Jun 29, 2012 kato explains in 2, 28, section 15 how to deduce it from rubins results on the main conjecture for imaginary quadratic fields. Theoretical and computational aspects of the birch and. 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. In each case, there is a main conjecture linking the tower to a padic lfunction. They were motivated by siegels mass formula for quadratic forms.
The origins of this conjecture can be traced back to numerical computations done by birch and swinnertondyer 5. Consequences of the birch and swinnertondyer conjecture. What is the birch and swinnertondyer conjecture, and what is known about it. On the conjecture of birch and swinnertondyer for elliptic curves with complex. Pdf the birch and swinnertondyer conjecture is a well known mathematics problem in the area of elliptic curve. Devlins exposition is meant for a broad audience and may be at the level you are looking for. In this case, for a generator p of eqeq tors, the padic birch and swinnertondyer conjecture predicts that log. On the conjecture of birch and swinnertondyer for an elliptic curve of rank 3 by joe p. The conjecture was made in the 1960s, and hundreds of people have thought about it for over 4 decades. An algorithm based on the work of cremona is developed in the parigp. Pdf report on the birch and swinnertondyer conjecture. Given an elliptic curve e and a prime p of good supersingular reduction, we formulate padic analogues of the birch and swinnertondyer conjecture using a pair of iwasawa functions l. The conjecture of birch and swinnertondyer springerlink. Whatever andrew wiles is doing secretly is, quite by definition, unknown to the rest of us.
Numerical evidence for the equivariant birch and swinnertondyer conjecture bley, werner, experimental mathematics, 2011. Kato explains in 2, 28, section 15 how to deduce it from rubins results on the main conjecture for imaginary quadratic fields. In the general case, the conjecture incorporates both a family of explicit congruence. The selmer groups at the central point lie deeper and are the subject of the birch and swinnertondyer conjecture.
Exceptional zero conjecture for symmetric powers of cm. Note that we are looking at selmer groups of modular forms at noncentral points. An elliptic curve is an irreducible curve of genus one with a specified point. Feb 25, 2017 what is the birch and swinnerton dyer conjecture, and what is known about it.
Empirical evidence for the birch and swinnertondyer conjecture. On the birch and swinnertondyer conjecture springerlink. A nontechnical account of the birchswinnertondyer conjecture. The birch and swinnerton dyer conjecture is a well known mathematics problem in the area of elliptic curve. In 2002, chris skinner and eric urban claimed a proof of a main conjecture for gl2.
Q, in other words for rational points on the curve. 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. On a conjecture of birch and swinnertondyer department of. The second conjecture of birch and swinnertondyer, which refines the first, is. On the conjecture of birch and swinnerton dyer for an elliptic curve of rank 3 by joe p. On the conjecture of birch and swinnertondyer for an. A birch and swinnertondyer conjecture for the mazurtate circle pairing bertolini, massimo and darmon, henri, duke mathematical journal, 2004. On birch and swinnertondyers conjecture for elliptic curves with complex multiplication. The galois group of the infinite tower, the starting field, and the sort of arithmetic module studied can all be varied.
I would point out though that the premise of the question is a bit misinformed. Introduced by andrew wiles manjul bhargava is the r. An elliptic curve, say e, can be represented by points on a cubic equation as below with certain a, b. Swinnerton dyer 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. But in special cases one can hope to say something. In this thesis, we give a survey on the birch and swinnertondyer conjecture. On the conjecture of birch and swinnertondyer for an elliptic curve of rank 3. I will discuss some recent joint work with yongxiong li which shows how one can use iwasawa theory at the neglected prime p2 to prove stronger versions of an old theorem of david rohrlich on the nonvanishing of birch swinnerton lvalues of the gross family of elliptic curves with complex multiplication, and discuss some possible generalisations. Eudml on the conjecture of birch and swinnertondyer. On atkin swinnerton dyer congruence relations authors.
Wenching winnie li, ling long, zifeng yang submitted on 17 nov 2003 v1, last revised 23 jun 2004 this version, v3. On the conjectures of birch and swinnertondyer and a. Here, daniel delbourgo explains the birch and swinnertondyer conjecture. This is probably a really silly question, but i was wondering if someone could explain the birch and swinnerton dyer conjecture to me in a simple way. There is a short nontechnical description of the birch and swinnertondyer conjecture in keith devlins book the millennium problems. Schrijver, construction of strongly regular graphs, twoweight codes and partial geometries by finite fields, combinatorica, 11, 1981, 6373. Swinnertondyer conjectured that if a given elliptic curve has an infinite number of solutions, then the associated. Elliptic curves and the conjecture of birchswinnertondyer.
