Some Finiteness Theorems for Abelian Varieties

Last time we saw (see Proposition 6.5 in those lecture notes) that an abelian variety A of dimension g over K, the fraction field of a henselian dvr R, acquires semistable reduction over K(A[`]) for ` equal to 4 or an odd prime, with ` not divisible by the residue characteristic. (The same circle of ideas yielded the non-obvious fact that over K the intersection of two “semistable field extensions for A” is another such, and that the minimal such extension has degree over K dividing an explicit constant N depending only on g, made explicit in Theorem 6.8 of those lecture notes.) As a consequence, for an abelian variety A over a global field K, A acquires everywhere semistable reduction over K(A[M ]) where M ∈ {12, 15, 20} is chosen so that char(K) does not divide M , so for any finite place v of K there is a factor of M having the form ` as above relative to the valuation ring of v. (If char(K) = 0 we can use M = 12, but in characteristics 2 and 3 we must use 15 and 20 respectively.) Moreover the Galois extension K(A[M ])/K is unramified at all v M∞ that are good for A. The conclusion is that for g-dimensional A overK, good outside a set of places S containing all v|M∞, A becomes everywhere semistable over an extension K ′/K that is unramified outside S and of degree bounded only in terms of g. By Hermite–Minkowski, there are finitely many such extensions K ′. Thus, there is a single finite extension K ′ = K ′ g,S over which all such A become semistable with bad places contained in the preimage S ′ of S. This reduces the problem of proving the Shafarevich conjecture for the triple (K,S, g) to the case of the triple (K ′, S ′, g) under the additional property of being everywhere semistable provided that an abelian variety A over K is determined up to finitely many possibilities (up to K-isomorphism) when its scalar extension AK′ over K ′ is known up to K ′-isomorphism. Thus, to justify the sufficiency of proving the Shafarevich conjecture in the more restrictive setting of everywhere semistable abelian varieties over number fields, it remains to prove: