Torsion conjecture: Difference between revisions
Content deleted Content added
stub |
indicate genus 1 case is done |
||
| Line 1: | Line 1: | ||
In [[mathematics]], the '''torsion conjecture''' or '''uniform boundedness conjecture''' for [[abelian variety|abelian varieties]] states that the [[order (group theory)|order]] of the [[torsion group]] of an abelian variety over a [[number field]] can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. |
In [[mathematics]], the '''torsion conjecture''' or '''uniform boundedness conjecture''' for [[abelian variety|abelian varieties]] states that the [[order (group theory)|order]] of the [[torsion group]] of an abelian variety over a [[number field]] can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field. |
||
The conjecture has been completely resolved in the case of [[elliptic curves]]. {{harvtxt|Mazur|1977}} proved uniform boundedness [[Mazur's torsion theorem|for elltiptic curves over the rationals]]. His work was later generalized by {{harvtxt|Merel|1996}} for general number fields. |
|||
| ⚫ | |||
| ⚫ | |||
| ⚫ | *{{ |
||
*{{cite journal|last=Mazur|first=Barry|authorlink=Barry Mazur|title=Modular curves and the Eisenstein ideal|volume=47|issue=1|pages=33-186|year=1977|doi=10.1007/BF02684339|mr=0488287|journal=[[Publications Mathématiques de l'IHÉS]]|ref=harv}} |
|||
| ⚫ | |||
[[Category:Abelian varieties]] |
[[Category:Abelian varieties]] |
||
Revision as of 13:50, 6 May 2015
In mathematics, the torsion conjecture or uniform boundedness conjecture for abelian varieties states that the order of the torsion group of an abelian variety over a number field can be bounded in terms of the dimension of the variety and the number field. A stronger version of the conjecture is that the torsion is bounded in terms of the dimension of the variety and the degree of the number field.
The conjecture has been completely resolved in the case of elliptic curves. Mazur (1977) proved uniform boundedness for elltiptic curves over the rationals. His work was later generalized by Merel (1996) for general number fields.
References
- Mazur, Barry (1977). "Modular curves and the Eisenstein ideal". Publications Mathématiques de l'IHÉS. 47 (1): 33–186. doi:10.1007/BF02684339. MR 0488287.
{{cite journal}}: Invalid|ref=harv(help) - Merel, Loïc (1996). "Bornes pour la torsion des courbes elliptiques sur les corps de nombres". Inventiones Mathematicae. 124 (1): 437–449. doi:10.1007/s002220050059. MR 1369424.
{{cite journal}}: Invalid|ref=harv(help)
 | This number theory-related article is a stub. You can help Wikipedia by expanding it. |