Modeling the distribution of Selmer groups, Shafarevich-Tate groups, and ranks of elliptic curves
Event details
| Date | 20.09.2012 |
| Hour | 11:15 › 12:30 |
| Speaker | Bjorn Poonen |
| Location | |
| Category | Conferences - Seminars |
Using only linear algebra over Z_p, we define a discrete probability distribution on the set of isomorphism classes of short exact sequences of Z_p-modules,
and then conjecture that as E varies over elliptic curves over a fixed global field, the distribution of 0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --
> 0 is that one. This one conjecture would have the following consequences:
1) Asymptotically, 50% of elliptic curves have rank 0 and 50% have rank 1.
2) Sha[p^infty] is finite for 100% of elliptic curves.
3) The Poonen-Rains conjecture on the distribution of Sel_p E holds.
4) Delaunay's conjecture a la Cohen-Lenstra on the distribution of Sha holds.
(This is joint work with M. Bhargava, D. Kane, H. Lenstra, and E. Rains.)
and then conjecture that as E varies over elliptic curves over a fixed global field, the distribution of 0 --> E(k) tensor Q_p/Z_p --> Sel_{p^infty} E --> Sha[p^infty] --
> 0 is that one. This one conjecture would have the following consequences:
1) Asymptotically, 50% of elliptic curves have rank 0 and 50% have rank 1.
2) Sha[p^infty] is finite for 100% of elliptic curves.
3) The Poonen-Rains conjecture on the distribution of Sel_p E holds.
4) Delaunay's conjecture a la Cohen-Lenstra on the distribution of Sha holds.
(This is joint work with M. Bhargava, D. Kane, H. Lenstra, and E. Rains.)
Links
Practical information
- General public
- Free
Organizer
- CIB
Contact
- Isabelle Derivaz-Rabii