Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize

Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize(scientificamerican.com)

58 points by digital55 62 days ago | 9 comments

nhatcher 62 days ago |

Oh wow! I wouldn't have expected this so many years later. Mordel's conjecture implies asva special case that for all n>=4 there are only a finite number of solutions to Fermat's equations with relative prime numbers. Brings me back!

delhanty 57 days ago | |

> Mordel's conjecture implies as a special case that for all n>=4 there are only a finite number of solutions to Fermat's equations with relative prime numbers

I just learnt that fact from Wikipedia's article on Mordel's conjecture (now Faltings' theorem), was curious whether the theorem could be strengthened to obtain a full proof of Fermat’s Last Theorem (FLT) that is genuinely different from the Taylor–Wiles proof (or its later variants) and so asked an AI (in this case Grok via Twitter).

Grok correctly told me "no it's not possible", but then surfaced (as an aside) a nice expository article on the Taylor–Wiles proof by Faltings from AMS notices in July 1995, which I thought I'd share here:

https://www.ams.org/notices/199507/faltings.pdf

011101101 57 days ago |

A point is that which has no breadth.

The line is a breadthless legth.

Mordell conjecture is that only circles or figure contain infinite points, whereas curves with exponents over 3 are finite accumulations.

ljsprague 57 days ago |

"He proved that if a curve’s equation has a variable raised to a power higher than 3, then it must have a finite number of [rational] points."

OgsyedIE 57 days ago | |

This must be an incorrect description of what has actually been proved, since x^4 is a counterexample.

raphlinus 57 days ago | | |

My understanding, which is to be taken with a grain of salt, is that there's an additional constraint, not stated in the Scientific American article, that the plane curve be irreducible. The example of x^4 is reducible, it's x^2 * x^2 among other thing. The actual conjecture is expressed in terms of genus, but this follows from the genus-degree formula.