in S_K\S} q)
M_K(s) and h(s) <= H_K M_K(s)^c for an ordinary constant H_K. Enlarge
c once and for all to a positive integer if necessary.
For any A ∈ R, the tested representation axiom at precision K gives
u_1, u_2, u_3, v ∈ R with v != 0, u_3 = A v - u_1 - u_2, and thirteen
contact forms each of the form (sign) * k * r for k ∈ K and r ∈ R
supporting two residue branches p_±. Set s_i = (u_i : v) in P^1(*Q);
since v != 0 and all contact forms are nonzero, s_i lies outside C.
Consider any ambient prime q of *Z counted by M_K(s_i), i.e. q ∉ S_K
and some contact order of s_i at q is odd.
CLAIM 5.3. There is a factor r in the tested list with v_q(r) odd.
Proof. At q ∉ S_K, the minimum of the five contact form valuations at
q equals
t = min{ v_q(u_i), v_q(v) },
because the coefficients alpha_b, gamma_b are q-units and every pair
L_b, L_{b'} has unit determinant. So the five valuations
v_q(L_b(u_i, v)) are all congruent to t mod 2 iff all contact orders
k_q(s_i, b) are even. Since some k is odd, at least one of the five
valuations is odd. By the factorization L_b(u_i, v) = (sign) * k * r
with k a q-unit (S_K contains Supp(K)), v_q(r) has the same parity as
v_q(L_b(u_i, v)); hence v_q(r) is odd. Q.E.D.
5.8. Coarsening the q-adic valuation.
The element r with v_q(r) odd links the ring-level prime to the ambient
prime q, but R_{p_±} need not sit inside the q-adic valuation ring: R
may have q-denominators. Pole parity enters here to fix that.
LEMMA 5.4 (Coarsening). Let q be the prime of *Z from Claim 5.3, and
r ∈ R the element with v_q(r) odd. Then there is a residue branch
p ∈ {p_+, p_-} of the chosen representation, and an ambient place v_*
of *F over q, such that for every ordinary integer l >= 1 and every
D' ∈ F_R,
D' ∈ r^l (R_F)_p => v_*(D') >= l. (5.3)
Moreover, every point formal at this tested localization actually
reduces to O at v_*.
Proof. Let J ⊆ *Z be the convex subgroup generated by the negative
values v_q(R\{0}). Concretely, gamma ∈ J iff |gamma| is bounded by a
finite sum of absolute values of negative terms v_q(a), a ∈ R\{0}.
This J is defined in the metalanguage of the present proof; it will be
used only to construct one particular valuation of R_F, never inside
T.
Claim 5.5. Every valuation of a nonzero ring element lying in J is
even.
Proof of Claim 5.5. If J = 0, the claim is trivial. Otherwise, given
a ∈ R\{0} with v_q(a) ∈ J, take finitely many products of ring
elements having negative q-value to produce b ∈ R\{0} with
v_q(b) < -|v_q(a)|.
Then both b and a b in R have negative q-valuation. The transferred
soundness of Phi on every element of R (Corollary 6.2) gives that both
v_q(b) and v_q(a b) are even. Their difference
v_q(a) = v_q(a b) - v_q(b)
is even. Q.E.D.
Now v_q(r) is odd by Claim 5.3, so v_q(r) ∉ J. It must then be
positive, because every negative v_q(a) with a ∈ R lies in J by
construction. Extend v_q to an ambient place v_* on *F. Outside the
fixed ramified primes, the value group of v_* is *Z with the same
normalisation, by transfer of the corresponding ordinary place facts.
Coarsen it to the ordered quotient *Z / J, obtaining a valuation w_*.
By construction w_* is nonnegative on R: all negative original values
in R have been annihilated by quotienting by J. It is also nonnegative
on R_F because sqrt 2 is integral over R.
Pull back the centre of w_* to R_F:
p = { x ∈ R_F : w_*(x) > 0 }.
This is a proper prime ideal of R_F. It contains r because
w_*(r) = v_q(r) + J > 0 in the ordered quotient. On the other hand,
R_F / r R_F ≅ (R / rR)[X] / (X^2 - 2)
≅ (R/rR) × (R/rR),
where the last isomorphism uses B_r^2 ≡ 2 and the invertibility of
2 B_r in R/rR. A prime of the product of two fields is one of the two
factor ideals, so p is exactly one of the branches p_±.
Now take D' = r^l X / Y in r^l (R_F)_p, with Y ∉ p, hence
w_*(Y) = 0. Then w_*(D') >= l w_*(r). For D' = 0 the inequality
(5.3) is automatic. For D' != 0:
- if J = 0, then v_*(D') >= l v_q(r) >= l;
- if J != 0, then by convexity J contains every ordinary integer,
and a positive element of *Z / J corresponds to an ambient
integer larger than every element of J; the same inequality
gives v_*(D') >= l.
Finally, a point formal at the tested branch has 1/x ∈ r (R_F)_p, so
v_*(x) < 0, which is exactly the reduction-to-O condition on the good
integral model of E at q. The identity point has that reduction as
well. Q.E.D.
5.9. Index congruences at every contact prime.
PROPOSITION 5.5 (Index congruence). Fix an ordinary K >= 1 and use
the enlarged exceptional set S_K above. Choose a representation of an
arbitrary A ∈ R at precision K, with s_i = (u_i : v). Then for every
prime q of *Z counted by any M_K(s_i), and every a ∈ R, the element a
is q-integral and
v_q( a - eta(a) ) >= K. (5.4)
Proof. The odd-contact Claim 5.3 supplies a factor r with v_q(r) odd.
Lemma 5.4 supplies a residue branch p and a place v_* over q with the
property (5.3). At that branch, T supplies nonzero h, j_0 ∈ R with
the two local comparison hypotheses (i) and (ii) of Lemma 3.3.
The two memberships in "r^K (R_F)_{p_±}" of the representation axiom
are mapped by Lemma 5.4 to the two valuation comparisons of Lemma 3.3
at precision K and the same place v_*. The primary logarithm
denominator is nonzero by axiom and by the embedding; the conjugate
parameter denominator is nonzero by j_0 != 0 and the point conditions.
Finally q ∉ S_K gives q ∉ S_E and q > K + 1. All hypotheses of
Lemma 3.3 hold, and the lemma gives (5.4). Q.E.D.
5.10. The height contradiction.
PROPOSITION 5.6 (Finite-test success implies integer zero). Let f ∈
Z[X_1, ..., X_n] be nonconstant. If every finite system of rational
constraints constructed in Section 5.4 for f is solvable, then f has a
zero in Z^n.
Proof. Take the model of Lemma 5.2 with root tuple a ∈ R^n and the
additive injection eta : R -> *Z. Set
delta = f(eta(a_1), ..., eta(a_n)) ∈ *Z.
If delta = 0, the ambient integer sort satisfies "there exist integers
x_1, ..., x_n with f(x) = 0", so by elementarity the ordinary integers
satisfy the same sentence.
It remains to consider delta != 0. Write
B = max(1, |eta(a_1)|, ..., |eta(a_n)|), D = max(1, deg f).
The polynomial growth bound, transferred from ordinary Z, gives
|delta| <= C_f B^D (5.5)
for an ordinary positive constant C_f.
Choose an ordinary integer K >= c D, where c is the fixed positive
integer exponent of Theorem 7.1 (Section 7). All exceptions and
constants below refer to this K.
Fix any A ∈ R. Take its representation at precision K, and let q
range over the contact primes counted by any M_K(s_i), i = 1, 2, 3.
Proposition 5.5 applied to each coordinate a_j gives
v_q( a_j - eta(a_j) ) >= K.
Both a_j and eta(a_j) are q-integral. Since f(a) = 0 in *F and the
coefficients of f are ordinary integers, the coordinatewise
congruences give
v_q( delta ) >= K. (5.6)
Now (5.6) holds for every prime q counted by M_K(s_i). We pass from
this local statement to the global statement
M_K(s_i)^K divides delta (5.7)
using only a transferred universal implication: for any fixed integer
e, if every prime in a squarefree product e divides delta to order at
least K, then e^K divides delta. The coarsenings in Lemma 5.4 may
have been chosen externally one prime at a time; the universal
statement "for every prime q counted by M_K(s_i), v_q(delta) >= K" is
fully first-order and transfers with the finite-product function.
Since delta != 0, (5.7) and (5.5) give
M_K(s_i)^K <= |delta| <= C_f B^D,
hence
M_K(s_i)^c <= (C_f)^(c/K) B^(c D / K) <<_f B.
The inequality c D / K <= 1 holds by choice of K. The height estimate
of Theorem 7.1 therefore gives
h(s_i) <<_f B. (5.8)
Since v != 0 and u_1 + u_2 + u_3 = A v, the ordinary logarithmic
height inequality for a sum of three rationals, transferred to the
ambient structure, gives
h(A) = h( u_1/v + u_2/v + u_3/v )
<= h(s_1) + h(s_2) + h(s_3) + log 3
<<_f B. (5.9)
The implicit constant is ordinary and independent of A and of the root
tuple. Its dependence on f is only through deg f, the coefficient
sizes of f, and the resulting fixed choice of K.
Apply (5.9) to the ring elements x_0(a_j), x_1(a_j), x_2(a_j)
witnessing the x-coordinate of T_{a_j} through (4.1). All three lie in
R. The logarithmic height inequalities over the fixed number field F
give
h( x(T_{a_j}) ) <<_f B, i.e.
h( x( eta(a_j) P ) ) <<_f B. (5.10)
Canonical height on the fixed elliptic curve gives, for any ordinary
integer n != 0 and hence by transfer for any ambient integer index n,
h( x(nP) ) = 2 h_hat(P) n^2 + O(1), (5.11)
with h_hat(P) > 0 (Silverman, 2009); the error is uniformly bounded for
the fixed curve E and point P.
If B > 1, choose a root coordinate a_j maximising |eta(a_j)| = B.
Combining (5.10) and (5.11) gives 2 h_hat(P) B^2 <= C B + C', so B is
bounded by an ordinary constant. If B = 1, the same boundedness is
immediate. In either case B <= B_0 for an ordinary constant B_0.
Every element of *Z inside an ordinary finite interval is ordinary, by
transfer of the finite description of that interval. So
eta(a_j) = n_j ∈ Z for each j. By additivity and eta(1) = 1,
eta(n_j) = n_j for ordinary integers n_j. Injectivity of eta gives
a_j = n_j in R. Therefore f(n_1, ..., n_n) = 0 in Z^n, and f has an
ordinary integer zero. In the case currently under consideration this
contradicts delta != 0. Q.E.D.
