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

Jordan S. Ellenberg, Christian Elsholtz, Chris Hall, Emmanuel Kowalski

• We consider, in the special case of certain one-parameter families of Jacobians of curves defined over a number field, the problem of how the property that the generic fiber of such a family is absolutely simple �spreads� to other fibers. We show that this question can be approached using arithmetic geometry or with more analytic methods based on sieve theory. In the first setting, non-trivial group-theoretic information is needed, while the version of the sieve we use is also of independent interest.