claimroutineinferred_dep

Corrected Galois representation on Tate module

Twists Galois action by

\beta

into a genuine representation

R_\ell=r_\ell^2

gives the representation shown modular

Claim·§4.7 Two-dimensional representations and modularity·inside thmHeight bounded by radical of contact primes

Statement (paper)

For

\ell=2,3

, the map

R_\ell(g)=\beta(g)\widetilde\mu_{\bar g}\circ g

is a continuous representation of

G_\Q

on the covariant rational Tate module of

A

with coefficients extended, commuting with

B

, and

R_\ell\simeq r_\ell\oplus r_\ell

for a two-dimensional representation

r_\ell

. With

G_{L'}=\ker(\beta|_{G_L})

, the field

L'/\Q

is Galois with

[L':\Q]\le 2m[L:\Q]\le CM

, and on

G_{L'}

the representation

R_\ell

coincides with the Tate-module representation of

A

with coefficients extended.

In plain English

For

\ell=2,3

, twisting the Galois action on the rational Tate module of

A

by the quasi-isogenies

\widetilde\mu_{\bar g}

and the cochain

\beta

gives a genuine continuous representation

R_\ell

G_\Q

commuting with

B

, which is two copies of a two-dimensional representation

r_\ell

. It agrees with the ordinary Tate-module representation on

G_{L'}

, where

L'/\Q

is Galois of degree at most

2m[L:\Q]\le CM

Why it's here

The quasi-isogenies from the non-CM claim only give a projective action, with defect the sign cocycle

\epsilon

. Multiplying by

\beta

from the sign-splitting lemma cancels that defect, so

R_\ell

is a true representation. Splitting it as

r_\ell\oplus r_\ell

yields the two-dimensional representation

r_\ell

whose modularity is used next, and the field

L'

of controlled degree is what makes the Faltings height bound possible.

Classified routine:

This is the standard Ribet/Pyle construction for abelian varieties of GL2-type (building blocks): the Tate-module action is twisted by the quasi-isogenies, and the sign cocycle is cancelled with a cochain. The result is a genuine representation that splits as two copies of a 2-dimensional one. Here it follows directly from the earlier quasi-isogeny and sign-splitting claims.

Uses

claimNon-CM case: quasi-isogenies and sign cocyclelemSign-splitting cochain with square-root character

Used by

claimThe 2-adic representation r_2 is modularclaimFaltings height of A is polynomial in M

Sources

PaperOpen

Section: §4.7 Two-dimensional representations and modularity
Page: —
Label: —
Line: sections/04-height.tex:856

Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

info
inferred_dep
outline: "Corrected two-dimensional representation and field L'" rests on claim-hei-height-4
info
inferred_dep
outline: "Corrected two-dimensional representation and field L'" rests on hei:sign-splitting