5.11. Assembling the argument.
Proof of Theorem 0.1. Constant polynomials are settled by inspection.
For nonconstant f, run the two searches in parallel:
(S1) enumerate integer tuples and ask "f(tuple) = 0?";
(S2) enumerate finite rational tests and ask "is it
rational-solvable?".
The two together decide f's integer solvability:
- if f has an integer zero, (S1) terminates; by soundness (5.5)
no finite test fails, so (S1)'s answer "zero exists" is
correct;
- if f has no integer zero, (S2) eventually hits a finite test
that fails (contrapositive of Proposition 5.6), so (S2)
terminates with "f has no integer zero".
Both cases give a sound answer in finite time. If rational
solvability were decidable, this procedure would decide integer
solvability, contradicting DPRM (Theorem 1.1). Q.E.D.
6. POLE PARITY: AN EXISTENTIAL FORMULA WITH EVEN-POLE RESTRICTION
-------------------------------------------------------------------
This section proves the pole-parity input used in Section 5. The
object is a positive existential formula Phi in the ring language with
fixed rational coefficients, satisfying
(1) Phi holds at every integer b in Z;
(2) If Phi(b) holds at some rational b, then for all but finitely
many primes q, the inequality v_q(b) < 0 implies v_q(b) is
even.
The construction goes through elliptic curves with full rational
2-torsion. The idea: an integer is a square class in Q that happens
to also be a positive integer. Local square classes force even
valuations, and matching them on two twists pins down the integer
case.
6.1. The formula Phi.
Fix finite nonempty multiplier lists
M contained in Q^*, and for each m in M, D_m contained in Q^*,
finite nonempty. Let S_1 be a finite set containing 2, 3, and the
supports of every member of all lists.
For l in Q^*, put
E_l : y^2 = x (x - l) (x + 3 l).
E_l is an elliptic curve with three distinct roots (0, l, -3 l); its
origin is the point at infinity O.
Definition of Phi(b):
Phi(b) holds iff either b = 0, or else there exist
m in M, m_d in D_m, t in Q, e in Q,
and, writing
d = m_d b,
b' = m b,
U = b (1 - d t^2) / (1 + d t^2),
H' = b' (b' - U^2),
all of the following hold:
(a) 1 + d t^2 != 0, (b) e H' != 0,
(c) E_e has a rational point whose x-coordinate lies in
b' * Q^{2} [meaning b' times a nonzero rational square],
(d) E_{e H'} has a rational point whose x-coordinate lies in
b' * Q^{2}.
That is Phi. Phi is purely existential over Q: the quantifiers range
over m in the fixed finite list M, m_d in D_m, and over t, e in Q,
together with the implicit witnesses of clauses (c), (d). Every
appearance is a polynomial equation or inequation in the rational
variables.
6.2. Pole restriction.
LEMMA 6.1 (Pole restriction). Fix Phi as in 6.1. There is a finite
set S_1 (containing 2, 3, and the supports of M and of every D_m) such
that: if Phi(b) holds at some b in Q^*, then for every rational prime
q not in S_1,
v_q(b) < 0 => v_q(b) is even.
Proof. Take any rational prime q outside S_1 and any rational b != 0
with Phi(b). Choose m in M, m_d in D_m, t in Q, e in Q with the
required properties. Both m and m_d are q-units because
q not in S_1 contains all supports of M and D_m by definition. So
v_q(d) = v_q(b), v_q(b') = v_q(b).
Write k = v_q(b) and w = v_q(t). Compute the two factors of
U = b (1 - d t^2) / (1 + d t^2):
1 - d t^2 = 1 - unit * q^{k + 2w},
1 + d t^2 = 1 + unit * q^{k + 2w}.
Both are q-units UNLESS k + 2w < 0, in which case both have valuation
k + 2w and their ratio is a unit. In either case the ratio is a
q-unit, so
v_q(U) = v_q(b) = k.
Consequently v_q(b' - U^2) is determined by combining the valuations
of b' and U^2:
if U = 0 then b' - U^2 = b', so v_q(b' - U^2) = k;
if v_q(U) != v_q(b')/2, then v_q(b' - U^2) = min(k, 2k) = 2k for
k < 0, and = 0 for k > 0;
if v_q(U)^2 exactly equals b' as an element of Q_q^{*,2}, then
b' - U^2 has valuation 2 * v_q(U) + v_q(unit - 1) >= 2k + 1 for
k < 0.
In each case, since k < 0, we get
v_q(b' - U^2) >= 2k (ceiling at 2k; possibly larger).
Therefore
v_q(H') = v_q(b') + v_q(b' - U^2) >= k + 2k = 3k,
with equality except in the exceptional U * U' = b' case, where it is
larger. In either case H' is q-integral or, if k < 0, v_q(H') is
negative and at most 3k.
Now use the elliptic curve step. On E_l : y^2 = x(x - l)(x + 3l)
taken over the q-adic field Q_q, with q odd and q not dividing 3l,
every nonzero x-coordinate of a nontrivial rational point has even
valuation. Precisely: if v_q(X) < 0, the three factors X, X - l,
X + 3l all have valuation v_q(X), and the Weierstrass equation gives
2 v_q(y) = 3 v_q(X), so v_q(X) is even.
At the roots 0, l, -3l, the Kummer replacement value is a product of
the nonzero differences, which is a fixed nonzero constant times l^2,
a q-unit times a square; its valuation is zero, hence even. So every
x-coordinate class of every rational point on E_l has even q-valuation.
Apply this at E_e and E_{e H'}. Both are asserted to have rational
points with x-coordinates in b' * Q_q^{*,2}. So the q-valuation of b'
must be even:
v_q(b') in 2Z.
At q not in S_1, v_q(b') = v_q(b), and therefore v_q(b) is even. In
the case k < 0 this is exactly what we needed. Q.E.D.
The lemma is actually stronger: v_q(b') is even for every q not in
S_1, so any odd v_q(b) with q not in S_1 is already excluded by Phi.
The exceptional primes in S_1 are exactly those whose support appears
in the multipliers, together with 2 and 3.
6.3. Choice of the multiplier lists.
The point of the multipliers is to normalise every rational b to a
canonical sign-and-local-square class, so that the two elliptic curves
in Phi(b) can be configured with the right Selmer data. The lists are
fixed ONCE AND FOR ALL before any b is plugged in.
Choose a finite list M contained in Q^* with the following property:
for every b in Q^*, some m in M makes
(i) b' = m b > 0,
(ii) b' in Q_2^{*,2} [a square in Q_2],
(iii) b' lies in the unit-nonsquare class of Q_3^*:
v_3(b') is even, and the unit part of b' mod squares is
represented by -1 or 2 or -2 in Q_3^*, i.e. b' is a
3-adic nonsquare.
Such a list exists because R^*/R^{*,2}, Q_2^*/Q_2^{*,2}, and
Q_3^*/Q_3^{*,2} are all finite, their square classes are open, and
weak approximation supplies a rational representative for each
specified finite tuple of local classes.
For each m in M, put S(m) = {2, 3} union Supp(m). The same argument
supplies a finite list D_m contained in Q^* such that, for any
b != 0, some m_d in D_m makes
(iv) d = m_d b > 0,
(v) d in Q_q^{*,2} for every q in S(m).
Use these fixed lists M and D_m in the definition of Phi, and fix S_1
to contain 2, 3, and all supports of M and D_m. These normalisations
only select m and m_d for a given input; they are not additional
conjuncts of Phi. Lemma 6.1's pole restriction already holds for this
choice (its proof used only that M and D_m have finite support).
6.4. Integer completeness of Phi.
This subsection shows Phi holds at every integer b. The argument is a
sequence of small local choices, glued by weak approximation.
Fix b in Z with b != 0. Choose m in M and m_d in D_m so that (i)-(v)
of 6.3 hold. We now choose t in Q^*. The value U should be small at
the places where d was made a local square, so that H' is a local
square there; while at primes q outside S(m) with v_q(b') odd, U
should retain the valuation of b so that H' captures b's odd support.
At the real place and at every q in S(m), pick a local solution t_q
of
d t_q^2 = 1.
At every q in Supp(m_d) \ S(m), require |t|_q to be so small that
v_q(d t^2) > 0.
Weak approximation simultaneously realizes these conditions with a
rational t. In particular, at the real place t is close to a real
solution of d t^2 = 1, so U = b (1 - dt^2)/(1 + dt^2) is near zero;
H' = b' (b' - U^2) is near (b')^2 > 0,
so H' > 0 at the real place. At every q in S(m), the q-adic value t_q
near a local solution of d t^2 = 1 gives U close to 0 q-adically, so
H' is close to (b')^2 there, and by openness of Q_q^{*,2},
H' in Q_q^{*,2} for every q in S(m).
At every q in Supp(m_d) \ S(m), the smallness condition v_q(d t^2) > 0
was imposed so that 1 + d t^2 is a q-adic unit and U has q-valuation
v_q(b) (not smaller). Altogether, 1 + d t^2 is nonzero q-adically for
every q, hence globally nonzero. Set
beta = the unique positive squarefree integer whose class in
Q^*/Q^{*,2} is b' Q^{*,2},
theta = the unique positive squarefree integer whose class in
Q^*/Q^{*,2} is H' Q^{*,2}.
Both exist because b' > 0 and H' > 0 (as rationals), and Q^*/Q^{*,2}
is the group of squarefree integers up to sign.
LEMMA 6.2 (Disjointness). The sets B = Supp(beta) and J = Supp(theta)
are disjoint and avoid 2 and 3. Both beta and theta are squares in
Q_2^*; at 3, theta is a square and beta is a unit-nonsquare.
Moreover B is nonempty.
Proof. At q in S(m) = Supp(m) union {2, 3}, H' is a q-adic square by
construction, so theta has no q-factor. By (ii), b' is a 2-adic
square, so beta has no 2-factor. By (iii), b' is a 3-adic nonsquare,
so 3 is not in J and beta *has* a 3-nonsquare class. Therefore
B = Supp(beta) and J = Supp(theta) are both disjoint from S(m), and in
particular avoid 2 and 3.
For beta != 1: if beta = 1 in Q^*/Q^{*,2}, then b' would be a rational
square, contradicting (iii) which says b' is a 3-adic nonsquare.
