Introduction to Iwasawa theory for elliptic curves (Q2768081)

The goal of this nicely written article is to introduce some of the ideas encompassed in the phrase ``Iwasawa Theory of Elliptic Curves''. It focusses on the control theorem of \textit{B. Mazur} [Rational points of abelian varieties with values in towers of number fields, Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] which states that the Selmer group of an abelian variety behaves well Galois-theoretically in a \({\mathbb{Z}_p}\)-extension for any prime \(p\) where the abelian variety has good ordinary reduction. In fact everything is done in the special case of elliptic curves. Although the proof of Mazur's control theorem presented in the article uses Galois cohomology, it does differ from previous proofs of \textit{B. Mazur} (loc. cit.) and \textit{Y. I. Manin} [Cyclotomic fields and modular curves, Russ. Math. Surv. 26, No. 6, 7-78 (1972); translation from Usp. Mat. Nauk 26, No. 6, 7-71 (1971; Zbl 0241.14014)] by avoiding the use of norm maps on formal groups of height 1. Instead the author uses the fact that if \(E\) is an elliptic curve with good ordinary reduction at a prime \(p\), then the \(p\)-primary subgroup of the Selmer group for \(E\) over finite extensions of \({\mathbb{Q}}\) (and certain infinite extensions) has a very simple and elegant description in terms of the Galois cohomology for the group \(E[{p^{\infty}}]\) of \(p\)-power torsion points of \(E\). The contents of the paper are as follows.NEWLINENEWLINENEWLINEIn the first chapter the author considers Mordell-Weil groups of infinite extensions of \({\mathbb{Q}}\) starting with the following necessary and sufficient condition (obtained by \textit{B. Mazur} (loc. cit.)) for the group \(E(K)\) of \(K\)-rational points of an elliptic curve \(E\) defined over \({\mathbb{Q}}\) over a Galois extension \(K\) of \({\mathbb{Q}}\) to be finitely generated: the torsion subgroup \(E(K)_{\text{tor}}\) of \(E(K)\) is finite and \({\text{rank}}(E(L))\) is bounded as \(L\) varies over all finite extensions of \({\mathbb{Q}}\subset K\). To check whether this condition is fulfilled is hard. However, the following result, which is a consequence of theorems of Kato and Rohrlich, gives an instance where this is verified. Suppose \(E\) is modular. Let \(\Sigma\) be a finite set of primes. Then \({\text{rank}}(E(L))\) is bounded as \(L\) varies over all finite abelian extensions of \({\mathbb{Q}}\) that are unramified outside \(\Sigma\). As a consequence one obtains that \(E({\mathbb{Q}}^{\text{ab}}_{\Sigma})\) is finitely generated, where \({\mathbb{Q}}^{\text{ab}}_{\Sigma}\) denotes the maximal abelian extension of \({\mathbb{Q}}\) unramified above \(\Sigma\).NEWLINENEWLINENEWLINEThe following theorem of Mazur is completely proved in the text. Let \(E\) be an elliptic curve defined over a number field \(F\), \(p\) a prime number and suppose that \(E\) has good ordinary reduction at every prime of \(F\) lying over \(p\). Suppose also that \(E(F)\) and \(\text{ Ш}_E(F)_p\) are finite, where \(\text{ Ш}_E(F)_p\) denotes the \(p\)-primary component of the Tate-Shafarevich group \(\text{ Ш}_E(F)\) of \(E/F\). Let \(K=\bigcup_nF_n\) be a \({\mathbb{Z}_p}\)-extension of \(F\). Then \({\text{rank}}(E(F_n))\) is bounded for \(n\geq 0\). The author also states two other theorems whose proofs are combinations of results by Rubin, Rohrlich, Gross-Zagier and the author, when \(F\) is an imaginary quadratic field. These results suggest the existence of more systematic patterns influencing the behavior of \(E(L)\) as \(L\) varies over finite extensions of \({\mathbb{Q}}\).NEWLINENEWLINENEWLINEChapter two is dedicated to Selmer groups. The previously cited description of the Selmer group is obtained as a consequence of the following results. Let \(K\) be an algebraic extension of \({\mathbb{Q}}\), \(v\) a prime of \(K\), \(K_v\) the completion of \(K\) at \(v\) and \(\kappa_v\) the inclusion \(E(K_v)\otimes({\mathbb{Q}}/{\mathbb{Z}})\hookrightarrow H^1(K_v,E(\overline{K}_v)_{\text{tor}})\). Suppose that \(E\) has good ordinary reduction at \(v\), and let \(\widetilde{E}\) be the reduction of \(E\) modulo \(v\). Denote by \(\pi\) the \(\text{Gal}(\overline{K}_v/K_v)\)-equivariant homomorphism \(E[{p^{\infty}}]\to\widetilde{E}[{p^{\infty}}]\) and \(\mathcal{F}[{p^{\infty}}]\subset E[{p^{\infty}}]\) the kernel of \(\pi\). This inclusion induces a map NEWLINE\[NEWLINE\varepsilon_v:H^1(K_v,\mathcal{F}[{p^{\infty}}])\to H^1(K_v,E[{p^{\infty}}]).NEWLINE\]NEWLINE The first result states that \(\text{im}(\kappa_v)=\text{im}(\varepsilon_v)_{\text{div}}\). The second result says that if \(K_v\) is an extension of \({\mathbb{Q}}_p\) of finite residue field \(k_v\) and if the profinite degree of \(K_v/{\mathbb{Q}}_p\) is divisible by \({p^{\infty}}\), then \(\text{im}(\kappa_v)=\text{im}(\varepsilon_v)\). This is true if \(K_v\) is a ramified \({\mathbb{Z}_p}\)-extension of \(F_v\), where \(F_v\) is a finite extension of \({\mathbb{Q}}_p\).NEWLINENEWLINENEWLINEChapter three describes the structure of \(\Lambda\)-modules, where \(\Lambda={\mathbb{Z}_p}[[T]]\) is the ring of formal power series over \({\mathbb{Z}_p}\) in one variable. Finally, chapter four proves Mazur's control theorem. This theorem states: if \(p\) is a prime number and \(E\) is an elliptic curve defined over a number field \(F\) having good ordinary reduction at every prime of \(F\) lying over \(p\), and \(F=\bigcup_nF_n\) is a \({\mathbb{Z}_p}\)-extension of \(F\), then the natural maps of \(p\)-primarycomponents of Selmer groups NEWLINE\[NEWLINE\text{Sel}_E(F_n)_p\to\text{Sel}_E(F_{\infty})_p^{\text{Gal}(F_{\infty}/ F_n)}NEWLINE\]NEWLINE have finite kernels and cokernels. It also says that their orders are bounded as \(n\to\infty\).NEWLINENEWLINEFor the entire collection see [Zbl 0974.00034].