Iwasawa theory of elliptic curves at supersingular primes over \(\mathbb Z_p\)-extensions of number fields.

*(English)*Zbl 1114.11053The goal of the paper is to study the Iwasawa theory of an elliptic curve \(E\) at a supersingular prime \(p\) along an arbitrary \(\mathbb{Z}_p\)-extension of a number field \(K\) in the case when \(p\) splits completely in \(K\). Unlike the case of an ordinary prime, in that case the trace map on the formal group \(\widehat{E}\) of \(E/\mathbb{Q}\) is not surjective along a ramified \(\mathbb{Z}_p\)-extension. Accordingly, the authors first investigate the interplay of the trace map and Galois theory and prove a control theorem that entertains the formal group of \(E\). This allows to obtain the information about the Selmer group of \(E\) from the local information regarding \(\widehat{E}\). The authors generalize the description of the Galois module structure of \(\widehat{E}(K_n)\) (\(K_n\) runs over the local cyclotomic \(\mathbb{Z}_p\)-extension of \(\mathbb{Q}_p\)) given in [S. Kobayashi. Invent. Math. 152, No. 1, 1–36 (2003; Zbl 1047.11105)], and use it to describe the kernel and cokernel of the trace map. The description of \(\widehat{E}\) is obtained in terms of generators and relations. The generators satisfy a compatibility property as the level varies; this allows to develop the Iwasawa theory in the supersingular case.

Under the assumption that \(p\) splits completely in a number field \(K\) the authors analyze \(\mathbb{Z}_p\)-extension of \(K\) and introduce algebraic \(p\)-adic \(L\)-functions. Then using the ideas of Kobayashi [loc. cit.] and R. Pollack [Exp. Math. 12, No. 2, 155–186 (2003; Zbl 1074.11061)] they intriduce plus/minus \(p\)-adic \(L\)-functions that lie in the Iwasawa algebra, and associate to them plus/minus \(\mu\) and \(\lambda\)-invariants. In the case where all these \(\mu\) and \(\lambda\)-invariants are zero, the authors prove that \(E(K_n)\) and \(\text{ Ш}(E/K_n)[p^\infty]\) are finite for all \(n\), describe the Galois structure of \(\text{ Ш}(E/K_n)[p^\infty]\), and obtain the formulas for its size.

The final part of the paper is devoted to the study of the arithmetic of \(E\) along the extension \(K_\infty/K\). It is shown that if the coranks of the Selmer groups are unbounded along this extension, then the algebraic \(p\)-adic \(L\)-functions vanish, the restricted Selmer groups are not cotorsion, and their coranks control the rate of growth of of the Selmer groups at each finite level. In the case where these coranks remain bounded, the authors prove that \(L\)-functions are nonzero, the restricted Selmer groups are cotorsion, and give asymptotic formulas for the growth of Selmer groups in terms of the Iwasawa invariants of the plus/minus \(p\)-adic \(L\)-functions thereby generalizing the results of B. Perrin-Riou [Exp. Math. 12, No. 2, 155–186 (2003; Zbl 1061.11031)].

Under the assumption that \(p\) splits completely in a number field \(K\) the authors analyze \(\mathbb{Z}_p\)-extension of \(K\) and introduce algebraic \(p\)-adic \(L\)-functions. Then using the ideas of Kobayashi [loc. cit.] and R. Pollack [Exp. Math. 12, No. 2, 155–186 (2003; Zbl 1074.11061)] they intriduce plus/minus \(p\)-adic \(L\)-functions that lie in the Iwasawa algebra, and associate to them plus/minus \(\mu\) and \(\lambda\)-invariants. In the case where all these \(\mu\) and \(\lambda\)-invariants are zero, the authors prove that \(E(K_n)\) and \(\text{ Ш}(E/K_n)[p^\infty]\) are finite for all \(n\), describe the Galois structure of \(\text{ Ш}(E/K_n)[p^\infty]\), and obtain the formulas for its size.

The final part of the paper is devoted to the study of the arithmetic of \(E\) along the extension \(K_\infty/K\). It is shown that if the coranks of the Selmer groups are unbounded along this extension, then the algebraic \(p\)-adic \(L\)-functions vanish, the restricted Selmer groups are not cotorsion, and their coranks control the rate of growth of of the Selmer groups at each finite level. In the case where these coranks remain bounded, the authors prove that \(L\)-functions are nonzero, the restricted Selmer groups are cotorsion, and give asymptotic formulas for the growth of Selmer groups in terms of the Iwasawa invariants of the plus/minus \(p\)-adic \(L\)-functions thereby generalizing the results of B. Perrin-Riou [Exp. Math. 12, No. 2, 155–186 (2003; Zbl 1061.11031)].

Reviewer: Vasyl I. Andriychuk (Lviv)

##### MSC:

11G05 | Elliptic curves over global fields |

11G07 | Elliptic curves over local fields |

11G40 | \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture |

11R23 | Iwasawa theory |

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