I szpiros conjecture is equivalent to the statement. In 1981, lucien szpiro made a famous conjecture that compares discriminant and conductors of elliptic curves. In number theory, szpiros conjecture relates the conductor and the discriminant of an elliptic curve. Lecture on the abc conjecture and some of its consequences, in. The abc conjecture was shown to be equivalent to the modified szpiro s conjecture. The abc conjecture consequences hodgearakelov theoryinteruniversal teichmuller theory. The abc conjecture implies asymptotic fermats last. Thus, in summary, it seems to the author that, if one ignores the delicate considerations that occur in the course of interpreting and combining the main results of the preparatory papers. Finally, in section 3 we state a primedepleted abc conjecture and show that it is a consquence of the primedepleted szpiro conjecture. In fact, an equivalent statement to abc is the socalled szpiro conjecture, which phrases the problem entirely in terms of an elliptic curve eover q and compares the minimal discriminant and the conductor f. Since szpiro s conjecture is easily seen to imply the abc conjecture.
Szpiro, 1981 let e be an elliptic curve over q which is a global minimal. This is essentially equivalent to the later abc conjecture that has been the topic of recent controversy, so we really should have been all this time arguing about szpiro, not abc. Pdf new 20070406 comments new 20190628 14 conformal and quasiconformal categorical representation of hyperbolic riemann surfaces. Whereas the abc conjecture describes an underlying mathematical phenomenon in terms of relationships between integers, szpiro s conjecture casts that. Proof claimed for deep connection between primes if it is true, a solution to the abc conjecture about whole numbers would be an astounding achievement. A new hope for a perplexing mathematical proof wired. The abc conjecture is also equivalent to szpiro s conjecture, which was proposed by the french mathematician lucien szpiro in.
Philosophy behind mochizukis work on the abc conjecture. In 1984, szpiro szp85a conjectured that xhas a small point, where a small point is an algebraic point of the curve x with height bounded from above in a certain way. Much like most of the other conjectures in this book, a proof of the abc. What szpiro is probably most famous for is the szpiro conjecture about elliptic curves which he first formulated in 1981. Lucien szpiro formulated a conjectural inequality called szpiro s conjecture relating some numerical invariants of elliptic curves, and sometime after the abc conjecture was introduced by david masser and joseph oesterle it was realized that for the generalized formulations over arbitrary number fields, the abc conjecture is equivalent to. In 1981, szpiro formulated a conjecture now known as szpiro s conjecture relating the discriminant of an elliptic curve with its conductor. Bogomolovs proof of the geometric version of the szpiro conjecture from the point of view of interuniversal teichmuller theory. We refer to section 2 for a precise formulation of szpiro s small points conjecture. Szpiros conjecture for function fields and mochizukis. He served as a distinguished professor at the cuny graduate center and formulated szpiro s conjecture, which is related to the abc conjecture. Dorian goldfeld, the abc conjecture is the most important. Lecture on the abc conjecture and some of its consequences. The abc conjecture was shown to be equivalent to the modified szpiros conjecture.
The first step is to translate arithmetic information on these objects further explanation needed to the setting of frobenioid. It is named for lucien szpiro who formulated it in the 1980s statement. It is thus tempting to try to prove the opposite implication, i. His conjecture inspired the abc conjecture, which was later shown to be equivalent to a modified form of szpiro s conjecture in 1988. Mochizuki has recently announced a proof of the abc conjecture. The abc conjecture originated as the outcome of attempts by oesterle and masser to understand the szpiro conjecture about elliptic curves, which involves more geometric structures in its statement than the abc conjecture. That the abc conjecture is intimately related to elliptic curves is well known. Shinichi mochizuki research institute for mathematical. In a general form, it is equivalent to the wellknown abc conjecture.
Davide castelvecchi at nature has the story this morning of a press conference held earlier today at kyoto university to announce the publication by publications of the research institute for mathematical sciences rims of mochizukis purported proof of the abc conjecture this is very odd. Can someone briefly explain the philosophy behind his work and comment on why it might be expected to shed light on questions like the abc conjecture. Elliptic curves and the abc conjecture university of vermont. It is one undisputed contribution of mochizukis to the subject of abc that will survive regardless of the ultimate outcome of the iut saga.
