Elliptic curves with complex multiplication and the conjecture of Birch and Swinnerton-Dyer. (English) Zbl 0991.11028

Viola, Carlo (ed.), Arithmetic theory of elliptic curves. Lectures given at the 3rd session of the Centro Internazionale Matematico Estivo (CIME), Cetraro, Italy, July 12-19, 1997. Berlin: Springer. Lect. Notes Math. 1716, 167-234 (1999).
Let \(E\) be an elliptic curve defined over an imaginary quadratic field \(K\), with complex multiplication by \(K\), and let \(L(E,s)\) be its \(L\)-function. The aim of this expository text is to prove the following.
(1) If \(L(E,1)\not=0\) then \(E(K)\) is finite.
(2) Moreover, if \(p>7\) is a prime number such that \(E\) has good reduction above \(p\), then the order of the \(p\)-part of the Shafarevich-Tate group of \(E\) can be explicitly computed, confirming the Birch and Swinnerton-Dyer conjecture in this case.
Part (1) was proved by J. Coates and A. Wiles [Invent. Math. 39, 223-251 (1977; Zbl 0359.14009)]. Note that part (2) has been improved by the author [Invent. Math. 103, 25-68 (1991; Zbl 0737.11030)], where the hypothesis of good reduction drops.
The text is self-contained enough; it often refers to J. W. S. Cassels [J. Lond. Math. Soc. 41, 193-291 (1966; Zbl 0138.27002)], S. Lang [Elliptic functions. 2nd ed., Springer (1987; Zbl 0615.14018)], G. Shimura [Introduction to the arithmetic theory of automorphic functions, Princeton Univ. Press (1994; Zbl 0872.11023)] and J. H. Silverman [The arithmetic of elliptic curves, Springer (1986; Zbl 0585.14026)] for the background, especially in the first four sections, where the necessary basic tools of the theory of elliptic curves are presented in a clear and concise way.
The rest of the text is devoted to the proof of (1) and (2). In Section 5, the fundamentals of complex multiplication are introduced.
Section 6 uses the results of the previous section to compute the Selmer group of an elliptic curve with complex multiplications, following Coates and Wiles (loc. cit.).
In Section 7 the elliptic units are introduced, and these ideas are generalised in Section 8 where the concept of Euler system is introduced, based on Kolyvagin.
Section 9 contains a description of Kolyvagin’s idea to bound the cardinality of certain ideal class groups. In Section 10, there is a proof of the theorem of Coates and Wiles (1). Section 12 contains a computation of the \(p\)-power Selmer group (with \(p\) as in (2)), hence proving the assertion (2). The proof needs some tools from Iwasawa theory, introduced in Section 11.
It should be pointed out that this text is nicely written and clear and is supported by several examples.
For the entire collection see [Zbl 0924.00032].


11G05 Elliptic curves over global fields
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G15 Complex multiplication and moduli of abelian varieties