propositionroutine

Elliptic curve over F has rank one

Elliptic curve

E(F)

has rank one, with

mE(F)=\mathbb{Z}P

fixes the generator P encoding integers as aP

Proposition·ell:rank·§2.1 Elliptic indices and the local comparison

Statement (paper)

The group

E(F)

has rank one. Consequently there exist a positive integer

m

and a nontorsion point

P\in E(F)

such that

mE(F)=\Z P.

We fix such

m

and

P

once and for all.

In plain English

The group

E(F)

has rank one, so there are a positive integer

m

and a nontorsion point

P\in E(F)

with

mE(F)=\Z P

. These

m

and

P

are fixed once and for all.

Why it's here

This fixes a generator

P

of (a multiple of) the Mordell–Weil group, which is what lets integers

a

be encoded as points

aP

on the curve, as used in the standard model where

T_a=aP

. The rank is computed by a two-isogeny descent: the image of

\alpha

has eight classes, the image of

\alpha'

is trivial, and the descent formula then gives rank one.

Classified routine:

A rank computation for the paper's specific elliptic curve E over F, done by a standard 2-isogeny descent (e.g. Silverman, The Arithmetic of Elliptic Curves, Ch. X). The existence of m and P then follows from the Mordell–Weil theorem.

Uses

Nothing else in the paper

Used by

lemIntegers form a model of the theory

Inside the proof

claimZ[√2] is a PID with square totally positive unitsclaimTwo-isogeny descent rank formulaclaimImage of alpha has exactly eight classesclaimTrivial image of alpha' forces rank one

Sources

PaperOpen

Section: §2.1 Elliptic indices and the local comparison
Page: —
Label: ell:rank
Line: sections/02-reduction.tex:27

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