Ribet's theorem

Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is part of number theory. It concerns properties of Galois representations associated with modular forms. It was proposed by Jean-Pierre Serre and proven by Ken Ribet. The proof was a significant step towards the proof of Fermat's Last Theorem (FLT). As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that FLT is true.

In mathematical terms, Ribet's theorem shows that if the Galois representation associated with an elliptic curve has certain properties, then that curve cannot be modular (in the sense that there cannot exist a modular form that gives rise to the same representation).

Statement
Let $f$ be a weight 2 newform on $Γ_{0}(qN)$ – i.e. of level $qN$ where $q$ does not divide $N$ – with absolutely irreducible 2-dimensional mod $p$ Galois representation $ρ_{f,p}$ unramified at $q$ if $q ≠ p$ and finite flat at $q = p$. Then there exists a weight 2 newform $g$ of level $N$ such that
 * $$ \rho_{f,p} \simeq \rho_{g,p}. $$

In particular, if $E$ is an elliptic curve over $$\mathbb{Q}$$ with conductor $qN$, then the modularity theorem guarantees that there exists a weight 2 newform $f$ of level $qN$ such that the 2-dimensional mod $p$ Galois representation $ρ_{f, p}$ of $f$ is isomorphic to the 2-dimensional mod $p$ Galois representation $ρ_{E, p}$ of $E$. To apply Ribet's Theorem to $ρ_{E, p}$, it suffices to check the irreducibility and ramification of $ρ_{E, p}$. Using the theory of the Tate curve, one can prove that $ρ_{E, p}$ is unramified at $q ≠ p$ and finite flat at $q = p$ if $p$ divides the power to which $q$ appears in the minimal discriminant $Δ_{E}$. Then Ribet's theorem implies that there exists a weight 2 newform $g$ of level $N$ such that $ρ_{g, p} ≈ ρ_{E, p}$.

Level lowering
Ribet's theorem states that beginning with an elliptic curve $E$ of conductor $qN$ does not guarantee the existence of an elliptic curve $E$ of level $N$ such that $ρ_{E, p} ≈ ρ_{E, p}$. The newform $g$ of level $N$ may not have rational Fourier coefficients, and hence may be associated to a higher-dimensional abelian variety, not an elliptic curve. For example, elliptic curve 4171a1 in the Cremona database given by the equation


 * $$E: y^2 + xy + y = x^3 - 663204x + 206441595$$

with conductor $43 × 97$ and discriminant $43^{7} × 97^{3}$ does not level-lower mod 7 to an elliptic curve of conductor 97. Rather, the mod $p$ Galois representation is isomorphic to the mod $p$ Galois representation of an irrational newform $g$ of level 97.

However, for $p$ large enough compared to the level $N$ of the level-lowered newform, a rational newform (e.g. an elliptic curve) must level-lower to another rational newform (e.g. elliptic curve). In particular for $p ≫ N^{N^{1+ε}}|undefined$, the mod $p$ Galois representation of a rational newform cannot be isomorphic to an irrational newform of level $N$.

Similarly, the Frey-Mazur conjecture predicts that for large enough $p$ (independent of the conductor $N$), elliptic curves with isomorphic mod $p$ Galois representations are in fact isogenous, and hence have the same conductor. Thus non-trivial level-lowering between rational newforms is not predicted to occur for large $p (p > 17)$.

History
In his thesis, originated the idea of associating solutions (a,b,c) of Fermat's equation with a different mathematical object: an elliptic curve. If p is an odd prime and a, b, and c are positive integers such that


 * $$a^p + b^p = c^p,$$

then a corresponding Frey curve is an algebraic curve given by the equation


 * $$y^2 = x(x - a^p)(x + b^p).$$

This is a nonsingular algebraic curve of genus one defined over $$\mathbb{Q}$$, and its projective completion is an elliptic curve over $$\mathbb{Q}$$.

In 1982 Gerhard Frey called attention to the unusual properties of the same curve, now called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to FLT would create a curve that would not be modular. The conjecture attracted considerable interest when Frey suggested that the Taniyama–Shimura conjecture implies FLT. However, his argument was not complete. In 1985 Jean-Pierre Serre proposed that a Frey curve could not be modular and provided a partial proof. This showed that a proof of the semistable case of the Taniyama–Shimura conjecture would imply FLT. Serre did not provide a complete proof and the missing bit became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, Kenneth Alan Ribet proved the epsilon conjecture, thereby proving that the Modularity theorem implied FLT.

The origin of the name is from the ε part of "Taniyama-Shimura conjecture + ε ⇒ Fermat last theorem".

Implications
Suppose that the Fermat equation with exponent $p ≥ 5$ had a solution in non-zero integers $a, b, c$. The corresponding Frey curve $E_{a^{p},b^{p},c^{p}}|undefined$ is an elliptic curve whose minimal discriminant $Δ$ is equal to $2^{−8} (abc)^{2p}$ and whose conductor $N$ is the radical of $abc$, i.e. the product of all distinct primes dividing $abc$. An elementary consideration of the equation $a^{p} + b^{p} = c^{p}$, makes it clear that one of $a, b, c$ is even and hence so is N. By the Taniyama–Shimura conjecture, $E$ is a modular elliptic curve. Since all odd primes dividing $a, b, c$ in $N$ appear to a $pth$ power in the minimal discriminant $Δ$, by Ribet's theorem repetitive level descent modulo $p$ strips all odd primes from the conductor. However, no newforms of level 2 remain because the genus of the modular curve $X_{0}(2)$ is zero (and newforms of level N are differentials on $X_{0}(N))$.