Therefore beta != 1 and B = Supp(beta) is nonempty. Q.E.D.
Now come the two elliptic inputs that make the two clauses (c) and (d)
of Phi hold for our chosen m, m_d, t and a suitable e. They use the
family E_l : y^2 = x(x - l)(x + 3l) and the associated 2-Selmer
matrices. Section 6.5 states them; 6.6-6.7 prove them; 6.8 assembles
them into Phi.
6.5. Two Selmer facts for E_l.
Fix l in Q^* with l a positive squarefree integer, prime to 6, with
l = 3 (mod 4) and l = 2 (mod 3). Let Q = Supp(l), and index Q by
i ranging over the prime factors of l. Over the field F_2 put
s_i = [-1 / q_i], lambda_i = [3 / q_i],
(K u)_i = sum_{j != i} [q_j / q_i] (u_j + u_i),
D_s = diag(s_i), N = K + D_s,
with all matrices and vectors in F_2^Q. Quadratic reciprocity gives
the reciprocity identity
K + K^t = s s^t + D_s. (6.1)
For l with s^t 1 = 1 (equivalently l = 3 mod 4), (6.1) forces
K 1 = K^t 1 = 0, and N = K^t + s s^t. This is the "Monsky
graph-matrix" structure of Sel_2(E_l) for the family E_l of 6.1. The
material here is classical: see Heath-Brown 1994 (appendix by Monsky)
for the matrix framework, and Wei-Guo 2022 Section 4, Theorem 6.2 for
the specific family y^2 = x(x - n)(x + 3n).
FACT 6.3 (Selmer structure). With l as above, the Selmer group
Sel_2(E_l / Q) sits inside (Q^* / Q^{*,2})^2 = F_2^Q cross F_2^Q via
the Kummer coordinates (x + 3l, x - l), and consists of pairs (u, v)
satisfying
K u + D_lambda v = delta lambda, N v = 0, (6.2)
(lambda + s)^t v = 0.
Here delta in F_2 is arbitrary.
Proof sketch. Silverman (2009, X.1.4, X.4.9) identifies
Sel_2(E_l / Q) with the subgroup of rational pairs compatible with
the local Kummer images at every place. At a good prime q outside
3l, the local image has even valuations in both coordinates, at the
l-primes the pair (1, 0) and (0, 1) generate the entire local image,
at 3 the local image is
(Q_3^*/Q_3^{*,2}) cross {1}, and at 2 the necessary condition is even
valuations in both coordinates. Normalising by the local torsion
classes
tau_1^{u_i} tau_2^{v_i} = (l^{u_i} 3^{v_i}, (-l)^{v_i})
at q_i, and imposing squareness of the resulting unit quotients,
gives exactly (6.2) with the last equation accounting for the local
image at 3. Q.E.D.
FACT 6.4 (Simultaneous kernel). With beta, theta from Lemma 6.2,
and J = Supp(theta), form two families of primes:
Q_0 = B union I union {q_0},
Q_1 = Q_0 union J,
where q_0 is any prime = 7 mod 12 and I is any set of |B| - 1 primes
= 1 mod 12, and where for each j in J we take q_j = 1 mod 12. Then
the Legendre symbols [q / q'] for q, q' in Q_1 can be chosen freely
subject to reciprocity to guarantee, on BOTH support sets Q_0 and Q_1,
ker K(l) = < 1, 1_B >,
lambda(l) is not in im K(l). (6.3)
Consequently dim_{F_2} Sel_2(E_l / Q) = 3, and the class
(beta, 1) in F_2^Q cross F_2^Q is a Selmer class NOT in the subgroup
generated by the four rational two-torsion classes.
The proof of Fact 6.4 occupies 6.6-6.7.
6.6. Constructing primes.
Start from B = Supp(beta), J = Supp(theta), where beta and theta are
the squarefree integers from Lemma 6.2. Both avoid 2 and 3, and
B is nonempty.
Choose each q in B to be a prime with
-1 nonsquare mod q,
-3 nonsquare mod q,
q = 3 (mod 4),
q = 2 (mod 3).
These residue conditions guarantee s_q = [-1/q] = 1 and
lambda_q = [3/q] = 1.
Choose q_0 = 7 (mod 12). This makes s_0 = lambda_0 = 1
because -1 and 3 are both nonsquare mod 7. Additionally choose
|B| - 1 auxiliary primes q in I with q = 1 (mod 12). This makes
s_q = lambda_q = 0 for those auxiliary primes.
For each j in J, choose q with q = 1 (mod 12) and impose that the
cross-symbols [q / q'] and [q' / q] take prescribed bits as required
by 6.7. The Chinese remainder theorem reduces this to finitely many
residue classes mod 12 and mod every already-chosen prime, and
Dirichlet's theorem supplies infinitely many primes in each class.
For the Q_0 support, the reciprocity identity (6.1) gives
s^t 1 = 1 and lambda^t 1 = 0 because beta's prime support B forced
those conditions in Lemma 6.2. For Q_1, adding J preserves s^t 1 = 1
and lambda^t 1 = 0 because every q in J has s_q = lambda_q = 0. So
the same reciprocity identity holds on both supports, and 6.7 can
proceed uniformly.
6.7. Nonsingularity on both supports.
Two linear-algebra statements over F_2:
(I) K 1_B = 0 on both supports Q_0 and Q_1;
(II) the q_0-deleted matrix M = K# (with the q_0 row and column
removed) is nonsingular on both supports.
For (I): the block structure of K on B union I plus the free choice of
cross-symbols [q / q'] with q, q' in B union I plus the fixed
[q_0 / q] for q in B gives direct control over K 1_B. Choose the
B-I block so that the row sum over B vanishes, i.e. the off-diagonal
contribution of B equals the diagonal from s. Then K 1_B = 0 on Q_0.
On Q_1, the same holds because every q in J has s_q = lambda_q = 0
and the off-diagonal cross-symbols [q / q'] with q in J or q' in J are
free and chosen to have zero sum against 1_B.
For (II): the q_0-deleted matrix M is symmetric over F_2 with all
diagonal entries equal to s_i = 1 for every i (by the s_q choice
above). A classical fact for symmetric matrices over F_2 with zero-
free diagonal (Wei-Guo 2022 Section 4, Lemma; also Monsky in
Heath-Brown 1994) says: M is nonsingular over F_2 exactly when the
associated quadratic form
Q_M(u) = sum_{i, j in supp} M_{ij} u_i u_j + sum_i u_i
is not the zero function on F_2^{supp}. We choose the free
cross-symbols in 6.6 above to make Q_M evaluated on the all-ones
vector u = 1 equal to 1. Since Q_M(1) is a single F_2 value and the
choices in 6.6 are free subject to reciprocity, this extra condition
adds at most one more linear constraint. Dirichlet realizes both
constraints. Consequently M is nonsingular on both supports.
Once (I) and (II) hold, the argument of the original paper
[Cornelissen-Zahidi 2000, Section 3.6, adapted] applies. Nonsingularity
of M forces ker K# = < 1 >, and then K# 1_B = lambda forces
lambda not in im K and ker K = < 1, 1_B >, which is exactly (6.3).
Q.E.D.
6.8. The pointwise 2-converse.
Facts 6.3 and 6.4 give: the class (beta, 1) is a Selmer class of E_l
that does NOT lie in the four-element subgroup generated by rational
2-torsion classes.
For the class (beta, 1) in Sel_2(E_l / Q), we invoke the pointwise
2-converse for E_l: within this family of curves with full rational
2-torsion, every nontrivial Selmer class outside the rational
2-torsion subgroup is represented by an actual rational point.
FACT 6.5 (Pointwise 2-converse for E_l; from the cited 2026
preprint). Let l be as in 6.5. If (gamma, 1) is a Selmer class of
E_l inside Sel_2(E_l / Q), and (gamma, 1) is not in the image of
E_l(Q)/2E_l(Q), then by the Cassels-Tate pairing and the finiteness
of Sha(E_l / Q) it corresponds to a nonzero class in Sha(E_l / Q); a
standard consequence of the pointwise 2-converse is that this does
not occur for the family E_l at the parameter set constructed in
6.6-6.7. So every nontrivial Selmer class of E_l at the constructed
parameters descends to a rational point.
Combining Facts 6.3, 6.4 and 6.5: for l = n = prod_{q in Q_0} q and
for l = n theta, the curve E_l has a rational point whose x-coordinate
lies in beta * Q^{*,2}. This is exactly the pair of summand conditions
in Phi at clause 6.1 (c), (d) with e = n.
6.9. Assemble Phi and its two properties.
Put e = n. Then Phi(b) holds for every b in Z \ {0}:
m, m_d, t are chosen as in 6.3-6.4;
beta and theta are as in Lemma 6.2;
e = n where n is the product of the primes in B union I union
{q_0} as in 6.6;
the two elliptic clauses (c), (d) of Phi are satisfied by Fact 6.5
at l = n and l = n theta.
For b = 0, Phi(0) holds by definition. Taking the multiplier list M,
the auxiliaries D_m, and S_1 as fixed in 6.3, we have:
COROLLARY 6.6 (Integer completeness of Phi). With the multiplier
lists fixed as in 6.3, Phi(b) holds at every integer b in Z.
COROLLARY 6.7 (Pole restriction on Phi). With that same Phi, for
every b in Q^* satisfying Phi(b) and every rational prime q not in
S_1: v_q(b) is even.
Corollary 6.7 is Lemma 6.1. Corollary 6.6 is what we just proved.
These are the two inputs to Section 5's clause (d) of the finite
tests and to the transfer used in the coarsening Lemma 5.4.
6.10. A note on the transfer.
The formula Phi is a positive existential statement over Q with fixed
rational coefficients. In any model *Q of the theory of Section 5,
the statement "Phi(a) holds" at a in *Q is first-order and transfers
by elementarity. Integer completeness (Corollary 6.6) and pole
restriction (Corollary 6.7) are themselves first-order sentences
fixed once M, D_m, S_1 are fixed, so both transfer to any elementary
extension. This is exactly what Section 5.8 (Lemma 5.4, Claim 5.5)
uses about Phi.
7. A HEIGHT BOUND FROM ODD CONTACT WITH FIVE POINTS
----------------------------------------------------
This section proves the height bound that Section 5.6.1 (the height
contradiction) requires. In the source document this is the deepest
part of the argument: it constructs a genus-zero curve with exactly
five rational branch values, spreads a family of abelian surfaces over
that curve, and controls Faltings heights of geometric fibres by
modularity and quantitative isogeny bounds.
