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# Difference between revisions of "Szpiro's conjecture"

From Citizendium, the Citizens' Compendium

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:<math> \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, </math> | :<math> \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \, </math> | ||

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==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 21:20, 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 *c*_{4}, *c*_{6} and conductor *f*, we have

## External links

- Szpiro and ABC, notes by William Stein