Non-simple abelian varieties in a family: geometric and analytic approaches
| dc.creator | Ellenberg, J. | |
| dc.creator | Elsholtz, C. | |
| dc.creator | Hall, C. | |
| dc.creator | Kowalski, E. | |
| dc.date | 2008-04-14 | |
| dc.date.accessioned | 2026-07-07T09:32:14Z | |
| dc.date.available | 2026-07-07T09:32:14Z | |
| dc.description | 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. | |
| dc.description | 19 pages | |
| dc.identifier | https://arxiv.org/abs/0804.2166 | |
| dc.identifier | http://arxiv.org/abs/0804.2166 | |
| dc.identifier.uri | http://salesiana.dossiersoluciones.com/handle/123456789/158743 | |
| dc.subject | Number Theory | |
| dc.subject | Algebraic Geometry | |
| dc.subject | 11G10 (Primary); 11N35, 14K15, 14D05 (Secondary) | |
| dc.title | Non-simple abelian varieties in a family: geometric and analytic approaches | |
| dc.type | text |