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Journal of the London Mathematical Society Advance Access originally published online on May 22, 2009
Journal of the London Mathematical Society 2009 80(1):135-154; doi:10.1112/jlms/jdp021
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© 2009 London Mathematical Society

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

Jordan S. Ellenberg

Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53705
USA
ellenber@math.wisc.edu

Christian Elsholtz

Department of Mathematics
Royal Holloway University of London
Egham
Surrey
TW20 0EX
United Kingdom
christian.elsholtz@rhul.ac.uk

Chris Hall

Department of Mathematics
University of Michigan
Ann Arbor MI 48109
USA
hallcj@umich.edu

Emmanuel Kowalski

ETH Zürich – D-MATH
Rämistrasse 101
8092 Zürich
Switzerland

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.

2000 Mathematics Subject Classification 11G10 (primary); 11N35, 14K15, 14D05 (secondary).

The first author was partially supported by NSF-Career grant DMS-0448750 and a Sloan Research Fellowship.

Received September 18, 2008; revised January 28, 2009; published online May 22, 2009.


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