Specialization of endomorphism rings of abelian varieties
摘要
Let k be a number field with algebraic closure k, let V be a variety defined over k, and let A be an abelian variety defined over the function field k(V). It is shown that for v in V{k) the absolute endomorphism ring End Ay of the fibre)) Ay is almost always)) isomorphic to the absolute endomorphism ring End A; and even that the exceptional set)) of such v, where there is no such isomorphism, is sparse)). More precisely, fix a projective embedding <^ of V over k and let hy be the associated absolute logarithmic Weil height. Then there is a constant A, depending only on the dimension of A, and a constant C, depending only on k, V, A and </?, with the following property. For any real d > 1 and h > 1 there exists a homogeneous polynomial of degree at most C'(max{d',/i})^, not vanishing identically on V, that vanishes at all exceptional v in V(k) with [k(v) :k] < d and h^(v) < h. For example, this implies that for any real H > 3 there are at most C(\ogH) x positive integers v < H for which the Jacobian of the curve y 5 = x(x -l){x -v) has complex multiplication; or, there are at most C'ff^log.fi^ sets of positive integers VQ, ... ,VQ < H for which the Jacobian of the curve y~ = vox 5 + + VQ has non-trivial endomorphisms.