Report on the Birch and Swinnerton-Dyer conjecture
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.
scientific article; zbMATH DE number 5853587
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