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Lectures on the conjecture of Birch and Swinnerton-Dyer. (English) Zbl 1285.11096

Popescu, Cristian (ed.) et al., Arithmetic of \(L\)-functions. Lecture notes from the IAS/Park City Mathematics Institute (PCMI) graduate summer school, Park City, UT, USA, 2009. Providence, RI: American Mathematical Society (AMS); Princeton, NJ: Institute for Advanced Study. (ISBN 978-0-8218-5320-7/hbk). IAS/Park City Mathematics Series 18, 169-209 (2011).
The paper is a clear and self-contained introduction to the Birch and Swinnerton-Dyer conjecture. Given an elliptic curve \(E\) defined over a global field \(k\), the author reviews the Mordell-Weil theorem in the first chapter and, after introducing the \(L\)-function attached to \(E/k\), states in the second chapter the Birch and Swinnerton-Dyer conjecture for \(E/k\). The presentation is enriched with lighting examples and discussions. The third chapter presents the results obtained toward this conjecture first in the case of function fields (by Artin and Tate), and then over number fields (by Kolyvagin, Gross-Zagier and Zhang). Four useful appendices complete the exposition.
For the entire collection see [Zbl 1226.11004].

MSC:

11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G05 Elliptic curves over global fields
11-02 Research exposition (monographs, survey articles) pertaining to number theory
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