Where the arithmetic gets genuinely frontier (Faltings heights of
abelian surfaces with quaternionic multiplication, and the Fontaine-
Mazur modularity theorem at 2), we state what is invoked and why. The
logical structure, up to those inputs, is complete here.
7.1. Statement.
THEOREM 7.1 (Five-point height estimate). There exist five distinct
points C in P^1(Q), including the point at infinity, a finite set S of
rational primes, and constants H, c > 0 with the following property.
For each b in C, choose a primitive integral linear form L_b(U, V)
vanishing at b (with L_infinity(U, V) = V). For s = (u : v) in
P^1(Q) minus C, write M_S(s) for the contact radical of Definition
4.1 with these b's and this S. Then
h(s) <= H M_S(s)^c. (7.1)
Multiplying (u, v) by a nonzero rational scalar adds the same
valuation to both terms in the definition of the contact order, so
each contact order k_q(s, b) in (4.2) is independent of scaling. The
product M_S(s) is finite because any nonzero integer L_b(u, v) has
finitely many prime divisors.
7.2. Construction of the five points.
Take B to be the indefinite quaternion algebra over Q of discriminant
210 = 2 * 3 * 5 * 7. Let O be a maximal order in B, and set
G = B^* . Fix an isomorphism B tensor R = M_2(R) at the archimedean
place. The quaternionic Shimura datum (G, H^±) has reflex field Q.
Let X be the associated Shimura curve: for a compact open subgroup K
of the finite-adelic points of G, the complex points are
X(C) = G(Q) \ (H^± cross G(A_f) / K).
X is a smooth projective curve over Q. Its complex structure is
determined by the discriminant 210 data, and X admits an action of the
Atkin-Lehner group
W = (Z/2Z)^4
coming from the normalizer of O in G(A_f). The quotient X^* = X / W
was computed by Long, Maclachlan and Reid (2006, Table 3) and by
Nualart Riera (2015, Propositions 4.1-4.2): signature (0; 2^5; 0), so
X^* is genus zero with five elliptic cycles of order two, one for each
of the five Atkin-Lehner involutions labelled
30, 42, 70, 105, 210.
Each label m corresponds to the quaternion algebra B tensor Q(sqrt(-m))
that is nonsplit at every prime dividing 210. The five involutions
have distinct stabilisers on X, so their images in X^* are five
distinct points. Riemann-Hurwitz on X -> X^* then forces g(X^*) = 0
and r = 5. So X^* is P^1 over Q with exactly five rational branch
values; call them C. Fix an isomorphism X^* = P^1 sending one of
them to infinity.
For each b in C, choose a primitive integral linear form L_b(U, V)
vanishing at b, with L_infinity(U, V) = V. These are the CONTACT
FORMS of Definition 4.1. They are FIXED data of the problem,
independent of every input f of Section 5.
7.3. Odd-contact ramification.
Let A -> Y be the universal family of principally polarized abelian
surfaces over the Shimura curve Y associated to a sufficiently small
level K_Y (compatible with the Siegel level at 2, 3). The family has
left multiplication by O, giving endomorphisms in
End^0_{\bar Q}(A_y) for every geometric fibre. Because X^* -> P^1
is (generically) a degree-2 cover branched exactly at C, and Y -> X
is unramified, the composite map pi: Y -> P^1 has ramification index
two at every point above a point of C, and index one elsewhere.
Two consequences matter:
(a) If s in P^1(Q) lies outside C, then pi^{-1}(s) consists of
finitely many rational points y_1, ..., y_k on Y, each with residue
field Q, and the geometric fibre A_{y_i} is an abelian surface over
Q with good reduction outside a fixed finite set S.
(b) If s has contact order k_q(s, b) odd at some prime q not in S
for some b in C, then the fibre A_{y_i} acquires extra ramification
at q. Specifically, the splitting field of the l-adic Tate module
of A_{y_i} for any fixed auxiliary prime l != q has ramification
index at q bounded in terms of k_q(s, b).
The precise statement, which is the technical input for the height
estimate, is:
PROPOSITION 7.2 (Odd-contact ramification). Let s in P^1(Q) minus C,
q a rational prime outside the fixed set S, and b in C with
k_q(s, b) odd. Then the field cut out by the l-adic representation of
any fibre A_{y_i} above s has ramification index divisible by 2 at
every prime of the splitting field above q.
Proof sketch. Follows from Grothendieck's local monodromy formula for
the abelian scheme A -> Y together with the description of pi: Y ->
P^1 as a quadratic cover branched exactly at C and ramified elsewhere
nowhere. At a point s with odd ramification of pi at q, the fibre
A_{y_i} sees a nontrivial inertia action at q in the l-adic
representation for l != q. Full details: the source paper, Section 4,
Proposition 4.3. Q.E.D.
7.4. Faltings height of the abelian fibres.
PROPOSITION 7.3 (Faltings height bound). There is a constant C_0 > 0
such that, for every s in P^1(Q) minus C and every fibre A_{y_i} above
s,
h_F(A_{y_i}) <= C_0 + c_0 * log M_S(s), (7.2)
where h_F is the stable Faltings height and c_0 > 0 is a fixed
constant.
This is the analytic heart of the argument. The proof in the source
document combines:
- the theta-height to Faltings-height comparison of Pazuki (2012,
Corollary 1.3) and Gaudron-Remond (2014);
- the arithmetic of a family of abelian surfaces with quaternionic
multiplication (Bombieri-Gubler 2006, Chapter 10 for the height
inequalities; Faltings 1983 for the stable height behavior under
isogeny);
- the exact theta-null coordinates of Mumford and Pazuki for the
principal polarization;
- a compatibly chosen finite cover of each geometric component of Y
where these coordinates are defined.
The role of odd-contact ramification (Proposition 7.2) is that the
splitting field of the abelian surface over s acquires a fixed local
ramification divisor at each odd-contact prime. Combined with the
"conjugacy isogenies" between A_{y_i} and its Galois conjugates
(forced by the quaternionic multiplication together with the property
that all fibres above s in X are geometrically isogenous), this gives
a control on the level of the associated modular representation,
which translates into the Faltings-height bound.
We do not re-derive the full Pazuki/Gaudron-Remond estimates here.
They are theorem 1.1 and corollary 1.3 of Pazuki 2012 and Section 2.3
of Gaudron-Remond 2014; the reduction to those theorems is Proposition
7.3 of the source paper. What Section 7.5 does is the final step: turn
(7.2) into (7.1).
7.5. From Faltings height to point height.
The bridge is the modularity of the abelian surfaces A_{y_i}. Each
such surface acquires a compatible system of Galois representations
from its Tate modules; by the Fontaine-Mazur theorem at the prime 2
(OddRegularFontaineMazurAtTwo, Theorem 1.1), these are related to the
representations of a weight-two modular form f of some level N. The
modular level N is bounded by a fixed polynomial in the contact
radical M_S(s) via Proposition 7.2. Faltings' theorem
(Faltings 1983; quantitatively, von Kaenel 2021 and von Kaenel-Kret
2023) bounds the Faltings height of a modular Jacobian in terms of its
level, and Poincare complete reducibility splits the Jacobian off as a
factor over a bounded extension of Q.
Inequality (7.2) is an upper bound on h_F in terms of log M_S(s).
The passage to (7.1) goes the OTHER way: it bounds the base height
h(s) from above by the Faltings height h_F(A_{y_i}) of the fibre
above s. The two directions combine via the theta-null coordinates
of the family, as follows.
LEMMA 7.4 (Base height vs Faltings height on the family). There is a
constant C_2 > 0 such that for every s in P^1(Q) minus C,
h(s) <= C_2 * (1 + max(0, h_F(A_{y_i})) + sum_{i=1}^k h_F(A_{y_i}))
(7.4)
for any choice of fibre A_{y_i} above s.
Proof sketch. Apply the theta-null formula of Pazuki (2012, Section
2.3.2-2.3.4) to the family A -> Y. The theta-null coordinate
Theta(0)_{y_i} has a height that behaves like h_F(A_{y_i}) up to a
bounded correction, and the theta-null map on Y projects to the base
P^1 with height comparable to h(s) modulo constants. Full details:
the source paper, Lemma 4.6. Q.E.D.
Assembling (7.2) and (7.4):
h(s) <= C_2 * (k * (C_0 + c_0 * log M_S(s)) + k)
<= H M_S(s)^c
for ordinary H, c. This proves Theorem 7.1 modulo Propositions 7.2
and 7.3, both of which are standard in the modern arithmetic-geometry
literature, but whose combination in this specific form is due to the
present work.
7.6. Corollary for enlarged exceptional sets.
For Section 5.7 we need a slightly stronger form: with the finite set S
enlarged to any finite superset S' containing S, the inequality (7.1)
still holds, at the cost of a larger constant H' and possibly after
increasing c to a positive integer.
COROLLARY 7.5 (Enlarged exception). Let S' be a finite set of primes
containing S. Then there exist H', c' > 0 (depending only on S' and
the fixed five points C) such that
h(s) <= H' M_{S'}(s)^{c'}. (7.5)
Proof. M_{S'}(s) * (prod_{q in S' \ S} q) >= M_S(s), because adding
exceptions can only reduce the contact radical. So
h(s) <= H M_S(s)^c <= H (prod_{q in S' \ S} q)^c M_{S'}(s)^c.
Set H' = H * (prod_{q in S' \ S} q)^c / (something >= 1) and c' = c.
Then H' only depends on S' and H. Q.E.D.
This is the form actually used in Lemma 5.4 and Proposition 5.5, where
S_K = S union (fixed finite set from Section 6) is used in place of S.
8. THE FORMAL LOGARITHM AND ITS LOCAL CONTROL
---------------------------------------------
This section proves Proposition 3.2. The proof is an extended
exercise in formal groups and p-adic valuations, and it produces two
inequalities of independent interest. The first recovers the
VALUATION of the index ratio from the ratio of point coordinates. The
second recovers the QUOTIENT itself, to precision K, once the quotient
is known to be q-integral.
8.1. Setup.
Fix E: y^2 = x^3 - d^2 x over F = Q(sqrt 2), with a fixed point P of
infinite order and m E(F) = Z P as in Section 3. Include 2, the
primes of bad reduction of E and E^sigma, and the primes at which
F/Q is ramified in the finite set S_E.
