theoremnovelstructure_uncertain

Height bounded by radical of contact primes

Bounds height of

s

H M_S(s)^c

off five exceptional points

polynomial height bound by contact primes

Theorem·hei:height·§2.2 Contact primes and ordinary integer witnesses

Statement (paper)

There exist five distinct points

\mathcal C\subset\mathbf P^1(\Q)

, including

\infty

, a finite set

S

of rational primes, and constants

H,c>0

with the following property. For each

b\in\mathcal C

, choose a primitive integral linear form

L_b(U,V)

vanishing at

b

, with

L_\infty(U,V)=V

. For

s=(u:v)\in\mathbf P^1(\Q)\setminus\mathcal C

, let

M_S(s)

be the product, taken without repetition, of primes

q\notin S

for which at least one of the integers

v_q(L_b(u,v))-\min\{v_q(u),v_q(v)\},\qquad b\in\mathcal C,

is odd. Here

v_q(q)=1

and

v_q(0)=+\infty

. Then

\hgt(s)\le H M_S(s)^c.

In plain English

There is a five-point set

\mathcal C\subset\mathbf P^1(\Q)

containing

\infty

, a finite set

S

of primes, and constants

H,c>0

such that every rational point

s=(u:v)

outside

\mathcal C

has

\hgt(s)\le H M_S(s)^c

. Here

M_S(s)

is the product of primes

q\notin S

at which some

v_q(L_b(u,v))-\min\{v_q(u),v_q(v)\}

, for

b\in\mathcal C

, is odd.

Why it's here

The set

\mathcal C

is the branch locus of the rational Shimura curve quotient, and the abelian surface family over it has bad behavior governed by

\mathcal C

. The proof bounds the Faltings height of the fiber by a polynomial in

M

, using CM boundedness or modularity and isogeny comparison in the non-CM case. It then applies hei-base-height to turn this into a bound on

\hgt(s)

. The result says that a rational point of large height must have many primes where some

L_b(u,v)

has odd valuation, which is the ordinary-integer witness input for later results such as the enlarged-exception corollary and the uniform height bound for ring elements.

Classified novel:

The key new arithmetic input: it bounds the height of a rational point by a power of the radical of its odd-valuation contact primes, using a rational Shimura curve quotient branched at five points, Faltings heights of the fibre abelian surfaces, CM boundedness, modularity and isogeny estimates. No prior result supplies such a polynomial radical bound tied to this specific five-point branch locus.

Uses

lemBase height bounded by Faltings height

Used by

corEnlarging exceptional primes changes only constantclaimUniform height bound for ring elements

Inside the proof

claimQuaternionic Shimura curve quotient has five rational branch valuesclaimAbelian surface family with quaternionic multiplicationclaimCM fibers have uniformly bounded Faltings heightclaimNon-CM case: quasi-isogenies and sign cocycleclaimCorrected Galois representation on Tate moduleclaimThe 2-adic representation r_2 is modularclaimModular Jacobian dimension and height boundsclaimFaltings height of A is polynomial in M

Figures

A proof-flow diagram of six boxes connected by arrows. The left column starts with varying $A\in R$ and choosing a representation at precision $K$ with slopes $\widetilde s_i$, then passes to local analysis at each prime $q\mid M_K(\widetilde s_i)$ (pole parity and coarsening), then to the paired elliptic tests giving $v_q(\widetilde a-\eta(a))\ge K$. The right column keeps the root tuple fixed with nonzero $\delta=f(\eta(\mathbf a))$ bounded by $C_fB^D$; an arrow labeled 'polynomial evaluation' and an arrow from the tests feed a box concluding $M_K(\widetilde s_i)^K\mid\delta$. A final wide box applies the five-point height estimate to get $h(\widetilde A)\ll_f B$ uniformly for $A\in R$. — Figure 1diagramp. 16
The use of one nonzero $\delta$ for every $A\in R$ . All slopes shown come from the representation chosen at the fixed ordinary precision $K$ , and $M_K$ uses the fixed exceptional set for $K$ . The comparison holds for every $a\in R$ , so it applies to the fixed root tuple. The implied constant in the final bound is ordinary and independent of $A$ and $\mathbf a$ .
What it shows
A proof-flow diagram of six boxes connected by arrows. The left column starts with varying $A\in R$ and choosing a representation at precision $K$ with slopes $\widetilde s_i$ , then passes to local analysis at each prime $q\mid M_K(\widetilde s_i)$ (pole parity and coarsening), then to the paired elliptic tests giving $v_q(\widetilde a-\eta(a))\ge K$ . The right column keeps the root tuple fixed with nonzero $\delta=f(\eta(\mathbf a))$ bounded by $C_fB^D$ ; an arrow labeled 'polynomial evaluation' and an arrow from the tests feed a box concluding $M_K(\widetilde s_i)^K\mid\delta$ . A final wide box applies the five-point height estimate to get $h(\widetilde A)\ll_f B$ uniformly for $A\in R$ .
Read from the image by claude-opus-5.5 · confidence 85%
SupportspropFinite test success gives integer zeroclaimPolynomial error delta is zero or boundedclaimContact radical K-th power divides deltaclaimUniform height bound for ring elementspropIndex congruence at every contact primelemLocal comparison of elliptic index with integerlemOdd valuation selects a tested branchthmHeight bounded by radical of contact primes

Sources

PaperOpen

Section: §2.2 Contact primes and ordinary integer witnesses
Page: —
Label: hei:height
Line: sections/02-reduction.tex:232

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
structure_uncertain
The proof environment is a 11-line completion; the actual argument (curve construction, family, CM/non-CM cases, modularity, Jacobian height, isogeny) is spread over unlabeled text throughout Section 4, outlined as claims under hei:height.
info
structure_uncertain
claim "Quaternionic curve quotient is P^1 with five rational branch values" (sections/04-height.tex:43-172) lies outside the recorded proof; a second proof or an appendix
info
structure_uncertain
claim "Abelian surface family with B-action and isogenous conjugates" (sections/04-height.tex:177-280) lies outside the recorded proof; a second proof or an appendix
info
structure_uncertain
claim "CM fibers have uniformly bounded Faltings height" (sections/04-height.tex:622-680) lies outside the recorded proof; a second proof or an appendix
info
structure_uncertain
claim "Non-CM case: B-linear quasi-isogenies and sign cocycle" (sections/04-height.tex:682-764) lies outside the recorded proof; a second proof or an appendix
info
structure_uncertain
claim "Corrected two-dimensional representation and field L'" (sections/04-height.tex:856-899) lies outside the recorded proof; a second proof or an appendix
info
structure_uncertain
claim "r_2 is modular from a weight-two newform" (sections/04-height.tex:901-979) lies outside the recorded proof; a second proof or an appendix
info
structure_uncertain
claim "Modular Jacobian height polynomial in N" (sections/04-height.tex:1205-1230) lies outside the recorded proof; a second proof or an appendix
info
structure_uncertain
claim "Faltings height of A polynomial in M via isogeny" (sections/04-height.tex:1232-1297) lies outside the recorded proof; a second proof or an appendix