Non-simple abelian varieties in a family: geometric and analytic approaches

Details Non-simple abelian varieties in a family: geometric and analytic approaches Non-simple abelian varieties in a family: geometric and analytic approaches

2008-04-14

arxiv.org/abs/0804.2166v1

Mathematics

Number Theory

19 pages

Scientific paper

Let $A_t$ be a family of abelian varieties over a number field $k$ parametrized by a rational coordinate $t$, and suppose the generic fiber of $A_t$ is geometrically simple. For example, we may take $A_t$ to be the Jacobian of the hyperelliptic curve $y^2 = f(x)(x-t)$ for some polynomial $f$. We give two upper bounds for the number of $t \in k$ of height at most $B$ such that the fiber $A_t$ 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.

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