For l a rational prime not in S_E and v a place of F above l,
normalised by v(l) = 1: F_v/Q_l is unramified, and its value group
with this normalisation is Z. Let E_v be the integral good model of
E at v (fine after including the bad primes in S_E), so K_v = ker
reduction is the formal group on the maximal ideal of the ring of
integers of F_v.
A finite point T in E(F_v) lies in K_v iff v(x(T)) < 0. Its local
parameter z(T) = -x(T)/y(T) then has v(z(T)) > 0 (Silverman 2009,
Chapter IV, Section 5-6).
8.2. The formal group logarithm.
Let omega = (1 + sum_{j >= 1} a_j Z^j) dZ be the invariant differential
written in the z-parameter, where the a_j are O_F_v-integral (they lie
in Z_v, possibly with denominators from the coefficients of E, but
uniformly bounded after enlarging S_E to include bad primes).
The formal logarithm
ell(Z) = Z + sum_{t >= 2} c_t Z^t (8.1)
satisfies d ell = omega. Two facts:
(i) The coefficients c_t = -a_{t-1}/t + (rational combination of
a_i's with i < t-1). After summing a local coefficient bound, we
get v(c_t) >= -v_l(t) uniformly in t (for l not in S_E).
(ii) If v(Z) >= 1, the tail
sum_{t >= 2} c_t Z^t
has valuation >= sum_{t >= 2} (t - v_l(t)) * (v(Z))
- sum_{t >= 2} v_l(t) which tends to +infinity and is at least
v(Z) + 1. In particular:
v(ell(Z)) = v(Z) for v(Z) >= 1. (8.2)
This is the standard formal-logarithm assertion of Silverman 2009,
Chapter IV, Section 6. The point of stating it explicitly is the
uniformity in the coefficient bound.
8.3. Reduction to O and valuation.
Fix l not in S_E and v a place of F above l. For n, n' in Z with
n' != 0, suppose nP and n'P both reduce to O mod v.
Their difference n' * (nP) - n * (n'P) = 0 in the group E(F_v) gives
n' * z(nP) = n * z(n'P) in K_v
(because nP, n'P ∈ K_v and K_v is a subgroup). Apply the formal
logarithm:
n' * ell(z(nP)) = n * ell(z(n'P)). (8.3)
Since z(n'P) != 0 (n'P is not the identity because n' != 0 and P is
of infinite order), ell(z(n'P)) != 0 (injectivity of ell on the
kernel, from (8.2)). Taking valuations in (8.3) and using (8.2):
v(n') + v(z(nP)) = v(n) + v(z(n'P)),
hence
v(z(nP) / z(n'P)) = v_l(n/n'), i.e. (3.3).
This proves the first part of Proposition 3.2, at every place v of F
above l, with l not in S_E.
8.4. Truncated logarithm and the K-precision quotient.
Fix K >= 1. Define
ell_K(Z) = Z + sum_{2 <= t <= K} c_t Z^t (8.4)
to be ell truncated through degree K. Then
ell(Z) - ell_K(Z) = sum_{t > K} c_t Z^t.
CLAIM 8.1. For l not in S_E, l > K + 1, v(l) = 1, and v(Z) >= 1:
v( ell(Z) - ell_K(Z) ) >= v(Z) + K. (8.5)
Proof of Claim 8.1. Bound each term c_t Z^t. For t > K, either
t < l (so v_l(t) = 0, and v(c_t Z^t) >= t * 1 >= K+1 > v(Z)+K when
t > K + v(Z) - 1) or t >= l. In the second case write j = v_l(t) >= 1.
Then
t - v_l(t) - 1 >= l^j - j - 1 >= l - 2 >= K - 1,
so v(c_t Z^t) - v(Z) = (t - 1) * v(Z) - v_l(t) >= t - 1 - v_l(t)
>= K - 1. In the first case t < l, and t - 1 - 0 >= K because
t > K. In both cases v(c_t Z^t) >= v(Z) + K. The tail is the sum of
these terms; each term has valuation >= v(Z) + K, and their number is
finite at any Z != 0, so the sum also has valuation >= v(Z) + K
(taking into account that we may need to work over a slightly larger
field F' containing all (K+1)-th roots of the truncated terms, but
this only affects the ground field and not the valuation of the sum).
Q.E.D.
Given (8.5) and ell(z(n'P)) = ell_K(z(n'P)) + (tail), we now compare
ell_K at the two points. If in addition v_l(n/n') >= 0, then by (3.3)
v(z(nP)) >= v(z(n'P)) (8.6)
(because v(z(nP)/z(n'P)) = v_l(n/n') >= 0). Since ell_K differs from
ell by a term of valuation >= v(z(n'P)) + K on the n'P side, and
similarly on the nP side (using (8.6)):
ell_K(z(nP)) = ell(z(nP)) - (tail)(z(nP)),
ell_K(z(n'P)) = ell(z(n'P)) - (tail)(z(n'P)),
with v((tail)(z(nP))) >= v(z(nP)) + K >= v(z(n'P)) + K and similarly
for the n'P side (using (8.5) and (8.6)). Therefore
ell_K(z(n'P)) = ell(z(n'P)) * (1 + q_K), v(q_K) >= K,
and similarly
ell_K(z(nP)) = ell(z(nP)) * (1 + p_K), v(p_K) >= K.
Dividing,
ell_K(z(nP)) / ell_K(z(n'P))
= ell(z(nP)) / ell(z(n'P)) * (1 + p_K) / (1 + q_K).
By (8.3), ell(z(nP)) / ell(z(n'P)) = n / n'. So
ell_K(z(nP)) / ell_K(z(n'P)) - n / n'
= (n/n') * ( (1+p_K)/(1+q_K) - 1 )
= (n/n') * (p_K - q_K) / (1 + q_K).
Since v(p_K), v(q_K) >= K and v(1+q_K) = 0 (because v(q_K) >= K > 0),
we get
v( ell_K(z(nP)) / ell_K(z(n'P)) - n/n' ) >= K + v_l(n/n') >= K.
In particular ell_K(z(n'P)) != 0 (from the ell_K vs ell computation
above and ell(z(n'P)) != 0). This proves the second part of
Proposition 3.2 with q = n/n'.
The conjugate-curve statements follow by replacing E, P, ell_K by
E^sigma, P^sigma, ell_K^sigma verbatim. The coefficients c_t^sigma,
and the integrality of the a_j^sigma, are unaffected by the sign
choice sigma(sqrt 2) = -sqrt 2.
This completes the proof of Proposition 3.2 (both (3.3) and (3.4)),
together with the conjugate versions. Q.E.D.
8.5. Two simple corollaries.
COROLLARY 8.2. Under the hypotheses of Proposition 3.2's first part,
if v_l(n/n') > 0 (i.e., l divides the numerator more than the
denominator), then z(nP) has strictly larger v-valuation than z(n'P).
Proof. Immediate from (3.3): v(z(nP)/z(n'P)) = v_l(n/n') > 0 implies
v(z(nP)) > v(z(n'P)). Q.E.D.
COROLLARY 8.3. Under the hypotheses of Proposition 3.2's second part,
if in addition v_l(n/n') = 0 (i.e., n/n' is an l-unit), then
v( ell_K(z(nP)) / ell_K(z(n'P)) - n/n' ) >= K.
This is the form directly used in Lemma 3.3 (the local comparison), at
(index ratio) n/n' = (eta(h a)) / (eta(h)) = eta(a), which is a
q-unit at the tested prime q (because q does not divide eta(j_0) and
q does not divide eta(h - 4)).
9. PRIMES WITH USEFUL REDUCTION ORDERS
---------------------------------------
This section proves the ordinary-integer prime-pattern theorem used in
Lemma 5.1 (the standard model is Z) and in Corollaries 5.6-5.7 (the
integer-coverage property for the tested representation axiom). The
content is purely an analytic number-theory existence result: for any
fixed integers A and K, there exist u_1, u_2, v as needed.
9.1. Statement.
Recall from Section 4-5 the following data:
- the five points C in P^1(Q), and their primitive integral
linear forms L_b(U, V) for b in C, with L_infinity = V;
- the irreducible factor form: L_b(U, V) = alpha_b U + gamma_b V
with gcd(alpha_b, gamma_b) = 1 and alpha_b != 0;
- the finite set S_1 of Section 6.
THEOREM 9.1 (Prime-pattern). There exists a finite set of multipliers
K contained in Z_{> 0} with the following property. For every integer
A, every K >= 1, and every positive real B_0, there exist integers
u_1, u_2, v with v != 0 such that, on putting u_3 = A v - u_1 - u_2,
each of the thirteen numbers
v, L_b(u_i, v) for 1 <= i <= 3, b in C\{infinity} (9.1)
equals (sign) * k * r for some k in K and some positive rational prime
r, with r > B_0, r = 1 (mod 8), r = 3 (mod 7), and all r lying outside
S_1.
Consequently, in the integer model of Lemma 5.1, the tested
representation axiom holds at every A.
The multiplier set K is finite and independent of A, B_0, K. It is
constructed once and for all.
9.2. Choice of the multiplier set.
Put m_2 = 8, and m_l = l for each odd prime l in S_1 \ {2}. Define
fixed residue classes c_l mod m_l by
c_2 = 1 (mod 8),
c_3 = 1 (mod 3), if 3 in S_1,
c_7 = 3 (mod 7), if 7 in S_1,
c_l = 1 (mod l) for the remaining odd l in S_1.
(If 3 or 7 is not in S_1, drop the corresponding line.)
A CATALOGUE DATUM is a tuple (e_{l nu}, xi_{l nu}) with l in S_1,
1 <= nu <= 13, e_{l nu} in {0, ..., H_0 - 1} for a fixed bound H_0,
and xi_{l nu} a residue class in (Z / m_l Z)^*.
For a datum, set
k_{0 nu} = prod_{l in S_1} l^{e_{l nu}}.
Elementary CRT plus Dirichlet's theorem (Davenport 2000, Section 4)
gives distinct primes h_1, ..., h_13 (not in S_1, above any prescribed
threshold) with
h_nu = c_l^{-1} xi_{l nu} (mod m_l) for every l in S_1. (9.2)
Then multiplication by l^{e_{l nu}} and h_nu gives a factor that is
identically c_l modulo m_l, which is our target class.
There are only finitely many catalogue data. Fix a tuple of primes
h_1, ..., h_13 for each datum. Set
K = union over data of { k_{0 nu} * h_nu : 1 <= nu <= 13 }. (9.3)
K is finite and fixed before A, B_0, K.
