lemmaroutine

Sign-splitting cochain with square-root character

Splits sign cocycle by cochain β\beta with β2=χ\beta^2=\chi, order m≤CMm\le CM
corrects projective Tate action and bounds the level
Lemma·hei:sign-splitting·§4.6 The endomorphism dichotomy and descent up to isogeny

Statement (paper)

There is a finite Dirichlet character χ\chi of order m≤CMm\le C M, unramified outside TT, and a continuous cochain β:GQ→μ2m\beta:G_\Q\to\mu_{2m} such that β2=χ,β(g)β(h)β(gh)=ϵ(g,h). \beta^2=\chi,\qquad \frac{\beta(g)\beta(h)}{\beta(gh)}=\epsilon(g,h). The cochain can be chosen trivial on inertia at every prime outside TT. Its restriction to GLG_L is a conjugacy-invariant character. The local conductor of χ\chi at any fixed prime is bounded independently of ss, and χ\chi is tame at odd primes.

In plain English

There is a finite Dirichlet character χ\chi of order m≤CMm\le CM, unramified outside TT, and 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)\beta(g)\beta(h)/\beta(gh)=\epsilon(g,h). The cochain β\beta can be chosen trivial on inertia at primes outside TT, and its restriction to GLG_L is a conjugacy-invariant character. The local conductor of χ\chi at any fixed prime is bounded independently of ss, and χ\chi is tame at odd primes.

Why it's here

This lemma splits the sign cocycle ϵ\epsilon as a coboundary of an explicit cochain β\beta with controlled order and ramification. That lets β\beta be used to correct the projective Tate-module action into a genuine representation, with level and ramification bounds that do not depend on ss. The proof builds χ\chi by matching local obstruction classes to [ϵ][\epsilon], bounds its order and conductor, lifts to β\beta, and then removes inertia outside TT by a quadratic twist.
Classified routine:
An adaptation of the standard technique for lifting projective Galois representations by splitting a sign 2-cocycle with a character (Tate's vanishing of H^2(G_Q, Q/Z), as made explicit in Serre's work on modular forms of weight one), here with bookkeeping of the order and conductor.

Uses

Nothing else in the paper

Used by

lemModular level is polynomially bounded in MclaimLarge primes in T are tameclaimUniform ramification cut for betaclaimCorrected Galois representation on Tate moduleclaimThe 2-adic representation r_2 is modular

Inside the proof

claimLocal square-root obstruction for charactersclaimGlobal character matching the obstruction classclaimOrder and conductor bounds for chiclaimCochain splitting epsilon, conjugation invariant on G_LclaimQuadratic twist removes inertia outside T

Sources

PaperOpen
Section
§4.6 The endomorphism dichotomy and descent up to isogeny
Page
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Label
hei:sign-splitting
Line
sections/04-height.tex:766
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