The birch and swinnertondyer conjecture proposal text amod agashe 1 project description this project lies in the area of number theory, and in particular in a subarea called arithmetic geometry. The conjecture of birch and swinnertondyer provides a mysterious bridge between them. Devlins exposition is meant for a broad audience and may be at the level you are looking for. An elliptic curve is an irreducible curve of genus one with a specified point. The birchswinnertondyer conjecture and heegner points.
Euler systems and re ned conjectures of birch swinnerton. Birch and swinnertondyer conjecture clay mathematics institute. Birch and swinnertondyer conjecture mathematics britannica. Conjecture 3 birch and swinnerton dyer, 1960s let be an elliptic curve over a global field, then the order of vanishing of at is equal to the rank of. By combining these two results we get the statement of the lemma. On the conjecture of birch and swinnertondyer for an elliptic curve. An introduction to the birch and swinnertondyer conjecture.
Let be smooth projective over of arbitrary dimension. Razar, some functions related to the derivatives of the lseries of an elliptic curve at s1. Mazur, tate, and teitelbaum gave a padic analogue of the birch and swinnerton dyer conjecture for elliptic curves. 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 conjecture of birch and swinnertondyer relates an analytic invariant of an elliptic curve the value of the lfunction, to an algebraic invariant of the curve the order of the tateshafarevich group. My understanding is that the birch and swinnerton dyer conjecture is an open problem. We prove the full birch and swinnertondyer conjecture for many specific elliptic curves of analytic rank zero and one and conductor up to 5000 by combining. The birch and swinnertondyer conjecture sunil chetty department of mathematics october 9, 2014 sunil chetty bsd. We give a brief description of the birch and swinnertondyer conjecture which is one of the seven clay problems. We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the padic lfunction of a modular abelian variety and showing that it satisfies the appropriate interpolation. The birch and swinnertondyer conjecture for abelian varieties.
They were motivated by siegels mass formula for quadratic forms. Near x 0 the first two terms in 5 become infinite, but we can combine them to obtain. On an analogue of the conjecture of birch and swinnertondyer. Base change and the birch and swinnertondyer conjecture cristian virdol department of mathematics kyushu university march 9, 2011 abstract in this paper we prove that if the birch and swinnertondyer conjecture holds for products of abelian varieties attached to hilbert newforms of parallel weight 2 with trivial central character, then the birch. The origins of this conjecture can be traced back to numerical computations done by birch and swinnertondyer 5. In mathematics, the birch and swinnertondyer conjecture describes the set of rational solutions to equations defining an elliptic curve. Lectures on the birchswinnertondyer conjecture semantic scholar. 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. Well, a proof of the conjecture would imply the algebraic rank something we.
On the conjecture of birch and swinnertondyer for an. In mathematics, the birch and swinnerton dyer conjecture describes the set of rational solutions to equations defining an elliptic curve. In this paper we prove that if the birch and swinnertondyer conjecture holds for abelian varieties attached to hilbert newforms of parallel weight 2 with trivial central character, then the birch. Birch and swinnertondyer conjecture sciencesean millennium. In this thesis, we give a survey on the birch and swinnerton dyer conjecture. Q, in other words for rational points on the curve. Recall that the mass formula provides a weighted count of integral quadratic forms within. On the conjecture of birch and swinnertondyer for abelian. Birch and swinnertondyer conjecture due to kato and. Computing the selmer group 61 the purpose of these notes is to present a reasonably selfcontained exposition of recent results concerning the birch and swinnertondyer conjecture for elliptic curves with complex multiplication.
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. A nontechnical account of the birchswinnertondyer conjecture. Mazur, tate, and teitelbaum gave a padic analogue of the birch and swinnertondyer conjecture for elliptic curves. Birch and swinnerton dyer conjecture, in mathematics, the conjecture that an elliptic curve a type of cubic curve, or algebraic curve of order 3, confined to a region known as a torus has either an infinite number of rational points solutions or a finite number of rational points, according to. We give a brief description of the birch and swinnerton dyer conjecture which is one of the seven clay problems. B, then the birch and swinnerton conjecture holds for a. Arthaud, n on birch and swinnertondyers conjecture for elliptic curves with complex multiplication. I cannot hope to do better than the 5page exposition by gunter harder and don zagier given here. The birch and swinnerton dyer conjecture has been proved only in special cases.
