Empirical evidence for the birch and swinnertondyer conjecture. The origins of this conjecture can be traced back to numerical computations done by birch and swinnertondyer 5. In the year 2000, landon clay, an american patron of mathematics. The birch and swinnertondyer conjecture, a computational. Pdf the birch and swinnertondyer conjecture is a well known mathematics problem in the area of elliptic curve. Like fermats last theorem, the riemann hypothesis looms on the horizon as the behemoth that interests almost every mathematician. The equivariant tamagawa number conjecture and modular symbols, pdf file, math.
Although kolyvagins conjecture is known in the case of elliptic curves of analytic rank one over k, the number m. The famous birchswinnertondyer conjecture, one of the clay millennium problems, predicts a relation between the rational points on an elliptic curve and a certain complex. Birchswinnertondyer conjecture from wolfram mathworld. Razar, some functions related to the derivatives of the lseries of an elliptic curve at s1. The birch and swinnertondyer conjecture for qcurves and. Indeed, the ppart of the birch and swinnertondyer conjectural formula for e k is equivalent to m. On atkinswinnertondyer congruence relations authors. An important problem in number theory is to understand the rational solutions to algebraic equations.
Rrgsqidenti es with the quotient of the multiplicative group r by t 1uand we check that in this case our conjecture recovers the classical birch and swinnertondyer conjecture for aover k. Birch and swinnertondyer conjecture hodge conjecture navierstokes equation p vs np problem poincar e conjecture. Q, in other words for rational points on the curve. On the conjecture of birch and swinnertondyer for an. Is andrew wiles secretly trying to solve bsd conjecture. The central object of study of this article is the division field qen that results by adjoining to q the coordinates of all ntorsion points on.
Congettura di birch e swinnertondyer curve ellittiche. There is a short nontechnical description of the birch and swinnertondyer conjecture in keith devlins book the millennium problems. Birch and swinnertondyer have conjectured that re is equal to the rank of the finitely generated group ef of points over f. Swinnertondyer conjectured that if a given elliptic curve has an infinite number of solutions, then the associated series has value 0 at a certain fixed point. Here, daniel delbourgo explains the birch and swinnertondyer conjecture. The birch and swinnertondyer conjecture for an elliptic curve over. Base change and the birch and swinnertondyer conjecture virdol, cristian, functiones et approximatio commentarii mathematici, 2012. The birch and swinnertondyer conjecture has been proved only in special cases. Nb that the reciprocal of the lfunction is from some points of view a more natural object of study. In 1976, coates and wiles showed that elliptic curves with complex multiplication having an infinite number of solutions have series which are zero at the relevant.
An elliptic curve, say e, can be represented by points on a cubic equation as below with certain a, b. Zx is a totally reducible polynomial of degree 3 without. Consequences of the birch and swinnertondyer conjecture. Since the rank part of the birch and swinnertondyer conjecture is a theorem for curves of analytic rank 0 or 1, this completely verifies the full. Introduced by andrew wiles manjul bhargava is the r. The birch and swinnertondyer conjecture for qcurves and odas period relations henri darmon department of mathematics and statistics, mcgill university, 805 sherbooke street west, montreal, canada h3a 2k6 email.
Congettura di birch e swinnertondyer curve ellittiche fattorizzazione discreta crittografia ing. Plan 1 introduction and statement of the birchswinnertondyer bsd conjectures for elliptic curves over q 2 numerical evidence1 and examples 1should be a short talk then rhb john cremona warwicknumerical evidence for the bsd conjecture3 31. Birch and swinnertondyer had given a formulation for rank 0 curves in which case we can talk about the value instead of the leading coefficient, and had also indicated that in the case of positive rank, heights of the generators of the mordellweil group seemed to play a role, but they never mentioned the regulator explicitly. When the solutions are the points of an abelian variety, the birch and swinnertondyer conjecture asserts that the size of the group of rational points is related to the behavior of an associated zeta function. Recall that the mass formula provides a weighted count of. Birch and swinnertondyer conjecture clay mathematics. Elliptic curves have a long and distinguished history that. The conjecture was made in the 1960s, and hundreds of people have thought about it for over 4 decades. Appointments 082015 visiting faculty, ecoevolutionary math group, ecole normale superieure, paris, france 20 assistant professor, department of. Wenching winnie li, ling long, zifeng yang submitted on 17 nov 2003 v1, last revised 23 jun 2004 this version, v3. The conjecture was chosen as one of the seven millennium prize problems listed by the clay mathematics institute, which has. Pdf report on the birch and swinnertondyer conjecture. Since it involves exact formulae rather than asymptotic questions, it has been tested numerically more extensively than any other conjecture in the history of number theory, and the numerical results obtained have always been in perfect accord with every aspect of the. But in special cases one can hope to say something.
