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Iwasawa theory for elliptic curves. (English) Zbl 0946.11027

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, 51-144 (1999).
Let \(E\) be an elliptic curve defined over a number field \(F\). This paper gives a thorough and detailed account of problems and results (quite a few are due to the author) in the Iwasawa theory for \(E\), concentrating on the study of the \(p\)-parts \(\text{Sel}_E (F_n)_p\) of the Selmer groups attached to \(E\) when one goes up the cyclotomic \(\mathbb{Z}_p\)-extension \(F_\infty= \bigcup_{n\geq 0}F_n\).
One main theme is built upon Mazur’s control theorem and Mazur’s conjecture [B. Mazur, Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)], both stated in the case of good ordinary reduction at \(p\). The theorem says (with obvious notations) that the kernels and cokernels of the natural maps \(\text{Sel}_E(F_n)_p \to\text{Sel}_E (F_\infty)^{\Gamma_n}\) are finite and uniformly bounded. The conjecture asserts that \(\text{Sel}_E(F_\infty)\) should be \(\Lambda\)-cotorsion. The conjecture is an immediate consequence of the theorem if \(\text{Sel}_E(F)_p\) is finite. A deeper result of Rubin-Kato-Rohrlich proves the conjecture if \(E\) is defined over \(\mathbb{Q}\) and \(F/\mathbb{Q}\) is abelian (this holds true also in the case of multiplicative reduction at \(p)\).
Variations around this theme are:
– Manin’s conjecture, which extends the Mazur control theorem to the case of multiplicative reduction. For \(F=\mathbb{Q}\), the conjecture follows from \(\log_p (q_E)\neq 0\), \(q_E\) being the Tate period for \(E\) \((\log_p(q_E)\neq 0\) because \(q_E\) is transcendental thanks to the so-called “Saint-Etienne theorem”)
– conjecture 1-8 of this paper, which states (without any assumption on the reduction type of \(E\) at \(p)\) that the \(\mathbb{Z}_p\)-corank of \(\text{Sel}_E (F_n)_p\) is bounded as \(n\) varies. In the case of good ordinary or multiplicative reduction, conjecture 1-8 follows easily easily from Mazur’s conjecture.
Assuming Mazur’s conjecture (that \(\text{Sel}_E (F_\infty)\) is \(\Lambda\)-cotorsion), another theme can be developed around the study of the characteristic ideal \((f_E(T))\) of the \(\Lambda\)-module \(X_E(F_\infty) =\operatorname{Hom} (\text{Sel}_E (F_\infty)\), \(\mathbb{Q}_p/ \mathbb{Z}_p)\). The elliptic “Main conjecture” (originally stated by Mazur) says that the generator \(f_E(T)\) can be chosen so as to satisfy a certain interpolation property. The special value \(f_E(0)\) is related to the Birch-Swinnerton-Dyer conjecture.
Other functorial properties of the module \(X_E(F_\infty)\), such as the “functional equation” for \(f_E(T)\) [R. Greenberg, Adv. Stud. Pure Math. 17, 97-137 (1989; Zbl 0739.11045)], can be expressed in the much more general context of Selmer groups attached to “ordinary” \(p\)-adic representations.
The paper is organized as follows: the first paragraph introduces the various themes and variations, and also a brief recap of the super-singular case; in the second paragraph, the author gives a new, simple and elegant description of the groups \(\text{Sel}_E(F_n)\) and \(\text{Sel}_E (F_\infty)\); he then applies this to give a new proof of Mazur’s control theorem (§3) and of a special case of Perrin-Riou and Schneider’s \(p\)-adic formula for \(f_E(0)\) (§4); last but not the least, §5, by far the longest, discusses a variety of examples and calculations. This carefully written paper contains a wealth of information which makes it a “must” for anyone interested in the subject.
For the entire collection see [Zbl 0924.00032].

MSC:

11R23 Iwasawa theory
11G05 Elliptic curves over global fields
11S25 Galois cohomology
11S40 Zeta functions and \(L\)-functions
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