claimroutineinferred_dep

Cochain splitting epsilon, conjugation invariant on G_L

Gives cochain β\beta with β2=χ\beta^2=\chi splitting ϵ\epsilon
cochain whose coboundary is the sign cocycle
Claim·§4.6 The endomorphism dichotomy and descent up to isogeny·inside lemSign-splitting cochain with square-root character

Statement (paper)

There is a continuous cochain β:GQ→μ2m\beta:G_\Q\to\mu_{2m} with β2=χ\beta^2=\chi and β(g)β(h)β(gh)=ϵ(g,h)\dfrac{\beta(g)\beta(h)}{\beta(gh)}=\epsilon(g,h) for all g,h∈GQg,h\in G_\Q. Its restriction to GLG_L is a character, and it is conjugacy-invariant: β(gug−1)=β(u)\beta(gug^{-1})=\beta(u) for all u∈GLu\in G_L and g∈GQg\in G_\Q.

In plain English

There is a continuous cochain β:GQ→μ2m\beta:G_\Q\to\mu_{2m} with β2=χ\beta^2=\chi whose coboundary is ϵ\epsilon, i.e. β(g)β(h)/β(gh)=ϵ(g,h)\beta(g)\beta(h)/\beta(gh)=\epsilon(g,h). On GLG_L it is a character, and it is invariant under conjugation by all of GQG_\Q.

Why it's here

Since ∂χ=[ϵ]\partial\chi=[\epsilon] from the previous claim, ϵ\epsilon is a coboundary after taking square roots, and this gives a square root β\beta of χ\chi that splits ϵ\epsilon as a cochain. The character and conjugacy-invariance properties on GLG_L are what later claims need. They adjust β\beta by a quadratic correction to kill inertia outside TT, which gives the cochain in the sign-splitting lemma.
Classified routine:
This follows from the previous claim by the standard connecting-homomorphism argument for the Kummer sequence 1→±1→μ_{2m}→μ_m→1. Since ∂χ=[ε], one picks a square-root lift of χ and corrects it by a cochain. The character property and conjugation invariance on G_L come from direct cocycle bookkeeping.

Uses

claimGlobal character matching the obstruction class

Used by

claimQuadratic twist removes inertia outside T

Sources

PaperOpen
Section
§4.6 The endomorphism dichotomy and descent up to isogeny
Page
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sections/04-height.tex:824
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Cochain beta splitting epsilon, conjugacy invariant on G_L" rests on claim-hei-sign-splitting-2