claimstandard

Two-isogeny descent rank formula

Two-isogeny descent:

|\mathrm{im}\alpha||\mathrm{im}\alpha'|=4\cdot2^{\mathrm{rank}}

ties the rank to sizes of the descent images

Claim·§5 The rank-one elliptic group and its local parameters·inside propElliptic curve over F has rank one

Statement (paper)

With

\varphi:E\to E'

and

\widehat\varphi:E'\to E

the two-isogenies whose composite is

[2]

, and

\alpha:E(F)\to F^*/F^{*2}

\alpha':E'(F)\to F^*/F^{*2}

the associated homomorphisms, we have

|\operatorname{im}\alpha|\,|\operatorname{im}\alpha'|=4\,2^{\rank E(F)}

In plain English

For the two-isogenies

\varphi:E\to E'

and

\widehat\varphi:E'\to E

with composite

[2]

, and their homomorphisms

\alpha

and

\alpha'

into

F^*/F^{*2}

, the product of the image sizes satisfies

|\operatorname{im}\alpha|\,|\operatorname{im}\alpha'|=4\cdot 2^{\operatorname{rank}E(F)}

Why it's here

This is the standard two-isogeny descent identity, which ties the rank of

E(F)

to the sizes of the images of

\alpha

and

\alpha'

. It lets the proof of the rank statement reduce to computing these images: once one is shown trivial, the other's size determines the rank, as used in the next claim.

Classified standard:

Standard 2-isogeny descent formula |im α||im α'| = 2^{r+2} (for curves with a rational 2-torsion point), as in Silverman–Tate, Rational Points on Elliptic Curves, Ch. III.

Uses

Nothing else in the paper

Used by

claimTrivial image of alpha' forces rank one

Sources

PaperOpen

Section: §5 The rank-one elliptic group and its local parameters
Page: —
Label: —
Line: sections/05-elliptic.tex:28

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