Szpiro's conjecture

In

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 (using notation from Tate's algorithm), we have

${\displaystyle \max\{\vert c_{4}\vert ^{3},\vert c_{6}\vert ^{2}\}\leq C(\varepsilon )\cdot f^{6+\varepsilon }.}$

abc conjecture

The abc conjecture originated as the outcome of attempts by Joseph Oesterlé and David Masser to understand Szpiro's conjecture,^[6] and was then shown to be equivalent to the modified Szpiro's conjecture.^[7]

Claimed proofs

In August 2012, Shinichi Mochizuki claimed a proof of Szpiro's conjecture by developing a new theory called inter-universal Teichmüller theory (IUTT).^[8] However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture,^[9][10][11] with Peter Scholze and Jakob Stix concluding in March 2018 that the gap was "so severe that … small modifications will not rescue the proof strategy".^[12][13][14]

See also

• Arakelov theory

References

Bibliography

• Zbl 0869.11051
• Szpiro, L. (1981). "Propriétés numériques du faisceau dualisant rélatif". Seminaire sur les pinceaux des courbes de genre au moins deux (PDF). Astérisque. Vol. 86. pp. 44–78.
  Zbl 0517.14006
  .
• Szpiro, L. (1987), "Présentation de la théorie d'Arakelov", Contemp. Math., Contemporary Mathematics, 67: 279–293,
  Zbl 0634.14012

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