The Frey curve
Recall that a Frey package \((a,b,c,p)\) is simply a prime \(p\geq5\) and nonzero pairwise-coprime integers \(a,b,c\) satisfying \(a^p+b^p=c^p\) and satisfying the congruences \(a\equiv3\pmod4\) and \(b\equiv0\pmod2\). We have shown above that if Fermat's Last Theorem is false, then a Frey package exists.
DefinitionFrey#
Given a Frey package \((a,b,c,p)\), the corresponding Frey curve (considered by Frey and, before him, Hellegouarch) is the elliptic curve over \(\mathbb{Q}\) defined by the equation \(Y^2=X(X-a^p)(X+b^p).\)
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Note that the roots of the cubic \(X(X-a^p)(X+b^p)\) are distinct because \(a,b,c\) are nonzero and \(a^p+b^p=c^p\).
Given a Frey package \((a,b,c,p)\) with corresponding Frey curve \(E\), the mod \(p\) Galois representation \(\rho_{E,p}\) associated to this package is the 2-dimensional representation of \(\operatorname{Gal}(\overline{\mathbb{Q}}/\mathbb{Q})\) on \(E(\overline{\mathbb{Q}})[p]\) described above. Frey's observation is that this mod \(p\) Galois representation has some very surprising properties. We will make this remark more explicit in the next chapter. Here we shall show how these properties can be used to finish the job.
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