An important problem in number theory is to understand the rational solutions to algebraic equations. 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. Nekovar and the birch and swinnertondyer conjecture article pdf available in proceedings of the american mathematical society 12610 october 1998 with 20 reads. But in special cases one can hope to say something. This criterion is expressed in terms of an analytic function, called the lfunction ls,e of the elliptic curve, which is a function in a variable s. Numerical evidence for the birch swinnertondyer conjecture.
They were motivated by siegels mass formula for quadratic forms. Ti on padic analogues of the conjectures of birch and swinnertondyer. One of the first nontrivial examples, cubics in two variables, leads to the theory of socalled elliptic curves. This conjecture is a natural strengthening of the birch and swinnertondyer conjecture. This is an expanded note prepared for a 40minute elementary introduction to the birch and swinnerton dyer 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. Arthaud, n on birch and swinnertondyers conjecture for elliptic curves with complex multiplication. Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. A nonabelian conjecture of birch and swinnertondyer type foranyotherpointx 2xz,wehave. The conjecture of birch and swinnertondyer springerlink. Pdf the birch and swinnertondyer formula for elliptic. The birch and swinnertondyer conjecture is a conjecture about the form. Base change and the birch and swinnertondyer conjecture virdol, cristian, functiones et approximatio commentarii mathematici, 2012. The birch and swinnertondyer conjecture clay mathematics institute. 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.
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. The origins of this conjecture can be traced back to numerical computations done by birch and swinnertondyer 5. A birch and swinnertondyer conjecture for the mazurtate circle pairing massimo bertolini henri darmon september 5, 2007 abstract let e be an elliptic curve over q attached to a newform f of weight two on. Razar, some functions related to the derivatives of the lseries of an elliptic curve at s1. In this thesis we use rankins method to evaluate the central critical value of the lseries attached to an elliptic curve e over q and certain odd irreducible 2dimensional artin representation tau. Abstract let e be an optimal elliptic curve over q of conductor n having analytic rank zero, i. 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. Numerical evidence for the birchswinnertondyer conjecture john cremona university of warwick bsd conference, cambridge 4 may 2011. Recall that the mass formula provides a weighted count of integral quadratic forms within. Plan 1 introduction and statement of the birchswinnertondyer bsd conjectures for elliptic curves over q 2 numerical evidence and examples. Heegner points on hijikatapizershemanske curves and the birch and swinnertondyer conjecture longo, matteo.
A nontechnical account of the birchswinnertondyer conjecture. The birch and swinnertondyer conjecture holds for a if and only if x picx,ors mx 2. The local neron pairing the purpose of this section is to prove 1. Is prof manjul bhargava close to solving birch and swinnerton. Elliptic curves and the conjecture of birchswinnertondyer.
Birch and swinnertondyer conjecture which emerges naturally from the discussion above. One of the crowning moments is the paper by andrew wiles which is difficult to understand let alone to appreciate the conjecture. We describe an algorithm to prove the birch and swinnertondyer conjectural formula for any given elliptic curve. We numerically verify the conjecture for these hyperelliptic curves. On the birch and swinnertondyer conjecture springerlink. A nonabelian conjecture of birch and swinnertondyer type. 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.
This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field of rational numbers. The birch and swinnertondyer conjecture for an elliptic curve over. Rankin lfunctions and the birch and swinnertondyer. 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. The birch and swinnerton dyer conjecture has been numerically verified for the jacobians of 32 modular hyperelliptic curves of genus 2 by flynn, lepr\evost, schaefer, stein, stoll and wetherell, using modular methods. Theoretical and computational aspects of the birch and. Proving the birch and swinnertondyer conjecture for specific elliptic.
This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field of. Eudml on the conjecture of birch and swinnertondyer. For any continuous irreducible finitedimensional complex representation tau of galqbarq, we. There is an alternating pairing, called the casselstate pairing iiieq. Iiieq qz whose kernel is precisely the divisible subgroup of iiieq. Eudml on padic analogues of the conjectures of birch. A note on height pairings, tamagawa numbers, and the birch and swinnertondyer conjecture s. Is andrew wiles secretly trying to solve bsd conjecture. Euler systems and the birchswinnerton dyer conjecture guido kings antonio lei david loeffler sarah zerbes these are the notes from my talk in essen in june 2014. By the mordellweil theorem, the group of krational points on a is finitely generated and as for elliptic curves, its rank is predicted by the birch and swinnertondyer conjecture. A basic consequence of this conjecture is the parity conjecture. The birch and swinnertondyer conjecture has been numerically verified for the jacobians of 32 modular hyperelliptic curves of genus 2 by flynn, lepr\evost, schaefer, stein, stoll and wetherell, using modular methods.
