On the Tate-Shafarevich groups over cyclotomic fields of an elliptic curve with supersingular reduction. I.

*(English)*Zbl 1033.11028The Tate-Shafarevich group \(\text Ш(E,K)\) of an elliptic curve \(E\) over a number field \(K\) measures the obstruction that occurs if one wants to compute the rank \(r\) of \(E\) over \(K\). More precisely, one has an exact sequence
\[
0 \rightarrow E'(K)/\phi(E/K) \rightarrow S^{(\phi)}(E/K) \rightarrow \text Ш(E/K)[\phi] \rightarrow 0,
\]
where \(\phi: E/K \rightarrow E'/K\) is an isogeny of \(E\) over \(K\) to another elliptic curve \(E'\) over \(K\). Here, \(E'\) is \(E\) and \(\phi\) is essentially the multiplication by a prime \(p \in \mathbf P\). Whereas the Selmer group \(S^{(\phi)}(E/K)\) is computable, almost nothing is known about the Tate-Shafarevich group \(\text Ш(E/K)\). This group is conjectured to be finite, but even its subgroup \(\text Ш(E/K)[\phi]\) is ascertained to be finite only in some special cases.

We denote the \(p\)-primary part of the Tate-Shafarevich group by \(\text Ш(E/K)[p]\). It is adaquate to apply Iwasawa theory to the elliptic curves \(E\) over say the rational field \(K=\mathbb{Q}\) and consider the asymptotic behaviour of \(\text Ш(E/K)[p]\) over the \(n\)th cyclotomic field \(K_n/\mathbb{Q}\) as \(n \to \infty\). Remember that \(K_n\) is a subfield of the cyclotomic \(\mathbb{Z}_p\)-extension \(K_{\infty}\) of \(\mathbb{Q}\). It has degree \[ [K_n:\mathbb{Q}] = p^n. \] If the \(p\)-primary part \(\text Ш(E/K_n)[p]\) is finite and has order \(p^{e_n}\), then there are numbers \(\lambda,\mu \in \mathbb{Z}_{\geq 0}\) and \(\nu \in \mathbb{Z}\) such that \[ e_n = \lambda n+\mu p^n+\nu \] for sufficiently large numbers \(n \in \mathbb{N}\) provided that the curve \(E\) has potentially ordinary reduction modulo \(p\). Of course, this is exactly the relation which holds for the \(p\)-class number of the field \(K_n\) and gives thus a justification for applying Iwasawa theory to elliptic curves. Iwasawa’s result is not known for elliptic curves \(E\) that do not have potentially ordinary reduction modulo \(p\). (However, the author, after having finished writing his article, found out that in the preprints of R. Pollak, “On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime” and of B. Perrin-Riou, “Arithmétique des courbes elliptiques a réduction supersingulière en \(p\)” some progress has been made also in this case.)

The main results (see Theorem 0.1) obtained in this important paper are as follows.

Let \(E\) be an elliptic curve having supersingular reduction at an odd prime \(p \in {\mathbf P}\). Suppose further that \[ \text{ord}_p(L(E,1)/\Omega_E) = 0,\tag{1} \] where \(L(E,s)\) denotes the \(L\)-function of the curve \(E\) over \(\mathbb{Q}\) and \(\Omega_E\) is the Néron period of \(E\) (so that the quotient \(L(E,\lambda)/\Omega_E\) has nothing to do with the prime \(p\)), \[ p \nmid \text{Tam}(E),\tag{2} \] where \(\text{Tam}(E) = \prod_{\ell \in {\mathbf P}} c_{\ell}\) is the product over all primes \(\ell \in {\mathbf P}\) of the Tamagawa numbers \[ c_{\ell} := [E({\mathbb{Q}}_{\ell}):E_0({\mathbb{Q}}_{\ell})] \] (that is to say: \(p\) does not divide the product \(\text{Tam}(E)\)).

