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Note on a conjecture of Szpiro. (English) Zbl 0742.14027

Les pinceaux de courbes elliptiques, Sémin., Paris/Fr. 1988, Astérisque 183, 19-23 (1990).
[For the entire collection see Zbl 0702.00011.]
The author shows that if \(E\) is any elliptic curve of conductor \(N\) and minimal discriminant \(D\) then there do not exist absolute constants \(C\) and \(k\) such that \(| D|\leq C\cdot N^ 6(\log(N))^ k\). In particular, he shows that one cannot hope to significantly improve the exponent in L. Szpiro’s discriminant conjecture [ibid. 7-18 (1990; see the preceding review)].

MSC:

14H52 Elliptic curves
11G05 Elliptic curves over global fields
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