claimroutineinferred_dep

Quadratic twist removes inertia outside T

Quadratic twist makes β\beta trivial on inertia outside TT
final adjustment of beta
Claim·§4.6 The endomorphism dichotomy and descent up to isogeny·inside lemSign-splitting cochain with square-root character

Statement (paper)

After multiplying β\beta by the quadratic character of Q(dβ)\Q(\sqrt{d_\beta}), for a suitable squarefree integer dβd_\beta whose odd prime factors are precisely the odd primes p∉Tp\notin T at which β∣Ip\beta|_{I_p} is nontrivial, the cochain β\beta is trivial on inertia at every odd prime outside TT. The corrected cochain still satisfies and the restriction properties, and takes values in μ2m\mu_{2m} with 2m≤CM2m\le CM.

In plain English

Multiplying β\beta by the quadratic character of Q(dβ)\Q(\sqrt{d_\beta}), where dβd_\beta is squarefree with odd prime factors exactly the odd primes p∉Tp\notin T at which β∣Ip\beta|_{I_p} is nontrivial, makes β\beta trivial on inertia at every odd prime outside TT. The corrected β\beta still satisfies β2=χ\beta^2=\chi and the coboundary relation, keeps the restriction properties on GLG_L, and takes values in μ2m\mu_{2m} with 2m≤CM2m\le CM.

Why it's here

This is the final adjustment in the proof of the sign-splitting lemma. Because β\beta from the previous claim is a character on GLG_L and LL is unramified outside TT, the inertia at odd p∉Tp\notin T can only act through a quadratic character. A quadratic twist therefore cancels it. Since the twist is a character, the coboundary relation is unchanged, and the order bound survives because the twist adds only a factor of 2 on top of μ2m\mu_{2m} values already within CMCM.
Classified routine:
A standard quadratic-twist adjustment: since β\beta restricted to GLG_L is a character with LL unramified outside TT, inertia at odd p∉Tp\notin T acts through a quadratic character, which is cancelled by twisting with the character of Q(dβ)\Q(\sqrt{d_\beta}). This is bookkeeping following from the previous claim and hei-hom-field.

Uses

claimCochain splitting epsilon, conjugation invariant on G_LlemBounded field for all geometric homomorphisms

Used by

Nothing in the paper

Sources

PaperOpen
Section
§4.6 The endomorphism dichotomy and descent up to isogeny
Page
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sections/04-height.tex:835
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Quadratic correction kills inertia outside T" rests on claim-hei-sign-splitting-4
  • info
    inferred_dep
    outline: "Quadratic correction kills inertia outside T" rests on hei:hom-field