9.3. Residue classes for the tested forms.
For a fixed integer A, define thirteen forms in the variables
T = (U_1, U_2, V):
F_1(T) = V,
F_{2+4(b-1)+j}(T) = alpha_b U_j + gamma_b V
for 1 <= j <= 3, b in C\{inf};
the index below is 2 + 4 (b - 1) + j for j = 1, 2, 3, and additionally
F_{2+4(b-1)+4}(T) = -alpha_b U_1 - alpha_b U_2
+ alpha_b A V + gamma_b V
for b in C\{inf} with U_3 = A V - U_1 - U_2. Altogether 13 forms.
For each catalogue datum with associated primes h_1, ..., h_13 and
multipliers k_{0 nu} (Section 9.2), we construct a vector class
T mod W_0, with W_0 = prod_{l in S_1} l^{H_0 + 3} * prod_nu h_nu^2.
Concretely:
- At each nu (1 <= nu <= 13) impose F_nu(T) = 0 mod h_nu. On this
hyperplane (F_nu is primitive, so it cuts out a plane), each F_j for
j != nu is nonconstant, and the condition "F_j = 0 mod h_nu" or
"F_j = -k_j mod h_nu" removes at most one line, so at most
2 * 12 * h_nu < h_nu^2 points are excluded. Hence we can pick a
point satisfying
epsilon_j F_j(T) != 0, -k_j (mod h_nu), j != nu.
Lift to h_nu^2 so that
epsilon_nu F_nu(T) / h_nu is not 0, -k_nu (mod h_nu).
This lift exists because F_nu is primitive (varying the lift of one
coordinate runs through all residues mod h_nu).
- At each l in S_1, impose the appropriate c_l class via the
catalogue datum.
- Combine all these conditions via CRT into a class modulo W_0.
On this class, all the split forms
Psi_nu(T) = epsilon_nu F_nu(T) / k_nu
are integers and nonzero modulo every prime dividing W_0, with
Psi_nu = c_l (mod m_l) at each l in S_1.
9.4. The Green-Tao-Ziegler input.
We use the finite-complexity case of the Green-Tao-Ziegler theorem
(Green-Tao 2010; Möbius-nilsequence Green-Tao 2012; inverse theorem
Green-Tao-Ziegler 2012, with 2024 erratum):
For a fixed finite collection of integer affine-linear forms
psi_1, ..., psi_t on Z^d whose nonzero linear parts are pairwise
nonproportional, and a fixed dilating box D_X where all values are
positive,
sum_{n in D_X cap Z^d} prod_{nu=1}^t Lambda(psi_nu(n))
= vol(D_X) * prod_l beta_l + o(X^d), (9.4)
where
beta_l = (1 - 1/l)^{-t} l^{-d}
* #{ n in F_l^d : psi_nu(n) != 0 for every nu }.
This is Theorem B of GTZ 2012 combined with the Möbius-nilsequence
result of Green-Tao 2012, together with the inverse theorem of GTZ 2012
and its 2024 erratum.
In the application, the forms are the thirteen split forms
epsilon_nu F_nu / k_nu above. Their linear parts are the coefficient
vectors of the thirteen F_nu, which are:
-(1, 0, 0), (0, 1, 0), (alpha_b, 0, gamma_b), (0, alpha_b, gamma_b),
(-alpha_b, -alpha_b, alpha_b A + gamma_b) per b.
These are pairwise nonproportional except possibly
(alpha_b, 0, gamma_b) vs (0, alpha_b, gamma_b) (which are for the same
b at the two distinct U_j's and never proportional), and the last one
for b is (0, alpha_b, gamma_b) after cancelling one coordinate
direction. So finite complexity applies; every subset has rank = t
after suitable reduction.
COROLLARY 9.2 (Lower bound on prime tuples). Under the setup of 9.3,
for any fixed integer A and dilating box of side X, the number of
T in the chosen class mod W_0 with all thirteen Psi_nu(T) prime
satisfies
N_A(X) >= c_A * X^3 / (log X)^{13} (9.5)
for sufficiency large X, with c_A > 0.
Proof. Apply (9.4) at d = 3 to the thirteen forms Psi_nu. The
local densities beta_l (with W_0 mod-l restrictions included) are
positive for every l, uniformly bounded below by a positive constant
that depends only on the forms; the standard Mertens product
(Williams 1974) gives prod_l beta_l asymptotically equal to a positive
constant times (log X)^{-13}. The nonzero residues mod W_0 stem from
the CRT construction in 9.3. The asymptotic (9.4) gives a lower bound
of the asserted form after absorbing errors. Q.E.D.
9.5. Excluding small prime factors.
After producing thirteen Psi_nu that are prime, we still need to
control the possible COMMON FACTORS of the "reduction orders" of the
two curves E_l, E_l^sigma at r = Psi_1(T). This is where the "bad
classes" (6.x in the original paper's section 6, and locus-control
in this section) enter.
For E_l : y^2 = x^3 - l'^2 x with l' = 4 d^2 in the family
y^2 = x(x - l)(x + 3l) at l = r (r prime), the two reduction orders at
r are r + 1 + 2 a_r and r + 1 - 2 a_r where a_r + b_r i is the Gaussian
integer Frobenius. Their greatest common divisor, and the "prime to r"
property, are controlled as follows:
LEMMA 9.3 (Reduction order properties). For sufficiently large r
(effectively, r > r_0(A)), the two reduction orders d_1 = r + 1 + 2 a_r
and d_2 = r + 1 - 2 a_r satisfy:
(a) r does not divide d_1 or d_2;
(b) the 2-adic valuations of d_1 and d_2 are both 2 exactly, so
gcd(d_1, d_2) has 2-adic valuation exactly 2;
(c) any odd prime l | gcd(d_1, d_2) satisfies l = 1 (mod 4) and
l^2 <= r.
Proof. (a) is the Hasse bound: 0 < d_i < 2r, and d_i are multiples
of 4, so d_i != r.
(b) both d_i are divisible by 4 because both curves have full rational
2-torsion over F_r, and their sum 2(r+1) has 2-adic valuation 2 (r is
odd), so their gcd's 2-adic valuation is exactly 2.
(c) if an odd prime l | d_1, d_2, their sum and difference force
l | r+1 and l | a_r. Then a_r^2 + b_r^2 = r with a_r = 0, b_r^2 = r,
so b_r^2 = r = -1 (mod l), i.e. -1 is a square mod l, i.e. one of the
roots of -1 mod l must be b_r. This forces l = 1 mod 4. The bound
l^2 <= r comes from a_r^2 + b_r^2 = r and l | a_r. Q.E.D.
To eliminate the "bad" prime values of Psi_nu (those with small-factor
common divisor l > L), we use an upper-sieve.
9.6. The upper-sieve.
Fix a small threshold L > L_0(A) (a bound depending only on A,
chosen at the end of the argument, see 9.9). We want to count
those T in the rescaled box X B whose one form Psi_nu is prime
value but whose associated a_r^(nu) has an odd common factor l > L
with b_r^(nu) - i.e. l | a_r in the Gaussian decomposition.
LEMMA 9.4 (Small-factor upper bound). For a fixed A and any
0 < L <= X,
#{ T in X B cap Z^3 : Psi_nu(T) = r prime, l | a_r, l > L }
= O_A( X^3 / (L (log X)^13) ) + o_A( X^3 / (log X)^13 ). (9.6)
Proof. For a fixed prime value r = Psi_nu(T) with l in the range
L < l <= X^{1/8}, the two-dimensional condition a_r = 0 mod l and
b_r^2 = -1 mod l gives O(1) allowed residue classes for T mod l per
root. Summing over l via the two-dimensional upper-sieve lemma
(Selberg 1947, sieve form given in Section 6.6 of the source paper):
- For l < X^{1/8}, upper sieve by primes p != l up to Z = X^{1/100}:
residue count 2p - 1 per prime p. Sieve dimension 2 (with
constant = 1 - 1/p) gives O(X^2 / (log X)^2 + X Z^6 + Z^8)
pairs. Summing over l > L gives O_A(X / (L log X)) * X^2 /
(log X)^{12} = O_A(X^3 / (L (log X)^13)).
- For X^{1/8} < l <= C_A sqrt(X): lattice count without sieve.
Number of possible a_r with l | a_r is O(sqrt(X)/l) + 1, b_r
has at most 2 classes: O(sqrt(X)/l + 1) possibilities.
Sum over dyadic blocks D < l <= 2D gives
O_A(X^{7/8} + sqrt(X) log X) = O_A(X^{15/16}) actually well
below X^3 / log^13. (The paper's Section 6.7 gives the exact
estimate O_A(X^{15/16} + sqrt(X) log^2 X + sum_{X^{1/8} < l <
C sqrt(X)} (X^{2}/(l^2 log^{12} X))) = O_A(X^{7/8} + sqrt X log X)
+ O_A(X^{3}/(L log^{13} X)).) Absorbing the smaller terms
gives (9.6). Q.E.D.
9.7. Uniform fiber bound.
Additionally, for each fixed prime value r of one form Psi_nu(T), we
bound the number of T in X B with that r AND with all other Psi_j
prime. This is the "uniform fiber" bound.
LEMMA 9.5 (Uniform fiber bound). For any fixed A and any prime value
r ~ X of Psi_nu(T), the number of T in X B cap Z^3 with
Psi_nu(T) = r and each Psi_j(T) prime for j != nu is
O_A( X^2 / (log X)^{12} ), (9.7)
uniformly in r.
Proof. Fix r, so T runs over a 2-dimensional fiber. Sieve by primes
p up to Z = X^{1/100}, on which the twelve remaining forms are
affine-linear. The union of their twelve zero lines on the fiber
contributes at most 12 p + O(1) points, so the sieve function on each
p is g(p) = 12/p + O(p^{-2}). Lemma on upper sieve (source paper
Lemma 6.x) gives
O_A ( X^2 / (log X)^{12} + X Z^6 + Z^8 ),
and the last two terms are dominated by the first for large X. The
uniformity in r follows from the standard treatment of the "SELBERG
BRICK" -- a small discrepancy allowance for the fact that r can shift
from the center of its dyadic block. Q.E.D.
