The arithmetic of elliptic curves – an update. (English) Zbl 1229.11088

This is a short survey on the progress that has been made on the arithmetic of elliptic curves in the past twenty-five years, with particular attention to the questions highlighted in J. Tate’s 1974 paper [Invent. Math. 23, 179–206 (1974; Zbl 0296.14018)].
Section headings are: 2. The \(L\)-function. 3. Modular forms. 4. The \(\ell\)-adic homology group. 5. Modular Galois representations. 6. The Mordell-Weil theorem. 7. The conjecture of Birch and Swinnerton-Dyer; 8. Heegner points on the curve \(X_0(N)\). 9. Heegner points and the Selmer group. 10. On the distribution of Frobenius classes. 11. Speculations on curves of higher rank.


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


Zbl 0296.14018