It is known that szpiro s conjecture, or equivalently the abc conjecture, implies langs conjecture giving a uniform lower bound for the canonical height of nontorsion points on elliptic curves. On the abc conjecture and some of its consequences by. Next we recite masons proof of an analogous assertion for polynomials at,bt,ct that implies, among other. International conference on recent advances in applied mathematics 2020 icraam2020, january 46, 2020, kuala lumpur, malaysia. He was born in paris in 1941 a jewish boy, he survived the nazi occupation as a hidden. Unlike 150year old riemann hypothesis or the twin prime conjecture whose age is measured in millennia, the abc conjecture was discovered. Hoshi about the suggested proof of the abc conjecture. Why abc is still a conjecture rims, kyoto university.
Szpiro, a french mathematician who often used to be a visitor at columbia, but is now permanently at the cuny graduate center, claimed in his. In the 1980s, he became well known for szpiro s conjecture, which gave rise to the famous abc conjecture. What the alphabet looks like when d through z are eliminated1,2 1. The wealth of consequences that would spring from a proof of the abc conjecture had convinced number theorists that proving the conjecture was likely to be very hard. Titans of mathematics clash over epic proof of abc conjecture. Whereas the abc conjecture describes an underlying mathematical phenomenon in terms of relationships between integers, szpiro s conjecture casts that same underlying relationship in terms of elliptic. The abc conjecture proclaims a link between the additive and the. In number theory, szpiro s conjecture concerns a relationship between the conductor and the discriminant of an elliptic curve.
This last example of the frobenius mutation and the associated core consti tuted by the. He formulated szpiro s conjecture and was a distinguished professor at the cuny graduate center and an emeritus director of research at the cnrs. Bennett and soroosh yazdani february 9, 2012 abstract szpiro s conjecture asserts the existence of an absolute constant k 6 such that if e is an elliptic curve over q, the minimal discriminant e of e is bounded above in modulus by the kth power of the conductor ne of e. Lucien szpiro obituary new york, ny new york times. In this note we show that a significantly weaker version of szpiro s conjecture, which we call primedepleted, suffices to prove langs conjecture. It is far too early to judge its correctness, but it builds on many years of work by him. The conjecture always seems to lie on the boundary of what is known and what is unknown, dorian goldfeld of columbia university has written. Our proofs require a number of tools from arakelov geometry, analytic number theory, galois representations, complexanalytic estimates on shimura curves, automorphic forms, known cases of the colmez conjecture, and results on generalized fermat equations. On the basis of consideration of szpiro s conjecture, dufour and kihel 12 in 2004 proved that halls conjecture weak form implies that n. Thesesimilaritiesare,insomesense,consequencesofthefact. The abc conjecture was made by masser and oesterle in 1985, inspired by work of szpiro. Some results of brocardramanujan problem on diophantine. It was known from the beginning that it would take experts months to understand his work enough to be able to verify the proof.
Szpiro s conjecture, which is closely related to the abc conjecture, states that we for every 0 there is a constant c such that mine c. It is equivalent to the general form of szpiro s conjecture. Notes on the oxford iut workshop by brian conrad mathbabe. Where can i find pdfs of shinichi mochizukis proof of the. Africacrypt 2020 presentation cairo, egypt, july 2022, 2020. In a slightly modified form, it is equivalent to the wellknown abc conjecture.
In fact, iut proves two other conjectures, by szpiro and vojta, which imply the abc. Lucien szpiro 23 december 1941 18 april 2020 was a french mathematician, known for his work in number theory, arithmetic algebraic geometry, and commutative algebra. Szpiro s conjecture abc consequence suppose e is an elliptic curve over q with conductor n and minimal discriminant. Have there been any updates on mochizukis proposed proof.
Over general number fields f, in the version where an f. Overview 1 the abc conjecture 2 elliptic curves 3 reduction of elliptic curves and important quantities associated to elliptic curves 4 szpiro s conjecture anton hilado uvm elliptic curves and the abc conjecture october 16, 2018 2 37. As the nature subheadline explains, some experts say author shinichi mochizuki failed to fix. In august 2012, a proof of the abc conjecture was proposed by shinichi mochizuki. Conjecture would have quite a few effects in other places in number theory.
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