NOTICE: Citizendium is still being set up on its newer server, treat as a beta for now; please see here for more.
Citizendium - a community developing a quality comprehensive compendium of knowledge, online and free. Click here to join and contribute—free
CZ thanks our previous donors. Donate here. Treasurer's Financial Report -- Thanks to our content contributors. --

Difference between revisions of "Szpiro's conjecture"

From Citizendium, the Citizens' Compendium
Jump to: navigation, search
(New article, my own wording from Wikipedia)
 
m (remove WPmarkup; subpages)
Line 1: Line 1:
 +
{{subpages}}
 
In [[number theory]], '''Szpiro's conjecture''' concerns a relationship between the [[conductor of an elliptic curve|conductor]] and the [[discriminant of an elliptic curve|discriminant]] of an [[elliptic curve]].  In a general form, it is equivalent to the well-known [[ABC conjecture]].  It is named for [[Lucien Szpiro]] who formulated it in the 1980s.
 
In [[number theory]], '''Szpiro's conjecture''' concerns a relationship between the [[conductor of an elliptic curve|conductor]] and the [[discriminant of an elliptic curve|discriminant]] of an [[elliptic curve]].  In a general form, it is equivalent to the well-known [[ABC conjecture]].  It is named for [[Lucien Szpiro]] who formulated it in the 1980s.
  
Line 17: Line 18:
 
==External links==
 
==External links==
 
* [http://modular.fas.harvard.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01/node2.html Szpiro and ABC], notes by William Stein
 
* [http://modular.fas.harvard.edu/mcs/archive/Fall2001/notes/12-10-01/12-10-01/node2.html Szpiro and ABC], notes by William Stein
 
[[Category:Conjectures]]
 
[[Category:Number theory]]
 
[[Category:Unsolved problems in mathematics]]
 
 
{{numtheory-stub}}
 

Revision as of 21:06, 27 October 2008

This article is developing and not approved.
Main Article
Talk
Related Articles  [?]
Bibliography  [?]
External Links  [?]
Citable Version  [?]
 
This editable Main Article is under development and not meant to be cited; by editing it you can help to improve it towards a future approved, citable version. These unapproved articles are subject to a disclaimer.

In number theory, Szpiro's conjecture concerns a relationship between the conductor and the discriminant of an elliptic curve. In a general form, it is equivalent to the well-known ABC conjecture. It is named for Lucien Szpiro who formulated it in the 1980s.

The conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with minimal discriminant Δ and conductor f, we have

The modified Szpiro conjecture states that: given ε > 0, there exists a constant C(ε) such that for any elliptic curve E defined over Q with invariants c4, c6 and conductor f, we have


References

  • S. Lang (1997). Survey of Diophantine geometry. Springer-Verlag, 51. ISBN 3-540-61223-8. 
  • L. Szpiro (1981). "Seminaire sur les pinceaux des courbes de genre au moins deux". Astérisque 86 (3): 44-78.
  • L. Szpiro (1987). "Présentation de la théorie d'Arakelov". Contemp. Math. 67: 279-293.


External links