Visibility and the birch and swinnertondyer conjecture. Grigor grigorov, andrei jorza, stefan patrikis, corina tarnitapatrascu and stephen donnelly, michael stoll. The birchswinnertondyer conjecture and heegner points 171 remark 3. Numerical evidence for the equivariant birch and swinnertondyer conjecture bley, werner, experimental mathematics, 2011.
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. Elliptic curves and the conjecture of birchswinnerton. The bsd conjecture let ebe an elliptic curve over q, so eq. For any continuous irreducible finitedimensional complex representation tau of galqbarq, we have. Bsd cconjecture, regulators and special values of lfunctions pdf. The conjecture was chosen as one of the seven millennium prize problems listed by the clay mathematics institute, which has. The conjecture of birch and swinnertondyer is one of the principal open problems of number theory today. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with non. Visibility and the birch and swinnertondyer conjecture for. 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. In mathematics, the birch and swinnertondyer conjecture describes the set of rational solutions to equations defining an elliptic curve.
Birch and swinnertondyer conjecture clay mathematics. Visibility and the birch and swinnertondyer conjecture for analytic rank zero amod agashe. The birch and swinnertondyer conjecture for qcurves and. Rankin lfunctions and the birch and swinnertondyer conjecture.
Jo inventiones mathematicae py 1986 vl 84 sp 1 ep 48 kw adic analogues of the birch and swinnerton dyer conjecture. Bloch department of mathematics, university of chicago, chicago, il 60637, usa introduction let g be an algebraic group defined over a number field k. A nonabelian conjecture of birch and swinnertondyer type minhyongkim bordeaux,july,2012. The birch and swinnerton dyer conjecture is a well known mathematics problem in the area of elliptic curve. Birch and swinnertondyer conjecture clay mathematics institute. Let e be a modular elliptic curve over a real quadratic. Consequences of the birch and swinnertondyer conjecture.
This conjectural claim is of huge importance as it implies that global duality theory for elliptic curves or abelian varieties in. Dedicated to martin taylor on the occasion of his 60th birthday. The birch and swinnertondyer conjecture for qcurves and oda. Is prof manjul bhargava close to solving birch and. Birch and swinnerton dyer 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. 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. The conjecture of birch and swinnerton dyer is one of the principal open problems of number theory today. Jo inventiones mathematicae py 1986 vl 84 sp 1 ep 48 kw adic analogues of the birch and swinnertondyer conjecture. Devlins exposition is meant for a broad audience and may be at the level you are looking for. In the calculation of the real period, there is a slight inaccuracy, which might give problems for curves with nonreduced components in the special fibre of their n. The motivation for this study is the twisted bsd conjecture which states that. A note on height pairings, tamagawa numbers, and the birch.
The birch and swinnerton dyer conjecture has been proved only in special cases. Twisted form of the birch and swinnertondyer conjecture. The birch and swinnertondyer conjecture is a well known mathematics problem in the area of elliptic curve. On the padic birch and swinnerton dyer conjecture for elliptic curves over number fields disegni, daniel, kyoto journal of mathematics, 2020 efficient verification of tunnells criterion bach, eric and ryan, nathan c. Assuming the birch and swinnertondyer conjecture or even the weaker statement that cnq is in. Empirical evidence for the birch and swinnertondyer conjecture. 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. Recall that the mass formula provides a weighted count of. Near x 0 the first two terms in 5 become infinite, but we can combine them. John cremona warwicknumerical evidence for the bsd conjecture4 31.
Assuming the birch and swinnertondyer conjecture or even the weaker statement that c nq is in. A nonabelian conjecture of birch and swinnerton dyer type minhyongkim bordeaux,july,2012. Ti on padic analogues of the conjectures of birch and swinnerton dyer. On the conjecture of birch and swinnertondyer for an elliptic curve of rank 3. There is a short nontechnical description of the birch and swinnertondyer conjecture in keith devlins book the millennium problems. We prove the full birch and swinnertondyer conjecture for many. An elliptic curve, say e, can be represented by points on a cubic equation as below with certain a, b. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems. The birchswinnertondyer conjecture and heegner points. Eudml on padic analogues of the conjectures of birch and. A polynomial relation fx, y 0 in two variables defines a curve c.
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