lemmanovelstructure_uncertainunresolved_ref

Prime patterns with controlled reduction orders

Finds thirteen numbers

\epsilon kr

with large good primes

r

, bounded

k

arithmetic input for the two conjugate local tests

Lemma·pri:pattern·§6 Prime patterns for the two conjugate local tests

Statement (paper)

There is a finite set

\mathcal K\subset\Z_{>0}

such that, for every

A\in\Z

and every real

B>0

, there exist integers

u_1,u_2,v

, with

v\ne0

, for which, on setting

u_3=Av-u_1-u_2

, each of the thirteen numbers

v,\qquad L_b(u_j,v) \quad(1\le j\le3,\ b\in\mathcal C\setminus\{\infty\})

has the form

\epsilon kr

, where

\epsilon\in\{1,-1\}

k\in\mathcal K

, and

r>B

is a positive rational prime. Every such prime splits in

F

and is a prime of good reduction for both

E

and

E^\sigma

. At either place of

F

above

r

, if

d_1,d_2

are the cardinalities of the reduction groups of

E,E^\sigma

over

\F_r

, then

\gcd(d_1,d_2)\mid4, \qquad r\nmid d_1d_2.

The set

\mathcal K

is independent of

A

and

B

In plain English

There is a finite set

\mathcal K

of positive integers, independent of

A

and

B

, such that for every

A\in\Z

and

B>0

one can find integers

u_1,u_2,v

with

v\ne0

and

u_3=Av-u_1-u_2

, for which the thirteen numbers

v

and

L_b(u_j,v)

are each of the form

\epsilon kr

with

\epsilon=\pm1

k\in\mathcal K

and

r>B

prime. Each such prime

r

splits in

F

, is a good reduction prime for

E

and

E^\sigma

, and the reduction group orders satisfy

\gcd(d_1,d_2)\mid4

and

r\nmid d_1d_2

Why it's here

This lemma supplies the arithmetic input for the two conjugate local tests: it lets one choose the thirteen test numbers so that each is a bounded multiple of a large prime with well-behaved reduction of

E

and

E^\sigma

. The proof combines a Green–Tao–Ziegler count of simultaneous prime values of the thirteen linear forms with congruence conditions fixing the orders as

r+1\pm2a

, and an upper-sieve count showing that primes whose two orders share a large odd factor are rare, so a good pattern survives. The uniform set

\mathcal K

and the order conditions are what the later integer-test and coverage results rely on.

Classified novel:

No earlier result supplies this: it pairs the Green–Tao–Ziegler count of simultaneous prime values of linear forms with congruence control of the orders of the conjugate elliptic curves E and E^σ, plus an upper-sieve bound on shared large odd factors. It is the paper's own arithmetic input for the conjugate local tests, built specifically for this construction.

Uses

Nothing else in the paper

Used by

corInteger coverage by prime patternscorSimultaneous integer tests at both places

Inside the proof

claimReduction orders r+1±2a and common factorsclaimFinite multiplier set and initial residue classclaimGreen-Tao-Ziegler lower bound for prime patternsclaimSieve product lower bound G(Z)claimFew primes with large common order factorclaimUniform upper sieve on affine fibersclaimSurviving prime pattern satisfies order conditions

Sources

PaperOpen

Section: §6 Prime patterns for the two conjugate local tests
Page: —
Label: pri:pattern
Line: sections/06-primes.tex:21

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
structure_uncertain
No proof environment; the proof is the prose of sections/06-primes.tex lines 121-699.
warn
unresolved_ref
Lower bound rests on the external Green-Tao-Ziegler linear-equations-in-primes theorem.