claimroutineinferred_dep

Normalized Selmer pairs with positive coordinates

Normalizes Selmer pairs: positive coordinates, bit-vector representatives

encodes Selmer classes as bit vectors

Claim·§3.3 A criterion for one twist·inside lemTwo-descent criterion for twist E_l

Statement (paper)

Every Selmer class

(a,b)

in Kummer coordinates

(x+3l,x-l)

has even valuations of both coordinates at

2

, and its first coordinate is positive at the real place. The sign of the second coordinate defines a surjective homomorphism from

\Sel_2(E_l/\Q)

\{\pm1\}

, so its kernel, the classes with both coordinates positive, has index two. Every class in this kernel has a unique representative of the form

\left(3^\delta\prod_iq_i^{u_i},\ \prod_iq_i^{v_i}\right)

with

\delta\in\F_2

and

u,v\in\F_2^Q

In plain English

Every Selmer class

(a,b)

in coordinates

(x+3l,x-l)

has even valuations at

2

in both coordinates and

a>0

at the real place. The sign of

b

gives a surjection

\Sel_2(E_l/\Q)\to\{\pm1\}

, and each class in its index-two kernel has a unique representative

\left(3^\delta\prod_iq_i^{u_i},\ \prod_iq_i^{v_i}\right)

with

\delta\in\F_2

and

u,v\in\F_2^Q

Why it's here

This handles the remaining places,

2

and

\infty

, so that together with the odd-prime local conditions every Selmer class is encoded by a pair of bit vectors

(\delta,u,v)

supported on

3

and

Q

. The later claim turns these vectors into the Rédei matrix equations. The surjectivity of the sign map uses the nontrivial torsion classes, and the index-two statement costs only one dimension, so the Selmer dimension can be read off from the positive-coordinate subgroup.

Classified routine:

Standard 2-descent bookkeeping for curves with full 2-torsion: local conditions at 2 and the real place plus the torsion class images (earlier claims) give the normalized representatives, as in classical Rédei-matrix/2-Selmer computations (e.g. Heath-Brown's congruent number descent).

Uses

claimTwo-torsion Kummer classes are independentclaimLocal Kummer images at odd primes and 3

Used by

claimSelmer conditions as Rédei matrix equations

Sources

PaperOpen

Section: §3.3 A criterion for one twist
Page: —
Label: —
Line: sections/03-parity.tex:359

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Conditions at 2 and real place; normalized Selmer pairs" rests on claim-par-descent-1
info
inferred_dep
outline: "Conditions at 2 and real place; normalized Selmer pairs" rests on claim-par-descent-2