Base change and the Birch-Swinnerton Dyer conjecture by Maruti Ram Murty

Cover of: Base change and the Birch-Swinnerton Dyer conjecture | Maruti Ram Murty

Published by Dept. of Mathematics, University of Toronto in Toronto .

Written in English

Read online

Subjects:

  • Birch-Swinnerton-Dyer conjecture.,
  • Group theory.

Edition Notes

Book details

StatementM. Ram Murty and V. Kumar Murty.
SeriesPreprint / University of Toronto, Dept. of Mathematics, Preprint (University of Toronto. Dept. of Mathematics)
ContributionsMurty, Vijaya Kumar, 1956-
Classifications
LC ClassificationsQA171 .M88 1990
The Physical Object
Pagination14 leaves.
Number of Pages14
ID Numbers
Open LibraryOL19136145M

Download Base change and the Birch-Swinnerton Dyer conjecture

Base change and the Birch and Swinnerton-Dyer conjecture Cristian Virdol Department of Mathematics Kyushu University March 9, Abstract Base change and the Birch-Swinnerton Dyer conjecture book this paper we prove that if the Birch and Swinnerton-Dyer con-jecture holds for products of abelian varieties attached to Hilbert new-forms of parallel weight 2 with trivial central character, then the Birch.

In this paper we prove that if the Birch and Swinnerton-Dyer conjecture holds for products of abelian varieties attached to Hilbert newforms of parallel weight 2 with trivial central character. 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 Kolyvagin-Gross-Zagier and the weak parity theorem of Tim and Vladimir Dokchitser. The second, third and fourth part of the essay represent an account, with detailed.

Birch and Swinnerton-Dyer Conjecture. Mathematicians have always been fascinated by the problem of describing all solutions in whole numbers x,y,z to algebraic equations like x 2 + y 2 = z 2.

Euclid gave the complete solution for that equation, but for more complicated equations this becomes extremely difficult. Indeed, in Yu. In mathematics, the Birch and Swinnerton-Dyer conjecture describes the set of rational solutions to equations defining an elliptic is an open problem in the field of number theory and is widely recognized as one of the most challenging mathematical problems.

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 whether an associated function is equal to zero or not equal to zero, respectively.

There is a short non-technical description of the Birch and Swinnerton-Dyer Conjecture in Keith Devlin's book The Millennium Problems. See Chapter 6, pages Devlin's exposition is meant for a broad audience and may be at the level you are looking for.

Base change and the Birch-Swinnerton Dyer conjecture book The Birch and Swinnerton-Dyer Conjecture is a well known mathematics problem in the area of Elliptic Curve. One of the crowning moments is the paper by Andrew Wiles which is difficult to understand let alone to appreciate the conjecture.

This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field of rational by: 1. $\begingroup$ This is probably not the kind of consequences you're looking for, but it is worthwhile to note that the Birch and Swinnerton-Dyer conjecture contains as a "mere" prerequisite the assertion that the Tate-Shafarevich group of elliptic curves is finite.

This conjectural claim is of huge importance as it implies that global duality theory for elliptic curves (or abelian varieties in. Serge Lang, Sur la conjecture de Birch-Swinnerton Dyer Roland Gillard, Unités elliptiques et fonctions L p -adiques M.

Razar, Some functions related to the derivatives of the L-series of an elliptic curve at s=1Cited by: the Birch-Swinnerton-Dyer conjecture for constant Abelian varieties, i.e.

those coming from base change from the nite constant eld. Work of Peter Schneider [Sch82b] and Werner Bauer [Bau92], which was completed in the article [KT03] of Kazuya Kato and Fabien Trihan in.

Birch-Swinnerton-Dyer and the Artin-Tate conjecture for the base a curve, see Gordon’s PhD thesis [Gor79]. Let be smooth projective over of arbitrary dimension. We prove the following partial generalisation, concerning the niteness part of the conjecture.

Theorem 3 (The Artin-Tate and the Birch-Swinnerton-Dyer conjecture). Let L/K be an extension of number fields, and A be an abelian variety defined over L.

Then e Birch and Swinnerton-Dyer conjecture holds for A if and only if the Birch and Swinnerton-Dyer conjecture lds for Res L/K know (Theorem of [M1]): C. Virdol / Journal of Number Theory () – Theorem Cited by: 3.

