claimroutineinferred_dep

Global character matching the obstruction class

Builds χ\chi unramified outside TT with ∂χ=[ϵ]\partial\chi=[\epsilon]
global character matching the obstruction class
Claim·§4.6 The endomorphism dichotomy and descent up to isogeny·inside lemSign-splitting cochain with square-root character

Statement (paper)

There is a global Dirichlet character χ\chi, unramified outside TT, whose square-root obstruction satisfies ∂χ=[ϵ]\partial\chi=[\epsilon] in H2(GQ,{±1})H^2(G_\Q,\{\pm1\}). Locally, at every finite place vv the obstruction ∂χv\partial\chi_v equals the local class of ϵ\epsilon, and the real invariants agree as well.

In plain English

There is a Dirichlet character χ\chi unramified outside TT whose square-root obstruction ∂χ\partial\chi equals the class [ϵ][\epsilon] in H2(GQ,{±1})H^2(G_\Q,\{\pm1\}). This holds place by place: at every finite place the local class ∂χv\partial\chi_v equals the local class of ϵ\epsilon, and the real invariants agree too.

Why it's here

This is the global construction step of the sign-splitting lemma. By the local criterion, matching ϵ\epsilon at each place reduces to prescribing the value χv(−1)\chi_v(-1), and ϵ\epsilon is locally trivial outside TT, so χ\chi can be built with ramification only in TT. The later claims then bound the order and conductor of χ\chi and use ∂χ=[ϵ]\partial\chi=[\epsilon] to produce the cochain β\beta with β2=χ\beta^2=\chi.
Classified routine:
Follows from the local criterion (previous claim) together with the Brauer group exact sequence / Hasse principle for H^2(G_Q,±1) and the existence of Dirichlet characters with prescribed local signs at finitely many primes (via CRT/Grunwald–Wang-type construction); a standard Galois cohomology argument.

Uses

claimLocal square-root obstruction for characters

Used by

claimOrder and conductor bounds for chiclaimCochain splitting epsilon, conjugation invariant on G_L

Sources

PaperOpen
Section
§4.6 The endomorphism dichotomy and descent up to isogeny
Page
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Label
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Line
sections/04-height.tex:806
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Dirichlet character chi matching obstruction class" rests on claim-hei-sign-splitting-1