It has to be defined over Q, of course.
> Or did Velu show that if there exists even
> one elliptic curve with a rational point of order N then
> FLT is false for exponent N (with N a prime power 9 or up)?
Yes.
> Also, does Velu make it clear when he did his work? (One
> would imagine he proved this implication before he became
> aware of Mazur's work, but not necessarily before Mazur
> proved his theorem.)
The talk was given in February 1976; he does not cite
Mazur, but surely was aware of his results.
[..Hurwitz..]
>> More generally, he reduces x^my^n + y^mz^n + z^mx^n = 0
>> to FLT for N = m^2 - mn + n^2.
> What are the hypotheses on m and n here?
positive integers, not both even.
franz