Non-simple abelian varieties in a family: geometric and analytic approaches
Abstract
Let At be a family of abelian varieties over a number field k parametrized by a rational coordinate t, and suppose the generic fiber of At is geometrically simple. For example, we may take At to be the Jacobian of the hyperelliptic curve y2 = f(x)(x-t) for some polynomial f. We give two upper bounds for the number of t ∈ k of height at most B such that the fiber At is geometrically non-simple. One bound comes from arithmetic geometry, and shows that there are only finitely many such t; but one has very little control over how this finite number varies as f changes. Another bound, from analytic number theory, shows that the number of geometrically non-simple fibers grows quite slowly with B; this bound, by contrast with the arithmetic one, is effective, and is uniform in the coefficients of f. We hope that the paper, besides proving the particular theorems we address, will serve as a good example of the strengths and weaknesses of the two complementary approaches.