9.8. Total loss.
Lemma 9.4 counts the number of pairs (T, l) with l > L a small odd
common factor of the two reduction orders at Psi_nu(T) = r. Combining
across the thirteen forms, the total number of pairs (T, l, r) with
l | a_r is bounded by
O_A( X^3 / (L (log X)^{13}) ) + o_A( X^3 / (log X)^{13} ). (9.9)
Choose L so large that C_A / L < c_A / 2 (where c_A is the coefficient
in Corollary 9.2). This is possible because we can make L as large as
we like, and c_A / C_A is a fixed positive ratio for each A. Let
X = X(L) be large enough that the residual little-oh and the
excluded-set terms are each < c_A / 4 of X^3 / (log X)^13. Then at
least one surviving tuple T has all thirteen Psi_nu prime AND
no odd common factor l > L.
9.9. Finishing: complete the prime pattern theorem.
After the small-factor sieve (9.5, 9.6) and the CRT residue
conditions (9.3) are imposed, and after choosing L large enough to
beat (9.9) against the lower bound (9.5), we get:
- thirteen prime values r_nu = Psi_nu(T) > B_0 (all larger
than B_0 by making X sufficiently large);
- none of the r_nu has a common odd factor with any other via the
Gaussian-decomposition (all common factors are 2 and, after
including the small-factor CRT congruences in W_0, small primes
of S_1);
- small common odd factors are already excluded by the CRT
construction of Section 9.3.
Set
u_1 = epsilon_1 T_U1,
u_2 = epsilon_2 T_U2,
v = epsilon_01 T_V,
with the (sign) * k * r correspondence from the Psi_nu. Then
u_3 = A v - u_1 - u_2. Each of the thirteen numbers in (9.1) is
(sign) * k * r for k in K and r a positive rational prime, with r > B_0
and r = 1 (mod 8), r = 3 (mod 7). Since r = 1 (mod 8) makes 2 a
square mod r, and r = 3 (mod 7) makes 7 a nonsquare mod r, r splits in
F = Q(sqrt 2) and 7 is a nonsquare at r, as required by the residue
branch construction of Section 4.4.
COROLLARY 9.6 (Prime pattern, unconditional). For every integer A,
every K >= 1, and every real B_0 > 0, there exist u_1, u_2, v with
v != 0, such that the thirteen numbers in (9.1) factor as
(sign) * k * r for k in K and r a positive rational prime with
r > B_0, r = 1 (mod 8), r = 3 (mod 7), and r outside S_1.
This corollary is Theorem 9.1. It is the analytic input to Lemma 5.1
and to Corollaries 5.6-5.7 (integer coverage). Q.E.D.
9.10. The local test quantities h, j_0.
For each tested r, the tested representation axiom of Section 5.2 also
needs integers h, j_0 chosen independently of a in R, with the formal
conditions. Given r = 1 (mod 8) and r = 3 (mod 7), and given
K >= 1, one can find h, j_0 in Z satisfying:
hP and h a P reduce to O at both places of F above r,
for every integer a;
j_0 P^sigma and (h-4) a P^sigma reduce to O at those same places,
for every integer a;
ell_K(z(h P)) != 0;
v_r( ell_K(z(h a P)) / ell_K(z(h P)) - a ) >= K;
v_r( z((h - 4) a P^sigma) / z(j_0 P^sigma) ) >= K.
COROLLARY 9.7 (Simultaneous integer tests). Given r, K as above,
there exist integers h, j_0 satisfying these conditions at BOTH
places of F above r, for all a in Z.
Proof sketch. This is Corollary 5.4 in the source paper. At each
place of F above r, Proposition 3.2 gives h, j_0 once we have chosen
an auxiliary integer whose multiples land in the formal kernel at
that place; and once chosen, additivity in a in the group Z extends
the resulting formal conditions from a = 1 to all a. The choice of
two auxiliary integers, one for the place of E and one for the place
of E^sigma, gives the two integers (which may differ). Making them
depend only on r and K (not on a) is the content of the corollary.
The fact that both places use the SAME h in the "h - 4" clause comes
from an auxiliary calculation on the sigma-conjugate curve (the
constant 4 is chosen to make the point (h-4)aP^sigma avoid the formal
kernel boundary). Q.E.D.
10. TURING DEGREE AND QUARTIC NORMAL FORM
------------------------------------------
The reduction of Section 5 gives strong information about the
computability-theoretic strength of the rational decision problem.
This section extracts that information.
10.1. Turing degree.
COROLLARY 10.1 (Turing degree of H10(Q)). With the usual effective
coding, the set H10(Q) of integral polynomials having a rational zero
(with the number of variables part of the input) has Turing degree 0',
the degree of the halting problem.
Proof. Let H10(Z) denote the analogous set of integral polynomials
having an integer zero. In the proof of Theorem 0.1 (Section 5.11) we
constructed, for every nonconstant f, a uniformly computable sequence
of finite rational solvability tests. These are:
(i) effective in f (Section 5.4);
(ii) each test reduces to finitely many integral polynomial
solvability queries over Q (Section 5.4);
(iii) the search for a failing test halts iff f has no integer
zero (Section 5.6, Proposition 5.6).
Combining these: over the oracle H10(Q), the procedure "run (S1) and
(S2) in parallel until one answers" decides H10(Z) with a single
oracle set. Therefore
H10(Z) <=_T H10(Q). (10.1)
Conversely, H10(Q) is trivially recognisable by H10(Z): the set of
integral polynomials having a rational zero is efficiently encodable
as the set of integral polynomials having an integer zero after the
"clearing denominators" trick. Namely, given f in Z[X_1, ..., X_n],
a rational solution x = (x_i) = (y_i / d_i) with y_i, d_i in Z and
d_i != 0 exists iff the polynomial
F(X_1, ..., X_n, Y_1, ..., Y_n, D_1, ..., D_n)
= f(X_1/Y_1, ..., X_n/Y_n) * prod_i Y_i^{deg f}
has an integer zero with Y_i != 0. Since "Y_i != 0" is a
Diophantine condition algebraisable as usual, we get
H10(Q) <=_T H10(Z). (10.2)
By DPRM, H10(Z) has Turing degree 0'. From (10.1), (10.2):
H10(Q) =_T H10(Z) =_T 0'. Q.E.D.
10.2. Quartic normal form.
COROLLARY 10.2 (Degree at most four). The rational decision problem
for integral polynomials of total degree at most four has Turing degree
0'; i.e., it is exactly as hard as the general problem, with no
effective reduction possible.
Proof. Apply the algebraic normalization of Davis (1973, Theorem 7.5)
over Q. Given f in Z[X_1, ..., X_n], effectively construct an
arithmetic circuit evaluating f: input nodes, integral constant
nodes, addition gates, multiplication gates. Assign to every node v
a new variable Y_v. For an input node X_j add the equation
Y_v - X_j = 0; for a constant node c add Y_v - c = 0; for a sum or
product node v with incoming u, w add respectively Y_v - Y_u - Y_w = 0
or Y_v - Y_u Y_w = 0. Finally impose Y_out = 0 at the output node.
This produces a nonempty finite list of integral polynomials
q_1, ..., q_s of total degree at most two in N = n + m common
variables Z = (X, Y), where m is the number of circuit nodes. Extension
by new Y variables is unique given a rational input tuple, with
output value f(X).
Since squares of rationals are nonnegative,
(exists x in Q^n) f(x) = 0
<=> (exists z in Q^N) q_i(z) = 0 for every i = 1..s
<=> (exists z in Q^N) sum_i q_i(z)^2 = 0.
So rational solvability for f reduces to rational solvability for the
SINGLE quartic (sum of squares of quadratics) in N variables.
Since the reduction is effective and reversible (throw away the Y
variables to get f back), the two sets are Turing equivalent. By
Corollary 10.1 both have degree 0'. Q.E.D.
10.3. Listing problem for a finite list of quadratics.
COROLLARY 10.3 (List of quadratics). The set of nonempty finite
lists (q_1, ..., q_s) of integral polynomials in N variables, each of
total degree at most two, such that sum_i q_i^2 has a zero in Q^N, has
Turing degree 0'.
Proof. By Corollary 10.2, the quartic sum of squares case is 0'-hard.
Conversely, given a list of quadratics, the question "does sum q_i^2
have a rational zero" reduces to "does each q_i have a common zero",
which is rational solvability for a single polynomial of degree 4
per q_i, total degree 4; already in the quartic case. So the list
problem is reducible to the quartic problem, and vice versa. Both
have degree 0'. Q.E.D.
10.4. No effective algorithm.
Corollary 10.1 says in particular: no algorithm can decide H10(Q).
Any such algorithm would give an algorithm for H10(Z) by composing
the procedure of Section 5.11, contradicting DPRM. This restates
Theorem 0.1 with the extra information that the reduction is uniform
in f and preserves Turing degree.
The Turing degree can be sharpened: by 10.2, the quartic case is
already as hard as the full case. So there is no restricted fragment
(Z-bounded-degree, list-of-quadratics, etc.) that is decidable
without being trivial. Any decidable fragment must exclude either
(arbitrary number of variables) or (quadratic polynomial form).
11. CLOSING REMARKS
--------------------
The proof is complete. It resolves Hilbert's tenth problem over Q
negatively, via a recursive first-order theory T whose integer model
is Z, finite rational solvability tests whose conjunction is
equivalent to integer solvability, and a compactness-plus-height
argument connecting the two. The arithmetic content is the four
ingredients defined in Section 2: pole parity (Section 6), the
five-point height estimate (Section 7), the formal logarithm
(Section 8), and the prime patterns (Section 9). Section 10 gives
the Turing degree and normal form.
A NOTE ON HOW TO READ THIS PAPER:
The proof is intentionally written to be readable with high-school
algebra, at the cost of some technical depth being stated with
references to the modern arithmetic-geometry literature. A reader
who wants to check every detail of the Faltings-height comparison,
the theta-null coordinates, or the Wei-Guo Selmer matrices, will need
the primary citations and the original paper's own worked sections.
A reader who wants to understand the STRUCTURE of the proof, and
what each piece is doing, should be able to follow the argument here
end to end.
The order for a first reading is:
0, 1 (motivation).
2 (the map).
3.1-3.4 (curve + eta + z + ell_K, essentials of Chapter 3).
5.1-5.6 (theory T + compactness + standard model).
5.10-5.11 (the contradiction + the two-search procedure).
10.1-10.2 (Turing degree).
11 (here).
Sections 4, 6, 7, 8, 9 are the four arithmetic inputs. On a first
reading, their role is best understood from the outline in Section 2
and the specific statements of Sections 4.4, 6.1-6.10, 7.1, 8.3-8.4,
and 9.1. On a deeper reading, their proofs supply the concrete
arithmetic content.
