The proof of fermat's last theorem it is come from the idea of G.frey which
Suggest to build an elliptic curve with coefficient are the solustion of fermat equation
$E : y^2=x(x-a^p)(x+b^p)$
And he try to find a contradition which do by others mathematician
Frey curve have no associated modular form(ribet prove this in 1986)
But wiles prove that every semi-stable elliptic curve over $Q$ have an associated modular form "contradiction" ⇒ that the equation $a^p+b^p=c^p$ have no integer solustion for p>2 and abc=/=0
The modular curve mean that there exist an q-expention$\Sigma$$a_{n}$$q^{n}$ where ($q=e^{2πinτ}$) , such that $N_{p}=p+1-a_{p}$ where $N_{p}$ is the number of solution of $E \pmod p$
$question$ why when we prove that frey curve have no associated modular form that mean the frey curve have no solustion modulo p but not in $Z$ ,why that $\implies$ that fermat have no integer solustion ?
Is that mean for every elliptic curve which have no modular form that $\implies$ this curve have no integer solution?
Best Answer
Everything is explained in the texts on Fermat last theorem. It is not about reducing the Frey curve modulo $p$ but instead it is about reducing its Galois representation.
From a point on the Fermat curve you get a your Frey curve $E$, an elliptic curve over $\Bbb{Q}$, assume $f=\sum_n a_n q^n$ is a modular form (where $a_p = p+1-\#E(F_p), (1-a_p p^{-s} + p^{1-2s}) \sum_k a_{p^k} p^{-sk} = 1$ and $a_n = \prod_{p^k \| n} a_{p^k}$) of weight $2$ and level $N$, let $f\bmod P$ be the formal power series obtained by reducing the coefficients $\bmod P$, this corresponds to the (Artin L-function of) the field extension $\Bbb{Q}(E[P])$ obtained by adding the $x,y$ of $E[P]$ (the $P$-torsion points of $E$), what they call the Galois representation on $E[P]$ (which as a group is $\cong (\Bbb{F}_P)^2$ so the Galois group is a subgroup of $GL_2(\Bbb{F}_P)$), the discriminant of $E$ has unusual properties from which they do the level lowering : $f \bmod P = g \bmod P'$ for some modular form $g$ of weight $2$ and level $M < N$, and they can do it several times making $M$ smaller and smaller until $M=2$ so $g$ is a modular form of weight $2$ and level $2$ : no such modular form exists.