claimroutine

Local square-root obstruction for characters

Character has a square root exactly when χv(−1)=1\chi_v(-1)=1 locally
local test for the square-root obstruction
Claim·§4.6 The endomorphism dichotomy and descent up to isogeny·inside lemSign-splitting cochain with square-root character

Statement (paper)

Let vv be a place of Q\Q and let χv\chi_v be a finite character of GQvG_{\Q_v} with values in Q‾ ×\overline\Q^{\,\times} and trivial coefficient action. Then its square-root obstruction ∂χv∈H2(GQv,{±1})\partial\chi_v\in H^2(G_{\Q_v},\{\pm1\}) vanishes if and only if χv\chi_v has a continuous character square root, which holds if and only if, via local reciprocity, χv\chi_v is trivial on the order-two subgroup {±1}\{\pm1\} of Qv×\Q_v^\times, that is, χv(−1)=1\chi_v(-1)=1. At the real place the same assertion holds, since GR=Z/2ZG_{\mathbf R}=\Z/2\Z. Moreover, at a prime outside TT the local class of ϵ\epsilon is zero.

In plain English

For a finite character χv\chi_v of GQvG_{\Q_v} (trivial action), the obstruction ∂χv∈H2(GQv,{±1})\partial\chi_v\in H^2(G_{\Q_v},\{\pm1\}) to a continuous character square root vanishes exactly when χv(−1)=1\chi_v(-1)=1 under local reciprocity; the same holds at the real place, and at primes outside TT the local class of ϵ\epsilon is zero.

Why it's here

This turns the existence of a square root of a character into a concrete local test, χv(−1)=1\chi_v(-1)=1, so a global character χ\chi can be chosen with ∂χ\partial\chi matching the class [ϵ][\epsilon] place by place. The vanishing of ϵ\epsilon locally outside TT means no condition is imposed there, which keeps χ\chi unramified outside TT in the later steps of the proof of the sign-splitting lemma.
Classified routine:
A standard local computation: under local reciprocity, the image of squaring on characters of Qv×\Q_v^\times is exactly the characters trivial on its 2-torsion {±1}\{\pm1\}. The connecting map comes from the Kummer-type squaring sequence. Vanishing of ϵ\epsilon outside TT is bookkeeping from how ϵ\epsilon was built.

Uses

Nothing else in the paper

Used by

claimGlobal character matching the obstruction class

Sources

PaperOpen
Section
§4.6 The endomorphism dichotomy and descent up to isogeny
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sections/04-height.tex:782
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