Epsilon theorem
Encyclopedia
In mathematics, Ribet's theorem (earlier called the epsilon conjecture or ε-conjecture) is a statement in number theory
concerning properties of Galois representations associated with modular form
s. It was proposed by Jean-Pierre Serre
and proved by Ken Ribet. The proof of epsilon conjecture was a significant step towards the proof of Fermat's Last Theorem
. As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that Fermat's Last Theorem
is true.
with integer coefficients in a global minimal form. Denote by δp, respectively, np, the exponent with which a prime p appears in the prime factorization of the discriminant Δ of E, respectively, the conductor N of E. Suppose that E is a modular elliptic curve
, then we can perform a level descent modulo primes ℓ dividing one of the exponents δp of a prime dividing the discriminant. If pδp is an odd prime power factor of Δ and if p divides N only once (i.e. np=1), then there exists another elliptic curve E' , with conductor N' = N/p, such that the coefficients of the L-series of E are congruent modulo ℓ to the coefficients of the L-series of E' .
The epsilon conjecture is a relative statement: assuming that a given elliptic curve E over Q is modular, it predicts the precise level of E.
If ℓ is an odd prime and a, b, and c are positive integers such that
then a corresponding Frey curve is an algebraic curve given by the equation
or, equivalently
This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q. In 1982 Gerhard Frey
called attention to the unusual properties of the same curve as Hellegouarch, now called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular. The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. 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 of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem.
, by the epsilon conjecture one can perform level descent modulo ℓ. Repeating this procedure, we will eliminate all odd primes from the conductor and reach the modular curve X0(2) of level 2. However, this curve is not an elliptic curve since it has genus zero, resulting in a contradiction.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
concerning properties of Galois representations associated with modular form
Modular form
In mathematics, a modular form is a analytic function on the upper half-plane satisfying a certain kind of functional equation and growth condition. The theory of modular forms therefore belongs to complex analysis but the main importance of the theory has traditionally been in its connections...
s. It was proposed by Jean-Pierre Serre
Jean-Pierre Serre
Jean-Pierre Serre is a French mathematician. He has made contributions in the fields of algebraic geometry, number theory, and topology.-Early years:...
and proved by Ken Ribet. The proof of epsilon conjecture was a significant step towards the proof of Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
. As shown by Serre and Ribet, the Taniyama–Shimura conjecture (whose status was unresolved at the time) and the epsilon conjecture together imply that Fermat's Last Theorem
Fermat's Last Theorem
In number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
is true.
Statement
Let E be an elliptic curveElliptic curve
In mathematics, an elliptic curve is a smooth, projective algebraic curve of genus one, on which there is a specified point O. An elliptic curve is in fact an abelian variety — that is, it has a multiplication defined algebraically with respect to which it is a group — and O serves as the identity...
with integer coefficients in a global minimal form. Denote by δp, respectively, np, the exponent with which a prime p appears in the prime factorization of the discriminant Δ of E, respectively, the conductor N of E. Suppose that E is a modular elliptic curve
Modular elliptic curve
A modular elliptic curve is an elliptic curve E that admits a parametrisation X0 → E by a modular curve. This is not the same as a modular curve that happens to be an elliptic curve, and which could be called an elliptic modular curve...
, then we can perform a level descent modulo primes ℓ dividing one of the exponents δp of a prime dividing the discriminant. If pδp is an odd prime power factor of Δ and if p divides N only once (i.e. np=1), then there exists another elliptic curve E' , with conductor N' = N/p, such that the coefficients of the L-series of E are congruent modulo ℓ to the coefficients of the L-series of E' .
The epsilon conjecture is a relative statement: assuming that a given elliptic curve E over Q is modular, it predicts the precise level of E.
History
In his thesis, Yves Hellegouarch came up with the idea of associating solutions (a,b,c) of Fermat's equation with a completely different mathematical object: an elliptic curve..If ℓ is an odd prime and a, b, and c are positive integers such that
then a corresponding Frey curve is an algebraic curve given by the equation
or, equivalently
This is a nonsingular algebraic curve of genus one defined over Q, and its projective completion is an elliptic curve over Q. In 1982 Gerhard Frey
Gerhard Frey
Gerhard Frey is a German mathematician, known for his work in number theory. His Frey curve, a construction of an elliptic curve from a purported solution to the Fermat equation, was central to Wiles' proof of Fermat's Last Theorem....
called attention to the unusual properties of the same curve as Hellegouarch, now called a Frey curve. This provided a bridge between Fermat and Taniyama by showing that a counterexample to Fermat's Last Theorem would create such a curve that would not be modular. The conjecture attracted considerable interest when Frey (1986) suggested that the Taniyama–Shimura–Weil conjecture implies Fermat's Last Theorem. 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 of this. This showed that a proof of the semistable case of the Taniyama-Shimura conjecture would imply Fermat's Last Theorem. Serre did not provide a complete proof and what was missing became known as the epsilon conjecture or ε-conjecture. In the summer of 1986, Ribet (1990) proved the epsilon conjecture, thereby proving that the Taniyama–Shimura–Weil conjecture implied Fermat's Last Theorem.
Implication of Fermat's Last Theorem
Suppose that the Fermat equation with exponent ℓ ≥ 3 had a solution in non-zero integers a, b, c. Let us form the corresponding Frey curve E. It is an elliptic curve and one can show that its discriminant Δ is equal to 16 (abc)2ℓ and its conductor N is the radical of abc, i.e. the product of all distinct primes dividing abc. By the Taniyama–Shimura conjecture, E is a modular elliptic curve. Since N is square-freeSquare-free integer
In mathematics, a square-free, or quadratfrei, integer is one divisible by no perfect square, except 1. For example, 10 is square-free but 18 is not, as it is divisible by 9 = 32...
, by the epsilon conjecture one can perform level descent modulo ℓ. Repeating this procedure, we will eliminate all odd primes from the conductor and reach the modular curve X0(2) of level 2. However, this curve is not an elliptic curve since it has genus zero, resulting in a contradiction.
See also
- abc conjectureAbc conjectureThe abc conjecture is a conjecture in number theory, first proposed by Joseph Oesterlé and David Masser in 1985. The conjecture is stated in terms of three positive integers, a, b and c , which have no common factor and satisfy a + b = c...
- Modularity theorem
- Fermat's Last TheoremFermat's Last TheoremIn number theory, Fermat's Last Theorem states that no three positive integers a, b, and c can satisfy the equation an + bn = cn for any integer value of n greater than two....
- Wiles' proof of Fermat's Last TheoremWiles' proof of Fermat's Last TheoremWiles's proof of Fermat's Last Theorem is a proof of the modularity theorem for semistable elliptic curves released by Andrew Wiles, which, together with Ribet's theorem, provides a proof for Fermat's Last Theorem. Wiles first announced his proof in June 1993 in a version that was soon recognized...
External links
- Ken Ribet and Fermat's Last Theorem by Kevin BuzzardKevin BuzzardKevin Mark Buzzard is a British mathematician and currently a Professor of Pure Mathematics at Imperial College London. He specialises in algebraic number theory....
June 28, 2008