Timeline for Estimating the size of reduction of rational points on $\mathbb{G}_m^2$
Current License: CC BY-SA 3.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 1, 2011 at 14:58 | vote | accept | Tzanko Matev | ||
| May 31, 2011 at 12:48 | comment | added | Joe Silverman | @ACL: The rank refers to the free rank of $\Gamma$ as a finitely generated abelian group, not to its rank after being reduced modulo p. | |
| May 31, 2011 at 12:47 | comment | added | Joe Silverman | Actually, our result is a little different in that we are looking at subgroups $\Gamma$ of $\mathbb{G}(\mathbb{Q})$. The rank means the free rank, which is the dimension of $\Gamma\otimes\mathbb{Q}$ as a $\mathbb{Q}$-vector space. We also deal with abelian varieties of arbitrary dimension, but looking again at our paper, I see that we didn't do the case of finitely generated subgroups of $\mathbb{G}^d(\mathbb{Q})$ for $d\ge2$. However, the argument that we use will easily generalize, and I suspect that the limsup formula remains the same. | |
| May 31, 2011 at 8:08 | comment | added | S. Carnahan♦ | @ACL: I believe "rank" here means "minimal number of generators". | |
| May 31, 2011 at 7:02 | comment | added | ACL | Joe: For any prime number $p$ (such that the two generators of $\Gamma$ are $p$-adic units), the group $\Gamma\mod p$ is finite, so it always has rank smaller than $2$! | |
| May 30, 2011 at 21:14 | history | answered | Joe Silverman | CC BY-SA 3.0 |