Szpiro's conjecture

From Infogalactic: the planetary knowledge core
Jump to: navigation, search

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

 \vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon }. \,

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 (see Tate's algorithm#Notation), we have

 \max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon }. \,

References

Lua error in package.lua at line 80: module 'strict' not found.

  • Lua error in package.lua at line 80: module 'strict' not found.
  • Lua error in package.lua at line 80: module 'strict' not found.
  • Lua error in package.lua at line 80: module 'strict' not found.


<templatestyles src="Asbox/styles.css"></templatestyles>