Timeline for Bad primes for algebraic curves
Current License: CC BY-SA 4.0
6 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Jun 22, 2021 at 5:10 | comment | added | User0829 | @Asvin Thank you very much! | |
| Jun 22, 2021 at 5:09 | comment | added | Asvin | I think Poonen's "Rational points on varieties" has a section on spreading out. The basic idea is simply that if you have a finite type morphism defined over $\mathbb Q$, there can only be finitely many primes in all the denominators and as long as you avoid those, you can extend integrally. And then you need to show that various things can be checked generically (like being an isomorphism). | |
| Jun 22, 2021 at 5:00 | comment | added | User0829 | @Asvin Thank you for your comment. Could you elaborate more on your second comment? I mean the "spreading out generic isomorphism" part. Any reference would be helpful for me. | |
| Jun 22, 2021 at 4:54 | comment | added | Asvin | For your second question, you can indeed say that they are isomorphic up to extending S points because they are isomorphic over the generic point and you can "spread out" generic isomorphisms. | |
| Jun 22, 2021 at 4:53 | comment | added | Asvin | In your elliptic curve example, you are really talking about your curves as if they live over $\mathbb Z$, not $\mathbb Q$. I think for curves, it follows from Neron-Ogg-Shafarevich that there is a "best" possible S to take that is minimal and it's exactly those primes for which the galois rep on tate module of jacobian of curve is unramified. | |
| Jun 22, 2021 at 4:47 | history | asked | User0829 | CC BY-SA 4.0 |