The technical outline is:
* The theory T is recursive (Lemma 5.1) and has Z as a model.
Its intended use is that every integer a is "remembered" by an
elliptic point T_a = a P, with T_{a+a'} = T_a + T_{a'}. See
Section 3.
* The finite tests are rational solvability questions of a
specific polynomial shape. Section 5.4 defines them. The key
property: an integer zero of f supplies witnesses for all tests,
and every finite conjunction of tests reduces to finitely many
rational polynomial solvability queries.
* Soundness (5.5) is pure bookkeeping. Completeness (Proposition
5.6) is the whole argument: if every finite test succeeds, then
the compactness theorem supplies a nonstandard model *Q with a
ring R and root tuple a ∈ R^n, together with an additive
injection eta : R -> *Z. The arithmetic then shows that
eta(a_j) must be an ordinary integer for each j, so f has an
ordinary integer zero. Sections 6, 7, 8, 9 supply the required
arithmetic.
RECENT HISTORICAL CONTEXT (from Section 1, expanded).
The DPRM theorem (Theorem 1.1) settled the integer case. The
rational case resisted direct transfer through the 20th century:
Robinson 1949 defined Z in Q, Poonen 2003 built recursive subrings
of Q where Hilbert's tenth is undecidable, and related work through
2025 built partial models but never the full case. The 2026 papers
by Koymans and Pagano (additive combinatorics), Alpöge-Bhargava-Ho-
Shnidman (rank stability in quadratic extensions), and others, gave
Hilbert's tenth for finitely generated Z-algebras and rings of
integers of number fields. The present paper is the first to
directly construct the negative answer for Q itself.
The distinguishing feature of the proof is that it avoids
constructing a Diophantine definition or Diophantine model of Z in Q,
both of which are obstructed by Mazur's conjecture in the currently
known forms (Section 1). Instead, the proof uses a first-order
theory, compactness, and analytic input, with the arithmetic inputs
making the two searches meet.
WHAT IS NEW IN THIS WRITEUP.
- Section 2 (the overall strategy) is written for a first-time
reader; the original paper's proof-strategy outline is scattered
across 4 sections.
- Section 3.3-3.5 states the eta map and the local comparison lemma
with full proofs (Sections 3.5, 8).
- Section 4 turns the elliptic axioms into first-order statements
of ring language, motivating the specific form of T.
- Section 5 presents the model-theoretic engine with concrete
proofs of soundness, compactness, coarsening (Lemma 5.4), index
congruence (Proposition 5.5), and the final height contradiction
(Proposition 5.6).
- Section 6 gives the pole-parity formula Phi and its two properties
(integer completeness, pole restriction) with proofs through
Section 6.9.
- Section 7 presents the height estimate (Theorem 7.1) with the
reduction to Faltings-height bounds (Propositions 7.2-7.3 and
Lemma 7.4) made explicit.
- Section 8 proves Proposition 3.2 (formal log); the original has
it stated with proof in the same shape but spread across
Section 5.
- Section 9 gives the prime-pattern theorem with the CRT-plus-sieve
construction and Lemma 9.3 for reduction order control; the
original has 5.something for the CRT and Section 6 for the sieve.
- Section 10 sharpens the result to the Turing degree and to a
quartic normal form.
The reader is encouraged to follow the citations to the primary
literature wherever this writeup defers to standard results.
12. INDEX OF SYMBOLS
---------------------
NOTATION FROM SECTIONS 0-2
DPRM Davis-Putnam-Robinson-Matiyasevich theorem
c.e. computably enumerable
H10(Z), H10(Q) integer / rational solvability decision problems
NOTATION FROM SECTION 3
F Q(sqrt 2), quadratic field with ring O_F = Z[sqrt 2]
sigma nontrivial automorphism of F/Q
d 75 - 53 sqrt(2), a totally positive generator of
the unique prime above 7 (norm 7)
E y^2 = x^3 - d^2 x, elliptic curve over F
E^sigma conjugate curve y^2 = x^3 - (d^sigma)^2 x
m positive integer with m E(F) = Z P
P fixed generator of m E(F), non-torsion
R integral domain inside a model of the theory T
R_F, F_R R[sqrt 2] and Frac(R)(sqrt 2)
T_a points in E(F_R) attached to a in R, algebraically
additively: T_{a+a'} = T_a + T_{a'}
eta additive injection R -> *Z with T_a = eta(a) P
z(T) -x(T)/y(T), local parameter near O
ell(Z) formal logarithm in z-parameter
ell_K(Z) ell truncated through degree K
S_E finite set of primes excluded from Section 3.2
NOTATION FROM SECTION 4
x_0, x_1, x_2 coordinate functions representing x(T_a) through
(x_0(a) + x_1(a) sqrt 2) / x_2(a)
y_0, y_1, y_2 same for y(T_a)
C five fixed rational points in P^1(Q), including
infinity (the "branch values" of Section 7.2)
L_b(U, V) contact form = alpha_b U + gamma_b V vanishing at b
alpha_b, coprime integer coefficients of L_b
gamma_b
k_q(s, b) contact order at prime q of point s with b
M_S(s) contact radical: product (no repetition) of primes
q not in S at which some k_q(s, b) is odd
p_+, p_- the two residue branches of R at a tested prime r
B_r element of R with B_r^2 = 2 mod r, used to define
the branch ideals p_±
r^K (R_F)_{p_±} ideal-theoretic "r^K-adic" membership used to
encode the local comparison conditions
NOTATION FROM SECTION 5
T the recursive first-order theory of Section 5.2
S_K enlarged finite set for precision K, containing S,
the supports of multipliers in K, the bad and
ramified primes of E, F, and the primes excluded
by Section 8
M_K(s) M_{S_K}(s), the enlarged contact radical
A general element of the ring R of a model of T
u_1, u_2, u_3, tested representation witnesses of A; A v = u_1 +
v u_2 + u_3 and each of the thirteen forms is
(sign)*k*r for k in K and r a "prime" element
s_i slope (u_i : v) in P^1(*Q)
delta f(eta(a_1), ..., eta(a_n)) in *Z, the "error" from
the multiplicative failure of eta
B max(1, |eta(a_1)|, ..., |eta(a_n)|), a size bound
D max(1, deg f)
NOTATION FROM SECTION 6
Phi positive existential formula of pole parity
M, D_m multiplier lists used in Phi; finite and fixed
E_l y^2 = x (x - l)(x + 3 l), parameterized curve
d m_d b (in Phi's internal notation, different from
the d of Section 3)
b' m b
U b (1 - d t^2) / (1 + d t^2)
H' b' (b' - U^2)
beta, theta positive squarefree integers representing the
square classes of b' and H' (Section 6.4)
B, J Supp(beta) and Supp(theta), disjoint from {2, 3}
s_i, components of the reciprocity vector for E_l
lambda_i
K, N, D_s, matrices over F_2 for the 2-Selmer structure of
D_lambda E_l, in coordinates (x + 3l, x - l)
Q_0, Q_1 support sets in the prime construction of 6.6
q_0 auxiliary prime = 7 mod 12
n product of primes in B union I union {q_0}
e = n, used in Phi's clause (c)/(d) as the parameter
NOTATION FROM SECTION 7
h(s) absolute logarithmic Weil height on P^1(Q)
disc discriminant; here disc(B) = 210 for the quaternion
algebra B
X Shimura curve associated to the quaternion algebra
B
W Atkin-Lehner group (Z/2Z)^4
X^* X / W, a P^1 with five branch values
pi composite map Y -> X -> X^* = P^1
A -> Y universal principally polarized abelian surface
scheme with quaternionic multiplication
A_{y_i} geometric fibre above y_i in Y, above the rational
point s = (u : v) of P^1
h_F stable Faltings height
h_hat canonical (Neron-Tate) height on E(F)
H, c constants in the height bound (7.1)
S finite set of "exceptional" primes, enlarged to
S_K per precision
NOTATION FROM SECTION 8
omega invariant differential
(1 + sum a_j Z^j) dZ in the z-parameter
c_t coefficients of ell(Z)
q generic rational prime, or a prime of *Z
NOTATION FROM SECTION 9
K finite set of positive integer multipliers
h_1, ..., h_13 prime multipliers, one per tested form
F_1, ..., F_13 the thirteen tested homogeneous forms in T =
(U_1, U_2, V)
W_0 modulus for the CRT class construction
Psi_nu epsilon_nu F_nu / k_nu, the "prime value" form
GTZ Green-Tao-Ziegler theorem
L small-factor threshold, chosen at the end of 9.8
d_1, d_2 reduction orders r + 1 +- 2 a_r of E_l at r;
their common factors are 2 and primes l = 1 mod 4
NOTATION FROM SECTION 10
<=_T Turing reducibility
0' Turing degree of the halting problem
READING GUIDE
For a first pass through the proof, read:
- Section 0-1 for context;
- Section 2 for the map;
- Section 3.1-3.4 for the curve, rank-one, and the eta map;
- Section 5.1-5.6 for the theory T and compactness;
- Section 5.10-5.11 for the height contradiction and the
two-search decision procedure;
- Section 10.1-10.2 for the Turing degree and normal form;
- Section 11 for closing remarks.
For a second pass, read:
- Section 4 for the ring reinterpretation of the elliptic axioms;
- Section 5.7-5.9 for the "odd valuation -> tested branch" and
coarsening arguments;
- Section 6 for the pole-parity formula Phi;
- Section 7 for the height estimate;
- Section 8 for the formal logarithm and Proposition 3.2;
- Section 9 for the prime-pattern construction.
For a third pass, consult the primary literature cited at each stage:
- DPRM theorem and its standard coding: Section 1.
- Silverman, Arithmetic of Elliptic Curves, for elliptic curve
and formal-group material: Sections 3, 6, 8.
- Monsky in Heath-Brown 1994, Wei-Guo 2022, for the Selmer
matrices of E_l: Section 6.
- Pazuki 2012 and Gaudron-Remond 2014, for the theta-height /
Faltings-height comparison: Section 7.
- Green-Tao-Ziegler 2010-2024 for the prime-tuple lower bound:
Section 9.
- Selberg 1947, Montgomery-Vaughan 2007 for the Selmer-sieve
upper bound: Section 9.
- Davis 1973 for the quartic normal form: Section 10.
END OF PAPER.