I have read in many places that the Frey curve (if it existed) $y^2=x(x-A)(x+B)$ (or equivalently, $y^2=x(x-A)(x-C)$ corresponds to the solutions of $a^n+b^n=c^n$, where $A=a^n/c^n$ and $B=b^n/c^n$.

However, the connection is not very clear to me. Can someone either explain it or point me to a source that would spell this out step by step?

Thank you!

Edit: Just to clarify, I am asking about the correspondence between the Frey curve and the solutions to Fermat's Last Theorem. I take it there is no simple relation between the two?

  • 1
    $\begingroup$ If FLT has a solution, construct the corresponding Frey curve. Wiles proved that every elliptic curve over $\mathbb{Q}$ is modular. Ribet showed that this curve is then modular of level 2, but it's known that there is no nonzero cusp form of level 2, contradiction. Thus FLT cannot have solution. $\endgroup$ – user27126 Oct 31 '13 at 20:12
  • $\begingroup$ I believe $A$ should simply be $a^n$ and likewise $B = b^n$ (see wikipedia). But with this, the correspondence is given in your question. If you give me three integers $a,b,c$ with $a^n + b^n = c^n$, then the corresponding Frey curve is $y^2 = x(x-A)(x+B)$. Likewise, if you have a Frey curve, then by definition there are integers $a$, $b$, $c$, and $n$ with $a^n = A$, $b^n = B$, and $a^n + b^n = c^n$. $\endgroup$ – RghtHndSd Oct 31 '13 at 20:46

I think the proof is very advanced and done by Ribet. If I have understood correctly, the connection is given in http://math.berkeley.edu/~ribet/Articles/invent_100.pdf


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