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

TL;DR

This paper compares arithmetic and analytic methods to bound the number of parameters in a family of abelian varieties over a number field where fibers are geometrically non-simple, highlighting their respective strengths and limitations.

Contribution

It provides two distinct bounds—one finite and one effective and uniform—on the number of non-simple fibers in a family of abelian varieties, illustrating the advantages of both approaches.

Findings

Arithmetic bound shows only finitely many non-simple fibers exist.

Analytic bound demonstrates slow growth of non-simple fibers with parameter height.

The paper exemplifies the complementary strengths of arithmetic and analytic methods.

Abstract

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.