Szpiro's conjecture, formulated by Lucien Szpiro in the 1980s, relates the conductor of an elliptic curve to its discriminant, and in a modified form is equivalent to the abc conjecture. It has been described as one of the most important unsolved problems in Diophantine analysis because of its many consequences elsewhere in number theory.
Facts
StatementThe conductor of an elliptic curve is related to its discriminant: the discriminant is bounded by a constant times the conductor to the power 6 plus epsilon. 1 Proposed Year Difficulty Hint Partially Attested
Progress Toward ResolutionOpen; a claimed proof by Shinichi Mochizuki via inter-universal Teichmuller theory (2012) is not accepted. 1 Mochizuki announced a proof in 2012, but the mathematical community has not accepted it, so the conjecture is treated as open. Classification
Resolution Status Prize Status
Prize Status (category) Connections
In Branch
Source Szpiro's Conjecture (Wikipedia)
Posed By
Source Szpiro's Conjecture (Wikipedia)
Sources
1. Szpiro's Conjecture (Wikipedia)
Lead section
the most important unsolved problem in Diophantine analysis
Infobox, conjecture date
1981
Lead section, first sentence
relates the conductor of an elliptic curve to its discriminant
Posed By: Lucien Szpiro, Lead sentence
It is named for Lucien Szpiro, who formulated it in the 1980s.
In Branch: Number Theory, Lead sentence
In number theory, Szpiro's conjecture relates the conductor of an elliptic curve to its discriminant.
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