number theory Number theory is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of mathematical objects constructed from integers (for example ...

, Szpiro's conjecture relates to the

conductor Conductor or conduction may refer to: Biology and medicine * Bone conduction, the conduction of sound to the inner ear * Conduction aphasia, a language disorder Mathematics * Conductor (ring theory) * Conductor of an abelian variety * Cond ...

and the discriminant of an

elliptic curve In mathematics, an elliptic curve is a smooth, projective, algebraic curve of genus one, on which there is a specified point . An elliptic curve is defined over a field and describes points in , the Cartesian product of with itself. If the ...

. In a slightly modified form, it is equivalent to the well-known ''abc'' conjecture. It is named for Lucien Szpiro, 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 ''Diophantine'' means pertaining to the ancient Greek mathematician Diophantus. A number of concepts bear this name: *Diophantine approximation *Diophantine equation *Diophantine quintuple In number theory, a diophantine -tuple is a set of p ...

" by

Dorian Goldfeld Dorian Morris Goldfeld (born January 21, 1947) is an American mathematician working in analytic number theory and automorphic forms at Columbia University. Professional career Goldfeld received his B.S. degree in 1967 from Columbia University. H ...

, in part to its large number of consequences in number theory including Roth's theorem, the Mordell conjecture, the

Fermat–Catalan conjecture In number theory, the Fermat–Catalan conjecture is a generalization of Fermat's Last Theorem and of Catalan's conjecture. The conjecture states that the equation has only finitely many solutions (''a'', ''b'', ''c'', ''m'', ''n'', ''k'') with ...

, and

Brocard's problem Brocard's problem is a problem in mathematics that seeks integer values of n such that n!+1 is a perfect square, where n! is the factorial. Only three values of n are known — 4, 5, 7 — and it is not known whether there are any more. ...

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'', :

\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), :

\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, formulated the ''abc'' conjecture which has been called "the most important unsolved problem in diophantine analysis". He is a member of Bourbaki Bourbaki(s) m ...

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 Diophanti ...

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

Consequences

Szpiro's conjecture and its modified form are known to imply several important mathematical results and conjectures, including Roth's theorem,

Faltings's theorem Faltings's theorem is a result in arithmetic geometry, according to which a curve of genus greater than 1 over the field \mathbb of rational numbers has only finitely many rational points. This was conjectured in 1922 by Louis Mordell, and know ...

, and a negative solution to the

Erdős–Ulam problem In mathematics, the Erdős–Ulam problem asks whether the plane contains a dense set of points whose Euclidean distances are all rational numbers. It is named after Paul Erdős and Stanislaw Ulam. Large point sets with rational distances The Erd ...

Claimed proofs

In August 2012,

Shinichi Mochizuki is a Japanese mathematician working in number theory and arithmetic geometry. He is one of the main contributors to anabelian geometry. His contributions include his solution of the Grothendieck conjecture in anabelian geometry about hyperbolic ...

claimed a proof of Szpiro's conjecture by developing a new theory called

inter-universal Teichmüller theory Inter-universal Teichmüller theory (IUT or IUTT) is the name given by mathematician Shinichi Mochizuki to a theory he developed in the 2000s, following his earlier work in arithmetic geometry. According to Mochizuki, it is "an arithmetic version ...

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

Peter Scholze Peter Scholze (; born 11 December 1987) is a German mathematician known for his work in arithmetic geometry. He has been a professor at the University of Bonn since 2012 and co-director at the Max Planck Institute for Mathematics since 2018. He ...

and Jakob Stix 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

* * * Conjectures Unsolved problems in number theory Abc conjecture {{numtheory-stub