Inside the proof of fermat's last theorem

Question

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?

Answer 1

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.