claimroutineinferred_dep

Twist scaling reduces to one parameter

Reduces to one

n

with class-

\beta

points on

E_n

and

E_{n\theta}

collapses two twist parameters to one

Claim·§3.2 Preparing integer witnesses·inside propPositive existential formula forcing even poles

Statement (paper)

Write

H'=\theta c^2

with

c\in\Q^\times

. For every

n\in\Q^\times

, the map

(x,y)\mapsto(c^2x,c^3y)

is an isomorphism

E_{n\theta}\to E_{nH'}

that preserves

x

-coordinate square classes. Hence, taking

e=n

in the formula, it suffices to find one

n

for which both

E_n

and

E_{n\theta}

have a rational point with nonzero

x

-coordinate of class

\beta

, the class of

b'

In plain English

Writing

H'=\theta c^2

with

c\in\Q^\times

, the map

(x,y)\mapsto(c^2x,c^3y)

is an isomorphism

E_{n\theta}\to E_{nH'}

that preserves

x

-coordinate square classes for every

n\in\Q^\times

. So it suffices to find one

n

such that both

E_n

and

E_{n\theta}

have a rational point with nonzero

x

-coordinate of class

\beta

, the class of

b'

, taking

e=n

in the formula.

Why it's here

The scaling by

c

identifies the twist by

nH'

with the twist by

n\theta

without changing square classes of

x

-coordinates. The two-parameter requirement, one condition for

E_n

and one for

E_{nH'}

, therefore collapses to a single parameter

n

with conditions on

E_n

and

E_{n\theta}

. This simplifies the construction of integer witnesses in the proof of the pole-parity proposition, where

\beta

and

\theta

are as constrained by the cross-square conditions on the prime sets

B

and

J

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Classified routine:

Direct computation: quadratic twists by classes differing by a square are isomorphic via the standard change of variables

(x,y)\mapsto(c^2x,c^3y)

, and since

c^2

is a square this preserves square classes of

x

-coordinates, giving a bookkeeping reduction inside the proof.

Uses

lemCross-square conditions on B, J, β, θ

Used by

Nothing in the paper

Sources

PaperOpen

Section: §3.2 Preparing integer witnesses
Page: —
Label: —
Line: sections/03-parity.tex:219

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Friction

info
inferred_dep
outline: "Twist scaling reduces to one parameter n" rests on par:cross-squares