View source for Szpiro's conjecture

{{MSC|11Gxx}}


A  conjectural 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.

The  conjecture states that: given $\epsilon > 0$, there exists a constant  $C(\epsilon)$ such that for any elliptic curve $E$ defined over  $\mathbb{Q}$ with minimal discriminant $\Delta$ and conductor $f$, we  have
$$
\vert\Delta\vert \leq C(\varepsilon ) \cdot f^{6+\varepsilon } \ .
$$
The  '''modified Szpiro conjecture''' states that: given $\epsilon > 0$,  there exists a constant $C(\epsilon)$ such that for any elliptic curve  $E$ defined over $\mathbb{Q}$ with invariants $c_4,c_6$ and conductor  $f$, we have
$$
\max\{\vert c_4\vert^3,\vert c_6\vert^2\} \leq C(\varepsilon )\cdot f^{6+\varepsilon } \ .
$$

==References==
* Lang, Serge; ''Survey of Diophantine geometry'' (1997), p. 51, Springer-Verlag, {{ZBL|0869.11051}} ISBN 3-540-61223-8 
* Szpiro, L.; ''Seminaire sur les pinceaux des courbes de genre au moins deux'',    Astérisque,  '''86'''  (1981), pp. 44-78, {{ZBL|0463.00009}}  
* Szpiro, L.; ''Présentation de la théorie d'Arakelov'', Contemp. Math.,  '''67'''  (1987), pp. 279-293, {{ZBL|0634.14012}}