lemmaroutine

Bounded splitting field for rational fibers

Gives bounded-degree field splitting the fiber, unramified outside SS and MM
controlled field for fibers of the abelian family
Lemma·hei:fiber-field·§4.3 Rational fibers and odd-contact ramification

Statement (paper)

For s∈P1(Q)∖Cs\in\mathbf P^1(\Q)\setminus\mathcal C, put M=MS(s)M=M_S(s). There is a Galois extension K0/QK_0/\Q of uniformly bounded degree splitting the fiber π−1(s)\pi^{-1}(s), unramified outside S∪{q:q∣M}S\cup\{q:q\mid M\}. Every fiber A=AyA=\mathcal A_y above ss is defined over K0K_0 and has good reduction at every place of K0K_0 outside SS.

In plain English

For a rational point ss of P1\mathbf P^1 outside the exceptional set C\mathcal C, with M=MS(s)M=M_S(s), there is a Galois extension K0/QK_0/\Q of uniformly bounded degree that splits the fiber π−1(s)\pi^{-1}(s) and is unramified outside SS and the primes dividing MM. Every fiber A=AyA=\mathcal A_y over ss is defined over K0K_0 and has good reduction at all places of K0K_0 outside SS.

Why it's here

This fixes one controlled number field over which all the abelian varieties in a given rational fiber live, with ramification confined to SS and the primes dividing MM. Later lemmas build on it: the field of definition of the geometric homomorphisms (Lemma 4.4) contains K0K_0, and the Galois representation r2r_2 is unramified outside finitely many primes because of this control on ramification and reduction.
Classified routine:
Follows by standard arguments: the splitting field of a fiber of a finite cover of P^1 has degree bounded by the cover's degree and ramifies only at bad primes and primes where s meets the branch locus (the Chevalley–Weil / specialization-of-ramification argument), with good reduction outside S coming from the smooth model of the family.

Uses

Nothing else in the paper

Used by

lemBounded field for all geometric homomorphismsclaimThe 2-adic representation r_2 is modular

Sources

PaperOpen
Section
§4.3 Rational fibers and odd-contact ramification
Page
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Label
hei:fiber-field
Line
sections/04-height.tex:303
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