claimroutineinferred_dep

Non-CM case: quasi-isogenies and sign cocycle

In non-CM case builds quasi-isogenies μg\mu_g with sign cocycle ϵ\epsilon
starts correcting the Galois action on Tate modules
Claim·§4.6 The endomorphism dichotomy and descent up to isogeny·inside thmHeight bounded by radical of contact primes

Statement (paper)

Assume CB=QC_B=\Q. Then D=BD=B and AA is absolutely simple. For each g∈Γ=Gal⁡(L/Q)g\in\Gamma=\operatorname{Gal}(L/\Q) there is a quasi-isogeny μg:gA→A\mu_g:{}^gA\to A defined over LL, with μ1=1\mu_1=1 and μg gα=αμg\mu_g\,{}^g\alpha=\alpha\mu_g for all α∈B\alpha\in B. Moreover μg gμh=c(g,h)μgh\mu_g\,{}^g\mu_h=c(g,h)\mu_{gh} with c(g,h)∈Q×c(g,h)\in\Q^\times and ∣c(g,h)∣=(deg⁡μg deg⁡μh/deg⁡μgh)1/4|c(g,h)|=\left(\deg\mu_g\,\deg\mu_h/\deg\mu_{gh}\right)^{1/4}. With ag=(deg⁡μg)−1/4a_g=(\deg\mu_g)^{-1/4} and μ~g=agμg∈Hom⁡L0(gA,A)⊗QQ‾\widetilde\mu_g=a_g\mu_g\in\operatorname{Hom}_L^0({}^gA,A)\otimes_\Q\overline\Q, one has μ~g gμ~h=ϵ(g,h)μ~gh\widetilde\mu_g\,{}^g\widetilde\mu_h=\epsilon(g,h)\widetilde\mu_{gh} with ϵ(g,h)=c(g,h)/∣c(g,h)∣∈{±1}\epsilon(g,h)=c(g,h)/|c(g,h)|\in\{\pm1\}, and ϵ\epsilon is a normalized cocycle on Γ\Gamma, which we inflate to GQG_\Q.

In plain English

If CB=QC_B=\Q, then D=BD=B and AA is absolutely simple, and each g∈Gal⁡(L/Q)g\in\operatorname{Gal}(L/\Q) has a BB-linear quasi-isogeny μg:gA→A\mu_g:{}^gA\to A over LL with μ1=1\mu_1=1. These satisfy μg gμh=c(g,h)μgh\mu_g\,{}^g\mu_h=c(g,h)\mu_{gh} with c(g,h)∈Q×c(g,h)\in\Q^\times and ∣c(g,h)∣=(deg⁡μgdeg⁡μh/deg⁡μgh)1/4|c(g,h)|=(\deg\mu_g\deg\mu_h/\deg\mu_{gh})^{1/4}. After normalizing by ag=(deg⁡μg)−1/4a_g=(\deg\mu_g)^{-1/4} the defect becomes a sign ϵ(g,h)∈{±1}\epsilon(g,h)\in\{\pm1\}, which is a normalized cocycle on Γ\Gamma, inflated to GQG_\Q.

Why it's here

Since the fibers are isogenous to their Galois conjugates and all such homomorphisms are defined over the bounded field LL, one can choose BB-linear isomorphisms up to isogeny between gA{}^gA and AA. These are unique up to a rational scalar, so the composition defect is a rational number. Rescaling by the degree removes its absolute value and leaves only a sign cocycle, which the next claim uses to correct the Galois action on Tate modules into a genuine representation.
Classified routine:
This is the standard construction of a Galois 2-cocycle from Galois-conjugate isogenies, as in Ribet's theory of Q-curves and abelian varieties of GL2-type (building blocks, Pyle; Ribet 1992/2004). Here it is adapted to quaternionic-multiplication surfaces, where Skolem–Noether and C_B=Q force each μ_g to be unique up to a rational scalar.

Uses

lemBounded field for all geometric homomorphismsclaimAbelian surface family with quaternionic multiplication

Used by

claimCorrected Galois representation on Tate module

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:682
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    outline: "Non-CM case: B-linear quasi-isogenies and sign cocycle" rests on hei:hom-field
  • info
    inferred_dep
    outline: "Non-CM case: B-linear quasi-isogenies and sign cocycle" rests on claim-hei-height-2