Timeline for Adjoining torsion points from abelian varieties
Current License: CC BY-SA 3.0
5 events
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| Dec 4, 2014 at 7:09 | comment | added | Pablo | @NoamD.Elkies this is a very nice and explicit construction. Following Silverman, and Dimitrov I would like to know if you can think of some natural "big" family of abelian varieties (with unbounded ranks) such that adding all their torsion points to the rationals will not result in an algebraically closed field? That is, something like the simple varieties... | |
| Dec 4, 2014 at 3:19 | comment | added | Joe Silverman | The following paper might help: MR1361754 Masser, D. W.(CH-BASL); Wüstholz, G.(CH-ETHZ) Factorization estimates for abelian varieties. Inst. Hautes Études Sci. Publ. Math. No. 81 (1995), 5–24. They give effective estimates for the degree of the isogeny and fields of definition for the factorization of $A$ into simple factors. So one might be able to use that to prove that almost all of the Jacobians of $y^2=PQ$, counted by height, are simple. It wouldn't follow directly, but might be the right tool to apply. (Just a thought.) | |
| Dec 4, 2014 at 1:04 | comment | added | Noam D. Elkies | Thanks. Yes, multiplying $P(x)$ by some auxiliary polynomial with degree of the appropriate parity must work, though proving it might be beyond the scope of a reasonable MO answer, unless some helpful result is already known (like a strong enough upper bound on the proportion of Jacobians of hyperlliptic genus-$g$ curves of height $\leq H$ that are not geometrically simple). | |
| Dec 3, 2014 at 23:23 | comment | added | Joe Silverman | Excellent. Now, to answer the further question, can you modify this so that the associated Jacobians are geometrically simple? Most of the ones you construct will be, but not all, I think. Maybe use $y^2=P(x)Q(x)$ for a generic $Q(x)$ so that $\deg(PQ)$ is odd? | |
| Dec 3, 2014 at 20:56 | history | answered | Noam D. Elkies | CC BY-SA 3.0 |