From this product, one naturally obtain a product of the values of riemann zeta function. Arithmetic of lfunctions the sources said the proof would help mathematicians solve the birch and swinnertondyer conjecture, one of seven problems which the u. My understanding is that the birch and swinnertondyer conjecture is an open problem. In certain special cases, there is a sort of island between the two worlds, which kolyvagin calls an euler system. On the birch and swinnertondyer conjecture for abelian. In bau92, werner bauer proved an analogue of schneiders result for the part of the conjecture, but only for abelian varieties with good. On an analogue of the conjecture of birch and swinnerton. They cover starks conjecture, the birch and swinnertondyer conjecture, and analytic and cohomological methods. Then the algebraic and analytic ranks of eare the same. The object of this note is to verify the conjecture of birch and swinnerton dyer. Alongside, it contains a discussion of some results that have been proved in the direction of the conjecture, such as the theorem of kolyvagingrosszagier and the weak parity theorem of tim and vladimir dokchitser. Lecture by fernando rodriguezvillegas supported by much experimental evidence, this conjecture relates the number of points on an elliptic curve.
The birch and swinnertondyer conjecture clay mathematics institute. Without loss of generality we may assume that e and ed are in short. Birchswinnertondyer and the artintate conjecture for the base a curve, see gordons phd thesis gor79. From the birch and swinnerton dyer conjecture to the. What is the birch and swinnertondyer conjecture in laymans. If the birch and swinnertondyer conjecture holds for b and a. In this paper we prove that if the birch and swinnerton dyer conjecture holds for abelian varieties attached to hilbert newforms of parallel weight 2 with trivial central character, then the birch. 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. John cremona warwicknumerical evidence for the bsd conjecture4 31. His proof used lfunctions in a fundamental way for the first time in number theory.
The birch and swinnertondyer conjecture is a well known mathematics problem in the area of elliptic curve. 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. We prove the following partial generalisation, concerning the niteness part of the conjecture. Report on the birch and swinnertondyer conjecture citeseerx. If the genus of c0 is greater than or equal to 2, then c0q is finite. 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 region known as a torus has either an infinite number of rational points solutions or a finite number of rational points, according to. This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field of.
From the birch and swinnerton dyer conjecture to the gl2 main conjecture. The birch and swinnerton dyer conjecture sunil chetty department of mathematics october 9, 2014 sunil chetty bsd. A formulation of p adic versions of the birch and swinnerton. The bridge predicted by the birch swinnertondyer conjecture can be constructed in two. An introduction to the birch and swinnertondyer conjecture brent johnson villanova university follow this and additional works at. The conjecture was made in the 1960s, and hundreds of people have thought about it for over 4 decades. Base change and the birch and swinnertondyer conjecture. It is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.
The conjecture of birch and swinnertondyer for elliptic. Sep 30, 2015 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. Class number formula and birch and swinnerton dyer conjecture. There is a short nontechnical description of the birch and swinnerton dyer conjecture in keith devlins book the millennium problems.
Euler systems and the birch and swinnertondyer conjecture. The birch and swinnertondyer conjecture, a computational. If we combine this result with the second corollary. The birch and swinnerton dyer conjecture is a well known mathematics problem in the area of elliptic curve. They are equivalent to the conjectures of perrinriou and bernardi. We also generalize work of kurihara and pollack to give a criterion for positive rank in terms of the value of the. There is a short nontechnical description of the birch and swinnertondyer conjecture in keith devlins book the millennium problems. The conjecture of birch and swinnertondyer for jacobians of. For this reason, we call the number the analytic rank of. This article explores the birch and swinnertondyer conjecture, one of. Eudml on the conjecture of birch and swinnertondyer.
In this thesis, we give a survey on the birch and swinnertondyer conjecture. Recent progress toward birch and swinnertondyer conjecture. Theorem 3 the artintate and the birchswinnertondyer conjecture. 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. Numerical evidence for the birch swinnertondyer conjecture. Bsd birch and swinnertondyer conjecture mathematics. Assume that there exists h an algebraic group defined over k and a.
But in special cases one can hope to say something. Recommended citation johnson, brent 2015 an introduction to the birch and swinnertondyer conjecture, rosehulman. We are now ready to state the conjecture of birch and swinnertondyer. On the birch and swinnertondyer conjecture springerlink. Empirical evidence for the birch and swinnertondyer conjecture. In this video i wanted to explain one of problems the birch and swinnerton dyer conjecture. Jun, 2014 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. Recall that the mass formula provides a weighted count of. One of the crowning moments is the paper by andrew wiles which is difficult to understand let alone to appreciate the conjecture. For example, you write i mean we can prove the birch and swinnerton dyer in a similar way.
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