Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
scientificamerican.com
Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
1–10 of 13 posts
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#2[dead]
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#3Oh 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!
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#4A 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.
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#5"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."
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#6"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."
This must be an incorrect description of what has actually been proved, since x^4 is a counterexample.
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#7"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."
This must be an incorrect description of what has actually been proved, since x^4 is a counterexample.
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.
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#8Earlier quoted context omitted.
This must be an incorrect description of what has actually been proved, since x^4 is a counterexample.
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.
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Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#9Earlier quoted context omitted.
This must be an incorrect description of what has actually been proved, since x^4 is a counterexample.
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.
The curve they mean y = x^4 is irreducible but the genus is 0 since it’s isomorphic to the affine line.
Re: Gerd Faltings, who proved the Mordell conjecture, wins the Abel Prize
#10Earlier quoted context omitted.
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.
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 projective plane, because it has a singularity at infinity.