The Hobbit: An Unexpected Journey
The Birch and Swinnerton-Dyer Conjecture by A. Wiles
A polynomial relation f (x, y) = 0 in two variables defines a curve C0 . If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f (x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. The set of all such points is denoted C0 (Q). If we consider a non-singular projective model C of the curve then topologically C is classified by its genus, and we call this the genus of C0 also.
Note that C0 (Q) and C(Q) are either both finite or both infinite. Mordell conjectured, and in 1983 Faltings proved, the following deep result
Theorem [F1]. If the genus of C0 is greater than or equal to two, then C0 (Q) is finite.
As yet the proof is not effective so that one does not possess an algorithm for finding the rational points. (There is an effective bound on the number of solutions but that does not help much with finding them.) The case of genus zero curves is much easier and was treated in detail by Hilbert and Hurwitz [HH]. They explicitly reduce to the cases of linear and quadratic equations. The former case is easy and the latter is resolved by the criterion of Legendre. In particular for a non-singular projective model C we find that C(Q) is non-empty if and only if C has p-adic points for all primes p, and this in turn is determined by a finite number of congruences. If C(Q) is non-empty then C is parametrized by rational functions and there are infinitely many rational points. The most elusive case is that of genus 1. There may or may not be rational solutions and no method is known for determining which is the case for any given curve. Moreover when there are rational solutions there may or may not be infinitely many. If a non-singular projective model C has a rational point then C(Q) has a natural structure as an abelian
group with this point as the identity element. In this case we call C an elliptic curve over
Q. (For a history of
A polynomial relation f (x, y) = 0 in two variables defines a curve C0 . If the coefficients of the polynomial are rational numbers then one can ask for solutions of the equation f (x, y) = 0 with x, y ∈ Q, in other words for rational points on the curve. The set of all such points is denoted C0 (Q). If we consider a non-singular projective model C of the curve then topologically C is classified by its genus, and we call this the genus of C0 also.
Note that C0 (Q) and C(Q) are either both finite or both infinite. Mordell conjectured, and in 1983 Faltings proved, the following deep result
Theorem [F1]. If the genus of C0 is greater than or equal to two, then C0 (Q) is finite.
As yet the proof is not effective so that one does not possess an algorithm for finding the rational points. (There is an effective bound on the number of solutions but that does not help much with finding them.) The case of genus zero curves is much easier and was treated in detail by Hilbert and Hurwitz [HH]. They explicitly reduce to the cases of linear and quadratic equations. The former case is easy and the latter is resolved by the criterion of Legendre. In particular for a non-singular projective model C we find that C(Q) is non-empty if and only if C has p-adic points for all primes p, and this in turn is determined by a finite number of congruences. If C(Q) is non-empty then C is parametrized by rational functions and there are infinitely many rational points. The most elusive case is that of genus 1. There may or may not be rational solutions and no method is known for determining which is the case for any given curve. Moreover when there are rational solutions there may or may not be infinitely many. If a non-singular projective model C has a rational point then C(Q) has a natural structure as an abelian
group with this point as the identity element. In this case we call C an elliptic curve over
Q. (For a history of
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