In number theory, Szpiro's conjecture relates to the

conductor Conductor or conduction may refer to: Music * Conductor (music), a person who leads a musical ensemble, such as an orchestra. * ''Conductor'' (album), an album by indie rock band The Comas * Conduction, a type of structured free improvisation ...

and the discriminant of an elliptic curve. In a slightly modified form, it is equivalent to the well-known ''abc'' conjecture. It is named for

Lucien Szpiro Lucien Serge Szpiro (23 December 1941 – 18 April 2020) was a French mathematician known for his work in number theory, arithmetic geometry, and commutative algebra. He formulated Szpiro's conjecture and was a Distinguished Professor at t ...

, who formulated it in the 1980s. Szpiro's conjecture and its equivalent forms have been described as "the most important unsolved problem in Diophantine analysis" by Dorian Goldfeld, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the Fermat–Catalan conjecture, and

Brocard's problem Brocard's problem is a problem in mathematics that asks to find integer values of n and m for which n!+1 = m^2, where n! is the factorial. It was posed by Henri Brocard in a pair of articles in 1876 and 1885, and independently in 1913 by Srinivasa ...

Original statement

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^.

Modified Szpiro conjecture

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 ''c''₄, ''c''₆ and conductor ''f'' (using notation from Tate's algorithm), we have :

\max\ \leq C(\varepsilon )\cdot f^.

''abc'' conjecture

The ''abc'' conjecture originated as the outcome of attempts by

Joseph Oesterlé Joseph Oesterlé (born 1954) is a French mathematician who, along with David Masser David William Masser (born 8 November 1948) is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known ...

and

David Masser David William Masser (born 8 November 1948) is Professor Emeritus in the Department of Mathematics and Computer Science at the University of Basel. He is known for his work in transcendental number theory, Diophantine approximation, and Dioph ...

to understand Szpiro's conjecture, and was then shown to be equivalent to the modified Szpiro's conjecture.

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). However, the papers have not been accepted by the mathematical community as providing a proof of the conjecture, with Peter Scholze and

Jakob Stix Jakob M. Stix (born in 1974) is a German mathematician. He specializes in arithmetic algebraic geometry (étale fundamental group, anabelian geometry and other topics). Stix studied mathematics in Freiburg and Bonn and received his doctorate in 2 ...

concluding in March 2018 that the gap was "so severe that … small modifications will not rescue the proof strategy". Web-page by Mochizuki describing discussions and linking consequent publications and supplementary material

References

Bibliography