Report on the Birch and Swinnerton-Dyer conjecture
type: Item
instance of: scholarly article
title: Report on the Birch and Swinnerton-Dyer conjecture
zbMATH Open document ID: 1273.11102
author: Massimo Bertolini
published in: Milan Journal of Mathematics
publication date: 2011-02-19T00:00:00Z
review text: The paper provides a nice survey on recent results on the Birch and Swinnerton-Dyer conjecture (BSD from now on). Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\), let \(L(E,s)\) be the associated \(L\)-series, then BSD states that \[ \text{ord}_{s=1}L(E,s)=\text{rank}_{\mathbb{Z}} E(\mathbb{Q})\;. \] A second statement relates the residue of the \(L\)-series at \(s=1\) to the order of the Tate-Shafarevich group of the curve (modulo some computable constants again depending only on the curve). Similar statements are conjectured over any number field \(K\). The first four sections provide a brief introduction to the conjecture and include the outline of a proof of the Mordell-Weil theorem, definition and basic properties of the \(L\)-series \(L(E,s)\) and a discussion of the Fermat-Pell equation which presents some analogies with the formulas of BSD. The final section introduces Shimura curves, Heegner points and modularity results in order to show how they have been employed to provide proofs of some cases of BSD, e.g. for curves of analytic rank 0 (i.e., such that \(L(E,1)\neq 0\)). The (sketched) proofs provide the crucial ideas and an outline of the strategy used to prove the main results of some important recent papers like \textit{K. Kato} [Astérisque 295, 117--290 (2004; Zbl 1142.11336)] and \textit{M. Bertolini} and \textit{H. Darmon} [Ann. Math. (2) 162, No.~1, 1--64 (2005; Zbl 1093.11037)]. The author concludes by hinting at the link between the BSD conjecture and the Iwasawa Main Conjecture for elliptic curves (and \(p\)-adic \(L\)-functions) over cyclotomic and anticyclotomic \(\mathbb{Z}_p\)-extensions: some recent results in this direction (namely \textit{C. Skinner} and \textit{E. Urban} [``The Iwasawa main conjectures for \(\mathrm{GL}_2\)'', avaliable at \url{http://www.math.columbia.edu/~urban/eurp/MC.pdf}]) should lead to concrete improvements in the understanding of BSD.
reviewed by: Andrea Bandini
Mathematics Subject Classification ID: 11G40
Mathematics Subject Classification ID: 11G05
Mathematics Subject Classification ID: 11G15
Mathematics Subject Classification ID: 11F33
Mathematics Subject Classification ID: 11R23
zbMATH DE Number: 5853587
zbMATH Keywords: elliptic curves
zbMATH Keywords: L-series
zbMATH Keywords: p-adic L-functions
zbMATH Keywords: Shafarevich-Tate groups
zbMATH Keywords: Main conjectures of Iwasawa theory
Wikidata QID: Q123143833
MaRDI profile type: Publication
full work available at URL: s00032-010-0123-6
OpenAlex ID: W2161350894
cites work: On the modularity of elliptic curves over \(\mathbb Q\): wild 3-adic exercises
cites work: Kolyvagin's descent and Mordell-Weil groups over ring class fields.
cites work: A rigid analytic Gross-Zagier formula and arithmetic applications. (With an appendix by B. Edixhoven)
cites work: Iwasawa's main conjecture for elliptic curves over anticyclotonic \(\mathbb Z_p\)-extensions
cites work: Q4657450
cites work: Generalized Heegner cycles and \(p\)-adic Rankin \(L\)-series. With an appendix by Brian Conrad
cites work: \(p\)-adic Rankin \(L\)-series and rational points on CM elliptic curves
cites work: Nonvanishing theorems for L-functions of modular forms and their derivatives
cites work: Notes on elliptic curves. II.
cites work: Partial Euler Products on the Critical Line
cites work: On the conjecture of Birch and Swinnerton-Dyer
cites work: Q4335074
cites work: Finiteness theorems for abelian varieties over number fields.
cites work: Heegner points and derivatives of \(L\)-series
cites work: Q4830109
cites work: Motivic L-Functions
cites work: Q3972742
cites work: On a Conjecture of Birch and Swinnerton-Dyer
cites work: On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields
cites work: Q5654155
cites work: Modular curves and the Eisenstein ideal
cites work: Rational isogenies of prime degree. (With an appendix by D. Goldfeld)
cites work: Arithmetic of Weil curves
cites work: On \(p\)-adic analogues of the conjectures of Birch and Swinnerton-Dyer
cites work: Bounds for the torsion of elliptic curves over number fields
cites work: Q3413444
cites work: Mean values of derivatives of modular \(L\)-series
cites work: Tate-Shafarevich groups and L-functions of elliptic curves with complex multiplication
cites work: The ``main conjectures'' of Iwasawa theory for imaginary quadratic fields
cites work: The Arithmetic of Elliptic Curves
cites work: The Iwasawa Main Conjectures for \(\mathrm{GL}_{2}\)
cites work: Ring-theoretic properties of certain Hecke algebras
cites work: Modular elliptic curves and Fermat's Last Theorem
cites work: Q3589185
cites work: Heights of Heegner points on Shimura curves.
cites work: Gross-Zagier formula for \(\text{GL}_ 2\).
cites work: Q3682555
DOI: 10.1007/S00032-010-0123-6
description: scientific article; zbMATH DE number 5853587