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Modular curves and arithmetic. (English) Zbl 0597.14023
Proc. Int. Congr. Math., Warszawa 1983, Vol. 1, 185-211 (1984).
[For the entire collection see Zbl 0553.00001.]
The author describes the recent advances and conjectures in the study of the Mordell-Weil groups of elliptic curves with complex multiplication, and more generally, of elliptic curves that admit Weil parametrizations. The exposition begins with a brief introduction to the theorem of Gross and Zagier giving a formula for the value at \(s=1\) of the derivative of the L-function attached to a Weil curve. The author then gives a description of Greenberg’s theory of variation of the Mordell-Weil rank over anti-cyclotomic extensions, and also describes Rohrlich’s theory of variation of the rank over the anti-cyclotomic tower, and over cyclotomic towers. Included in this exposition is an introduction to the p-Selmer group, and the Tate-Shafarevich group of an elliptic curve. Of course, the Iwasawa theory of elliptic curves is a central tool in the study of many of the topics mentioned above, and so the author provides a description of some of the important results in this field. This leads, naturally, to the p-adic theory, including the p-adic height pairing of Schneider, the p-adic height pairings of Mazur and Tate, the p-adic L- functions, and p-adic Heegner measures.
The author concludes this paper by asking if there is a p-adic version of the Gross-Zagier theorem. The question has recently been given an affirmative answer by B. Perrin-Riou (see her forthcoming papers). Using both the Gross-Zagier theorem, and Perrin-Riou’s p-adic version, Karl Rubin has recently proved that if E/\({\mathbb{Q}}\) is an elliptic curve with complex multiplication, and if the Mordell-Weil rank of E is at least 2 then the L-function of E has order of vanishing at least 2.
We should mention that since the Warsaw congress there has been one other striking result in the theory of c. m. curves. Karl Rubin has recently proved the finiteness of the Tate-Shafarevich group for c. m. elliptic curves whose L-functions do not vanish at \(s=1\).
Reviewer: S.Kamienny

14H25 Arithmetic ground fields for curves
14K22 Complex multiplication and abelian varieties
11F27 Theta series; Weil representation; theta correspondences
14H52 Elliptic curves
14G25 Global ground fields in algebraic geometry
14G20 Local ground fields in algebraic geometry
14H45 Special algebraic curves and curves of low genus
11G15 Complex multiplication and moduli of abelian varieties