The Birch and Swinnerton-Dyer Conjecture is a well known mathematics problem in the area of Elliptic Curve. One of the crowning moments is the paper by Andrew Wiles which is difficult to. Abstract. The conjecture of Birch and Swinnerton-Dyer is one of the principal open problems of number theory today.

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 Cited by: THE BIRCH AND SWINNERTON-DYER CONJECTURE ANDREW WILES A polynomial relation f(x,y) = 0 in two variables defines a curve C 0.

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 File Size: KB. They cover Stark's conjecture, the Birch and Swinnerton-Dyer conjecture, and analytic and cohomological methods. Arithmetic of L-functions The sources said the proof would help mathematicians solve the Birch and Swinnerton-Dyer conjecture, one of seven problems which the U.

Abstract. The aim of these lectures is to introduce the Birch and Swinnerton-Dyer conjectures in its entirety. One part of these conjectures predicts the equality of two different ‘ranks’ associated to an elliptic curve defined over a number : Sudhanshu Shekhar, R.

Sujatha. We are now ready to state the conjecture of Birch and Swinnerton-Dyer. The Birch and Swinnerton-Dyer Conjecture By the theorem of Mordell, it is known that for an elliptic curve E over the rationals Q, the set E(Q) is finitely generated.

More explicitly: E(Q) ∼= Zr ⊕T () for some non-negative integer r, and T a finite abelian Size: KB. The Birch and Swinnerton-Dyer conjecture (henceforth abbreviated BSD conjecture) gives a conjectural answer to this latter problem. In order to state the conjecture, we need to define the L-function of an elliptic curve E.

If p is a prime number, then let N. I cannot hope to do better than the 5-page exposition by Gunter Harder and Don Zagier given here: Abstract: Mazur, Tate, and Teitelbaum gave a p-adic analogue of the Birch and Swinnerton-Dyer conjecture for elliptic curves.

We provide a generalization of their conjecture in the good ordinary case to higher dimensional modular abelian varieties over the rationals by constructing the p-adic L-function of a modular abelian variety and showing that it satisfies the appropriate interpolation Cited by: 2.

check that in this case our conjecture recovers the precise form of the geometric Birch and Swinnerton-Dyer conjecture that is given by Kato and Trihan in [16]. In general, by refining the methods of [16] we are able to prove that, whenever there exists a prime ℓ such that the ℓ-primary component of the Tate-Shafarevic group of AFile Size: KB.

The Birch{Swinnerton-Dyer conjecture. The origins of this conjecture can be traced back to numerical computations done by Birch and Swinnerton-Dyer ([5]). They were motivated by Siegel’s mass formula for quadratic forms. Recall that the mass formula. The Birch and Swinnerton -Dyer Conjecture is a well known mathematics problem in the area of Elliptic Curve.

One of the crowning moments is the paper by Andrew Wiles which is difficult to understand let alone to appreciate the conjecture. This paper surveys the background of the conjecture treating the ranks of the elliptic curves over the field ofCited by: 1.

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 Mordell-Weil group seemed to play a role, but they never mentioned the regulator explicitly.

$\endgroup. why does one refrain using the techniques used in the case above, sharply i mean,what are the things that prevent one from proving the conjecture by using the class formula,in case of elliptic curves,i mean can anyone enumerate the reasons why does one fail to prove Birch and Swinnerton Dyer conjecture by using the Class formula.

A Conjectura de Birch-Swinnerton-Dyer está relacionada a outros dois problemas diofantinos famosos: o famoso problema fermatiano e a chamada Conjectura ABC [criada por David Masser e Joseph Oesterlé, em ]. Uma consequência da conjectura citada é a existência de um algoritmo para decidir se um dado número "n" é congruente ou não.

Title: The Birch-Swinnerton-Dyer Conjecture. Authors: Jae-Hyun Yang (Submitted on 14 Nov ) Abstract: We give a brief description of the Birch and Swinnerton-Dyer Conjecture which is one of the seven Clay problems. Comments: International Conference on Related Subjects to Clay Problems ; held in Jeonju, Republic of Korea (July Cited by: 3.

