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

Post by thornhill » Tue, Mar 24, 2026, 7:45 PM UTC

jlev1 wrote: post #10

thornhill wrote: post #9

Earlier quoted context omitted.

The curve they mean y = x^4 is irreducible but the genus is 0 since it’s isomorphic to the affine line.

The correct description is “a smooth curve of genus at least 2”. https://en.wikipedia.org/wiki/Faltings%27_theorem The reason for the confusion is that a smooth, projective plane curve of degree d has genus (d-1)(d-2)/2, which is 2 or greater starting at d=4. Hence the phrasing in the article, which is missing the “smooth, projective” hypothesis. The equation y = x^4 doesn’t define a smooth curve when extended to the…

Post by delhanty » Tue, Mar 24, 2026, 11:51 PM UTC

nhatcher wrote: post #3

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!