(3) The Galois action \[ \rho_E[p]:G_{\mathbb{Q}} \rightarrow \operatorname{Aut}(E[p]) \] on the \(p\)-torsion group \(E[p]\) of \(E\) is surjective (here \(\operatorname{Aut}(E[p]) \cong Gl_2(\mathbb{F}_p))\). Under these three assumptions the Pontrjagin dual \((\text Ш(E/K_{\infty})[P])^{\vee}\), which is considered here instead of the \(p\)-Tate-Shafarevich group \(\text Ш(E/K_{\infty})[p]\), itself is isomorphic to \(\Lambda\) as a \(\Lambda\)-module, where \(\Lambda := \mathbb{Z}_p[[ \text{Gal}(K_{\infty}/\mathbb{Q})]]\). (In the case in which \(E\) has ordinary reduction at \(p\), a conjecture of Mazur states that the Pontrjagin dual \((\text{Sh}(E/K_{\infty}) [p])^{\vee}\) is a torsion \(\Lambda\)-module. This remarkable conjecture was proved by Rubin (in the CM-case) and Kato (in the non-CM-case)).

Furthermore, the rational point groups \(E(K_n)\), \(n \in {\mathbb{Z}}_{\geq 0}\), have all rank zero, and the \(p\)-primary parts \(\text Ш(E/K_n)[p]\) are finite.

(These results are said to be easily derivable also from a deep unpublished theorem of Kato.)

Moreover, it is shown in this paper that if \(p^{e_n}\) is the order of \(\text Ш(E/K_n)[p]\) and if the rational numbers \[ \lambda = -\frac{1}{2} \quad \text{and}\quad \mu = \frac{p}{p^2-1} \] are admitted for \(\lambda\) and \(\mu\), then one has \[ e_n = \lfloor \lambda n+\mu ^n \rfloor, \] where \(\lfloor x \rfloor\) is the greatest integer below \(x \in \mathbb{R}\). Finally, the structure of \(\text Ш(E/K_n)[ p]\) as a finite Abelian group can be completely given (see Theorem 7.4). A conjecture of Mazur and Tate asserting that a certain modular element is in the Fitting ideal of the Pontrjagin dual of the Selmer group \(S(E/K_n)\) is generalized here in such a way that it gives more information than the usual Iwasawa main conjecture. This generalization follows from the last part of the theorem of this paper.

An essential ingredient of the proof of Theorem 0.1 is Mazur’s control theorem. Unfortunately it does not hold in this case for the whole Selmer group \(S(E/K_{\infty})\). But it holds for a certain subgroup \(S_0(E/K_{\infty})\) (see definition 4.1) and this suffices for the present purpose. In addition, the proof of the theorem makes extensive use of the work of the authors, Coates, Greenberg, Kato, Mazur, Rubin, Tate and others so that the details of proof have not been checked by the reviewer.

For part II of this paper the author announces that he will study the behaviour of \(\text Ш(E/K_n)[p]\) without the assumption (1).

We denote the \(p\)-primary part of the Tate-Shafarevich group by \(\text Ш(E/K)[p]\). It is adaquate to apply Iwasawa theory to the elliptic curves \(E\) over say the rational field \(K=\mathbb{Q}\) and consider the asymptotic behaviour of \(\text Ш(E/K)[p]\) over the \(n\)th cyclotomic field \(K_n/\mathbb{Q}\) as \(n \to \infty\). Remember that \(K_n\) is a subfield of the cyclotomic \(\mathbb{Z}_p\)-extension \(K_{\infty}\) of \(\mathbb{Q}\). It has degree \[ [K_n:\mathbb{Q}] = p^n. \] If the \(p\)-primary part \(\text Ш(E/K_n)[p]\) is finite and has order \(p^{e_n}\), then there are numbers \(\lambda,\mu \in \mathbb{Z}_{\geq 0}\) and \(\nu \in \mathbb{Z}\) such that \[ e_n = \lambda n+\mu p^n+\nu \] for sufficiently large numbers \(n \in \mathbb{N}\) provided that the curve \(E\) has potentially ordinary reduction modulo \(p\). Of course, this is exactly the relation which holds for the \(p\)-class number of the field \(K_n\) and gives thus a justification for applying Iwasawa theory to elliptic curves. Iwasawa’s result is not known for elliptic curves \(E\) that do not have potentially ordinary reduction modulo \(p\). (However, the author, after having finished writing his article, found out that in the preprints of R. Pollak, “On the \(p\)-adic \(L\)-function of a modular form at a supersingular prime” and of B. Perrin-Riou, “Arithmétique des courbes elliptiques a réduction supersingulière en \(p\)” some progress has been made also in this case.)

The main results (see Theorem 0.1) obtained in this important paper are as follows.