The Birch and Swinnerton-Dyer Conjecture Sunil Chetty Department of Mathematics October 9, Sunil Chetty BSD. Motivation Elliptic curves L-functions of Elliptic Curves Progress and Application Curves Common Strategies Definition There are two objects which could be used to define a curve.

On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer. Mazur; J. Tate; J. Teitelbaum. Inventiones mathematicae () Volume: 84, page ; ISSN: ; /e; Access Full Article top Access to full text.

How to cite topCited by: An Introduction to the Birch and Swinnerton-Dyer Conjecture Brent Johnson Villanova University Follow this and additional works at: Recommended Citation Johnson, Brent () "An Introduction to the Birch and Swinnerton-Dyer Conjecture," Rose-HulmanAuthor: Brent Johnson.

On an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes over higher dimensional bases over nite elds Timo Keller Decem Abstract We formulate an analogue of the conjecture of Birch and Swinnerton-Dyer for Abelian schemes with everywhere good reduction over higher dimensional bases over finite fields.

the ‘order of vanishing’ part of the Birch and Swinnerton-Dyer conjecture for Artin-Hasse- Weil L-series and, in addition, plays a key role in a complementary article of the rst two authors in which the conjecture formulated here is proved, modulo the standard niteness.

Assuming the Birch and Swinnerton-Dyer conjecture (or even the weaker statement that Cn(Q) is infinite ⇔ L(Cn,1) = 0) one can show that any n ≡ 5,6,7 mod8 is a congruent number and moreover Tunnell has shown,again assuming the conjecture,that forn odd and square-free n is a congruent number ⇐⇒ #{x,y,z ∈ Z:2x2 + y2 +8z2 = n}.

Numerical evidence for the Birch–Swinnerton-Dyer conjecture John Cremona University of Warwick BSD conference, Cambridge 4 May Plan 1 Introduction and statement of the Birch–Swinnerton-Dyer (BSD) conjectures for elliptic curves over Q 2 Numerical evidence and examples John Cremona (Warwick)Numerical evidence for the BSD conjecture2 / Modular symbols and the (equivariant) conjecture of Birch and Swinnerton-Dyer chris wuthrich 8th July 1 Introduction These is the draft for my lecture notes for the Birch and Swinnerton-Dyer summer school in Alghero A part of this was presented in a slightly di erent order in Sardinia.

I intend to add a bit more at a later stage. Iwasawa Theory and the “Main Conjecture” 50 Computing the Selmer Group 61 The purpose of these notes is to present a reasonably self-contained expo-sition of recent results concerning the Birch and Swinnerton-Dyer conjecture for elliptic curves with complex multiplication.

The goal is the following the-orem. Theorem. EULER SYSTEMS AND THE BIRCH{SWINNERTON-DYER CONJECTURE GUIDO KINGS / ANTONIO LEI / DAVID LOEFFLER / SARAH ZERBES These are the notes from my talk in Essen in June 1.

The BSD conjecture Let Ebe an elliptic curve over Q, so. Taniyama conjecture for this curve, we show that its ¿-series L(s) has a triple zero at * = 1 and compute limy_] L(s)/(s - I)3 to 28 decimal places; its value agrees with the product of the regulator and real period, in accordance with the Birch-Swinnerton-Dyer conjecture if III is trivial.

Numerical Evidence for the Equivariant Birch and Swinnerton-Dyer Conjecture Bley, Werner, Experimental Mathematics, ; Base change and the Birch and Swinnerton-Dyer conjecture Virdol, Cristian, Functiones et Approximatio Commentarii Mathematici, Author: Cusak P.Taniyama conjecture for this curve, we show that its L-series L(s) has a triple zero at s = 1 and compute lim, _I L(s)/(s - 1)3 to 28 decimal places; its value agrees with the product of the regulator and real period, in accordance with the Birch-Swinnerton-Dyer conjecture if III is trivial.The Conjecture of Birch and Swinnerton-Dyer relates an analytic invariant of an elliptic curve -- the value of the L-function, to an algebraic invariant of the curve -- the order of the Tate--Shafarevich group.

Gross has refined the Birch--Swinnerton-Dyer Conjecture in the case of an elliptic curve with complex multiplication by the full ring of integers in a quadratic imaginary by: 1.

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