We prove the full birch and swinnertondyer conjecture for many. Devlins exposition is meant for a broad audience and may be at the level you are looking for. The conjecture of birch and swinnertondyer springerlink. The second conjecture of birch and swinnertondyer, which refines the first, is where the. A polynomial relation fx, y 0 in two variables defines a curve c0. They were motivated by siegels mass formula for quadratic forms. One of the first nontrivial examples, cubics in two variables, leads to the theory of socalled elliptic curves. Empirical evidence for the birch and swinnertondyer conjecture robert l. Arthaud, n on birch and swinnertondyers conjecture for elliptic curves with complex multiplication. We formulate a refined version of the birch and swinnertondyer conjecture for abelian varieties over global function fields. Pdf this article begins with a description of the basic problems in the arithmetic theory of elliptic curves, which motivate the statement of the. Elliptic curves and the conjecture of birchswinnerton. The birch and swinnertondyer conjecture by the theorem of mordell, it is known that for an elliptic curve e over the rationals q, the set eq is.
Whatever andrew wiles is doing secretly is, quite by definition, unknown to the rest of us. This is probably a really silly question, but i was wondering if someone could explain the birch and swinnertondyer conjecture to me in a simple way. This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field of rational numbers. On the birch and swinnertondyer conjecture springerlink. Contents preface iii 1 numerical veri cation of bsd for hyperelliptic curves of higher genus over q 1 1. Wiles didnt need bhargavas result to become motivated to work on the birc. The birch and swinnertondyer conjecture is a striking example of conjectures in number theory, specifically in arithmetic geometry, that has abundant numerical evidence but not a complete general solution. One of the crowning moments is the paper by andrew wiles which is difficult to understand let alone to appreciate the conjecture. Professor william stein mathematics the current state of knowledge about the birch and swinnertondyer conjecture relies on some of the deepest and most di cult mathematical endeavors, including the modularity. Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. In the general case, the conjecture incorporates both a family of explicit congruence. This was extended to the case where f is any finite abelian extension of k by arthaud 1978. On the conjecture of birch and swinnertondyer springerlink. What is the birch and swinnertondyer conjecture, and what is known about it.
Birch and swinnertondyer conjecture, in mathematics, the conjecture that an elliptic curve a type of cubic curve, or algebraic curve of order 3, confined to a. Explore anything with the first computational knowledge engine. Matematika tekutin v pohybu eduard feireisl matematick y ustav av cr, praha 185. This conjectural claim is of huge importance as it implies that global duality theory for elliptic curves or abelian varieties in. I would point out though that the premise of the question is a bit misinformed. The birch and swinnertondyer conjecture bilkent university. The conjecture of birch and swinnertondyer is one of the principal open problems of number theory today. On the conjectures of birch and swinnertondyer and a. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. Theoretical and computational aspects of the birch and. The birch and swinnertondyer conjecture clay mathematics institute. Elliptic curves with abelian division fields request pdf. The birch and swinnertondyer conjecture is a well known mathematics problem in the area of elliptic curve. On the conjecture of birch and swinnertondyer for an elliptic curve of rank 3.
The object of this note is to verify the conjecture of birch and swinnertondyer. Empirical evidence for the birch and swinnertondyer. On the conjecture of birch and swinnertondyer for elliptic curves. Birch and swinnertondyer conjecture asserts that rankeq ords1leq,s1. Numerical evidence for the birch swinnertondyer conjecture. Pdf on a refinement of the birch and swinnertondyer. This refinement incorporates both families of congruences between the.
Then the algebraic and analytic ranks of eare the same. A polynomial relation fx, y 0 in two variables defines a curve c. Welcome to chaos math 8674 topics in number theory. On the conjecture of birch and swinnertondyer for an elliptic curve. More specifically, we will discuss the conjecture of birch and swinnertondyer, which predicts deep connections between the lfunction of an elliptic curve and its arithmetic, and the vast conjectural generalizations for. Lecture by fernando rodriguezvillegas supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve mod p to the rank of the group of rational points.
71 997 432 344 581 292 779 1416 536 692 495 565 721 821 1453 1239 643 515 91 1355 212 1305 368 1208 64 738 1102 1051 852 1049 941 928 1073 155 26 464