lemmaroutineinferred_dep

Bounded field for all geometric homomorphisms

Gives one bounded field L⊃K0L\supset K_0 defining all geometric homomorphisms
single Galois group for level and cocycle arguments
Lemma·hei:hom-field·§4.5 A field of definition for homomorphisms

Statement (paper)

There is a Galois extension L/QL/\Q containing K0K_0, of uniformly bounded degree, over which all geometric homomorphisms between all Galois conjugates of AA are defined. It is unramified outside TT.

In plain English

There is a Galois extension L/QL/\Q containing K0K_0, of uniformly bounded degree and unramified outside TT, over which every geometric homomorphism between any Galois conjugates of AA is defined.

Why it's here

This lemma extends the controlled field K0K_0 from the fiber lemma to a single field LL that also trivializes all homomorphisms among the conjugates of AA. The bounded degree and ramification confined to TT let later arguments on levels, tameness, Faltings height, and the quasi-isogeny cocycle work over one fixed Galois group Gal⁡(L/Q)\operatorname{Gal}(L/\Q).
Classified routine:
Follows from the fiber-field lemma together with the standard fact that homomorphisms between abelian varieties are defined over an extension of bounded degree, unramified where the varieties have good reduction (Silverberg 1992 on fields of definition of endomorphisms, plus Néron–Ogg–Shafarevich), then taking the Galois closure.

Uses

lemBounded splitting field for rational fibers

Used by

lemModular level is polynomially bounded in MclaimQuadratic twist removes inertia outside TclaimLarge primes in T are tameclaimCM fibers have uniformly bounded Faltings heightclaimNon-CM case: quasi-isogenies and sign cocycle

Sources

PaperOpen
Section
§4.5 A field of definition for homomorphisms
Page
—
Label
hei:hom-field
Line
sections/04-height.tex:584
Leannot aligned

Phase 2 aligns this node to a Lean declaration.

Friction

  • info
    inferred_dep
    critique: The field L contains "the splitting field K_0 from Lemma " (04-height.tex:575-576). The proof (line 605) uses that K_0 is unramified outside T and that the conjugates have good reduction, both from that lemma.