Let \(E\) be an elliptic curve having supersingular reduction at an odd prime \(p \in {\mathbf P}\). Suppose further that \[ \text{ord}_p(L(E,1)/\Omega_E) = 0,\tag{1} \] where \(L(E,s)\) denotes the \(L\)-function of the curve \(E\) over \(\mathbb{Q}\) and \(\Omega_E\) is the Néron period of \(E\) (so that the quotient \(L(E,\lambda)/\Omega_E\) has nothing to do with the prime \(p\)), \[ p \nmid \text{Tam}(E),\tag{2} \] where \(\text{Tam}(E) = \prod_{\ell \in {\mathbf P}} c_{\ell}\) is the product over all primes \(\ell \in {\mathbf P}\) of the Tamagawa numbers \[ c_{\ell} := [E({\mathbb{Q}}_{\ell}):E_0({\mathbb{Q}}_{\ell})] \] (that is to say: \(p\) does not divide the product \(\text{Tam}(E)\)).

(3) The Galois action \[ \rho_E[p]:G_{\mathbb{Q}} \rightarrow \operatorname{Aut}(E[p]) \] on the \(p\)-torsion group \(E[p]\) of \(E\) is surjective (here \(\operatorname{Aut}(E[p]) \cong Gl_2(\mathbb{F}_p))\). Under these three assumptions the Pontrjagin dual \((\text Ш(E/K_{\infty})[P])^{\vee}\), which is considered here instead of the \(p\)-Tate-Shafarevich group \(\text Ш(E/K_{\infty})[p]\), itself is isomorphic to \(\Lambda\) as a \(\Lambda\)-module, where \(\Lambda := \mathbb{Z}_p[[ \text{Gal}(K_{\infty}/\mathbb{Q})]]\). (In the case in which \(E\) has ordinary reduction at \(p\), a conjecture of Mazur states that the Pontrjagin dual \((\text{Sh}(E/K_{\infty}) [p])^{\vee}\) is a torsion \(\Lambda\)-module. This remarkable conjecture was proved by Rubin (in the CM-case) and Kato (in the non-CM-case)).

Furthermore, the rational point groups \(E(K_n)\), \(n \in {\mathbb{Z}}_{\geq 0}\), have all rank zero, and the \(p\)-primary parts \(\text Ш(E/K_n)[p]\) are finite.

(These results are said to be easily derivable also from a deep unpublished theorem of Kato.)

Moreover, it is shown in this paper that if \(p^{e_n}\) is the order of \(\text Ш(E/K_n)[p]\) and if the rational numbers \[ \lambda = -\frac{1}{2} \quad \text{and}\quad \mu = \frac{p}{p^2-1} \] are admitted for \(\lambda\) and \(\mu\), then one has \[ e_n = \lfloor \lambda n+\mu ^n \rfloor, \] where \(\lfloor x \rfloor\) is the greatest integer below \(x \in \mathbb{R}\). Finally, the structure of \(\text Ш(E/K_n)[ p]\) as a finite Abelian group can be completely given (see Theorem 7.4). A conjecture of Mazur and Tate asserting that a certain modular element is in the Fitting ideal of the Pontrjagin dual of the Selmer group \(S(E/K_n)\) is generalized here in such a way that it gives more information than the usual Iwasawa main conjecture. This generalization follows from the last part of the theorem of this paper.

An essential ingredient of the proof of Theorem 0.1 is Mazur’s control theorem. Unfortunately it does not hold in this case for the whole Selmer group \(S(E/K_{\infty})\). But it holds for a certain subgroup \(S_0(E/K_{\infty})\) (see definition 4.1) and this suffices for the present purpose. In addition, the proof of the theorem makes extensive use of the work of the authors, Coates, Greenberg, Kato, Mazur, Rubin, Tate and others so that the details of proof have not been checked by the reviewer.

For part II of this paper the author announces that he will study the behaviour of \(\text Ш(E/K_n)[p]\) without the assumption (1).

Reviewer: Horst G. Zimmer (Saarbrücken)

##### MSC:

11G05 | Elliptic curves over global fields |

11R23 | Iwasawa theory |

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

11G07 | Elliptic curves over local fields |

14F20 | Étale and other Grothendieck topologies and (co)homologies |

14F40 | de Rham cohomology and algebraic geometry |

14G25 | Global ground fields in algebraic geometry |

14H25 | Arithmetic ground fields for curves |

14H52 | Elliptic curves |

14L05 | Formal groups, \(p\)-divisible groups |