Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0 - MathOverflow

Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0

2009-10-27T05:40:48Z

The projective curve $3x^3+4y^3+5z^3=0$ is often cited as an example (given by Selmer) of a failure of the Hasse Principle: the equation has solutions in any completion of the rationals $\mathbb Q$, but not in $\mathbb Q$ itself.

I don't think I've ever seen a proof of the latter claim — is someone able to provide an outline? What are the necessary tools?

Answer by Thomas Riepe for Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0

2009-10-27T06:37:39Z

I think I saw a proof of that in Cassel's "Diophantine Equations with special reference to elliptic curves" and in some surveys by Mazur in the Bull. AMS (perhaps this, but I have in the moment no time to look).

Answer by David Lehavi for Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0

2009-10-27T07:01:23Z

The "standard" technique for killing the Hasse priniciple for elliptic curves is to show that the Tate-Shafarevich group has a copy of (Z/mZ)^2 for some m - see chapter X in Silverman's the arithmetic of Eliptic curves, both for the theory and examples. All the examples which Silverman presents ar with m = 2. Selmers example requires m = 3, which requires (much) more computations. Poonen has an example on his web page of a family of elliptic curves violating the Hasse principle, and containing Selmers example, but you'd have to dive through a labirinth of references.

Answer by Ho Chung Siu for Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0

2009-10-27T07:04:59Z

My friend has written an introduction to algebraic number theory before, which contains a short proof of this statement, but I didn't check its validity.

[Edit: Update the link of the document] http://www.2shared.com/document/2d6M7kNU/Introduction_to_Algebraic_Numb.html p.41 of the document, or p.45 of the pdf.

Answer by Kevin Buzzard for Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0

2009-11-20T18:59:13Z

There's a proof in Cassels' little blue book on elliptic curves which the OP might find more to his taste than some others mentioned here.

Answer by KConrad for Proof of no rational point on Selmer's Curve 3x^3+4y^3+5z^3=0

2010-01-15T14:39:06Z

This problem is in Cassels' book "Local Fields" and I wrote up a solution once along those lines, for an algebraic number theory class. See

http://www.math.uconn.edu/~kconrad/blurbs/gradnumthy/selmerexample.pdf,

but I should advise that it comes out seeming pretty tedious. Solutions that involve elliptic curves are more conceptual. Others have already provided pointers to references for that approach.