Base change and the birch and swinnertondyer conjecture virdol, cristian, functiones et approximatio commentarii mathematici, 2012. 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. Birch and swinnertondyer conjecture clay mathematics institute. Razar, some functions related to the derivatives of the lseries of an elliptic curve at s1. One of the first nontrivial examples, cubics in two variables, leads to the theory of socalled elliptic curves. In this thesis, we give a survey on the birch and swinnerton dyer conjecture. Explore anything with the first computational knowledge engine. This is an expanded note prepared for a 40minute elementary introduction to the birch and swinnertondyer conjecture presented at the farewell party for chenyu chi, who was leaving harvard after his 8 years as a graduate student and a junior fellow here. Roland gillard, unites elliptiques et fonctions l padiques. Assuming the weil taniyama conjecture for this curve, we show that its lseries ls has a triple zero at s 1.
This article is within the scope of wikiproject mathematics, a collaborative effort to improve the coverage of mathematics on wikipedia. Remarks on the birchswinnertondyer conjecture request pdf. Verification of the birch and swinnertondyer conjecture for elliptic curves with complex multiplication unpublished ls1981 j. Yang, the birchswinnertondyer conjecture, proceedings of the 2002 international conference on related subjects to clay problems, the institute of pure and applied mathematics, cheonbuk. 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. The birch and swinnertondyer conjecture is a well known mathematics problem in the area of elliptic curve. Elliptic curves and the conjecture of birch swinnertondyer, sarah zerbes ucl an important problem in number theory is to understand the rational solutions to algebraic equations. I will discuss some recent joint work with yongxiong li which shows how one can use iwasawa theory at the neglected prime p2 to prove stronger versions of an old theorem of david rohrlich on the nonvanishing of birchswinnerton lvalues of the gross family of elliptic curves with complex multiplication, and discuss some possible generalisations.
Zx is a totally reducible polynomial of degree 3 without. Birch and swinnertondyer conjecture clay mathematics. Is prof manjul bhargava close to solving birch and swinnerton. Yang, the birch swinnerton dyer conjecture, proceedings of the 2002 international conference on related subjects to clay problems, the institute of pure and applied mathematics, cheonbuk. As yet the proof is not effective so that one does not possess an algorithm for finding the rational. On the conjecture of birch and swinnertondyer for an elliptic curve. 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. One of the crowning moments is the paper by andrew wiles which is difficult to.
Birchswinnertondyer conjecture from wolfram mathworld. Then the algebraic and analytic ranks of eare the same. Recall that the mass formula provides a weighted count of. 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. E is an elliptic curve defined over q, then the algebraic rank of e equals. It relates the rank of the free part of the finitely generated,group of the elliptic curve to the vanishing order of the corresponding hasseweil lfunction at 3 1. Although kolyvagins conjecture is known in the case of elliptic curves of analytic rank one over k, the number m. Elliptic curves have a long and distinguished history that. Recall that the mass formula provides a weighted count of integral quadratic forms within. 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. Miller department of mathematics princeton university abstract. If the genus of c0 is greater than or equal to 2, then c0q is finite. If you would like to participate, please visit the project page, where you can join the discussion and see a list of open tasks. They are equivalent to the conjectures of perrinriou and bernardi.
We show that in many cases these elements are visible in an abelian surface or abelian 3fold. In mathematics, the birch and swinnertondyer conjecture describes the set of rational solutions to equations defining an elliptic curve. On atkinswinnertondyer congruence relations authors. The equivariant tamagawa number conjecture and modular symbols, pdf file, math. The conjectures of birch and swinnertondyer, and of tate. The conjecture of birch and swinnerton dyer is one of the principal open problems of number theory today. On the conjecture of birch and swinnertondyer for elliptic curves. Pdf this article begins with a description of the basic problems in the arithmetic theory of elliptic curves, which motivate the statement of the. John cremona warwicknumerical evidence for the bsd conjecture4 31. The birch and swinnertondyer conjecture for qcurves and oda. Bernoulli lecture nonvanishing of some birchswinnerton. This chapter explains the conjecture that birch and swinnertondyer made about ranks of elliptic curves the bsd rank conjecture. One of the crowning moments is the paper by andrew wiles which is difficult to understand let alone to appreciate the conjecture. There is a short nontechnical description of the birch and swinnerton dyer conjecture in keith devlins book the millennium problems.
The conjecture of birch and swinnertondyer is one of the principal open problems of number theory today. Kolyvagin conjecture and the structure of selmer groups. The birch and swinnertondyer conjecture clay mathematics institute. In the last few years, it has become increasingly evident that the study of the. We numerically verify the conjecture of birch and swinnerton dyer concerning the analytic and geometric rank of an elliptic curve.
On the conjecture of birch and swinnertondyer springerlink. 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. Numerical evidence for the birch swinnertondyer conjecture. The birch and swinnertondyer conjecture, a computational. 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 conjecture.
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