Szpiro's conjecture: Difference between revisions
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* {{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=51 }} | * {{cite book | author=S. Lang | authorlink=Serge Lang | title=Survey of Diophantine geometry | publisher=[[Springer-Verlag]] | year=1997 | isbn=3-540-61223-8 | pages=51 }} | ||
* {{cite journal | author=L. Szpiro | title=Seminaire sur les pinceaux des courbes de genre au moins deux | journal=Astérisque | volume=86 | issue=3 | year=1981 | pages=44-78 }} | * {{cite journal | author=L. Szpiro | title=Seminaire sur les pinceaux des courbes de genre au moins deux | journal=Astérisque | volume=86 | issue=3 | year=1981 | pages=44-78 }} | ||
* {{cite journal | author=L. Szpiro | title=Présentation de la théorie d'Arakelov | journal=Contemp. Math. | volume=67 | year=1987 | pages=279-293 }} | * {{cite journal | author=L. Szpiro | title=Présentation de la théorie d'Arakelov | journal=Contemp. Math. | volume=67 | year=1987 | pages=279-293 | zbl=0634.14012 }} | ||
==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 | ||
Revision as of 15:19, 11 January 2013
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
- Szpiro and ABC, notes by William Stein