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?