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On the \(p\)-adic Birch, Swinnerton-Dyer conjecture for non-semistable reduction. (English) Zbl 1013.11029
Let \(E/{\mathbb{Q}}\) be an elliptic curve without complex multiplication. The main result of the paper is to prove an analogue of Kato’s result on the \(p\)-adic Birch and Swinnerton-Dyer conjecture (originally done in the case where \(E\) is ordinary at \(p\)) in the case where \(E\) is potentially ordinary at \(p\). The main difficulty is to show that Kato’s Euler systems really map to the \(p\)-adic \(L\)-function.
Let \(F_{\infty}\) be the \({\mathbb{Z}}_p\) extension of \({\mathbb{Q}}\), \(\Gamma=\text{Gal}(F_{\infty}/{\mathbb{Q}})\), \(\text{Gal}({\mathbb{Q}}(\mu_{p^{\infty}})/{\mathbb{Q}})=\Gamma\times\Delta\) and \(\text{Sel}(E/F_{\infty})\) the corresponding Selmer group. It is known [B. Mazur, Invent. Math. 18, 183-266 (1972; Zbl 0245.14015)] that the Pontryagin dual \(\text{Sel}(E/F_{\infty})^{\wedge}\) of \(\text{Sel}(E/F_{\infty})\) is a compact, finitely generated module over the Iwasawa algebra \({\mathbb{Z}}_p[[\Gamma]]\). In the 1970’s Mazur conjectured that \(\text{Sel}(E/F_{\infty})^{\wedge}\) encoded the formulae predicted by Birch and Swinnerton-Dyer for the Hasse-Weil \(L\)-function \(L(E,s)\).
In the current paper the author makes the following hypothesis: either \(E\) is potentially ordinary at \(p\geq 5\) or \(E\) is potentially ordinary at \(p=3\) with semistable reduction over a quadratic extension of \({\mathbb{Q}}_3\). This hypothesis implies the existence of a character \(\varepsilon\) of conductor \(p\) such that \(\text{Frob}_p^{-1}\in G_{{\mathbb{Q}}_p/I_p}\) has a unique eigenvalue \(\alpha\) on the subspace \(H^1_{\text{ét}}(E_{\overline{\mathbb{Q}}},\overline{\mathbb{Q}}_{\ell})\otimes\varepsilon^{-1}\) fixed by the inertia group \(I_p\) at a prime \(\ell\neq p\). It was also proved in [D. Delbourgo, Compos. Math. 113, 123-154 (1998; Zbl 0911.11050)] that there exists a unique element \(L_p^{an}\in{\mathbb{Z}}_p[[\Gamma\times\Delta]][\mu_p,p^{-1}]\) such that for all characters \(\chi\neq\varepsilon\) of \(p\)-power conductor \[ \chi(L_p^{an})=\frac{p^m}{\alpha^m\sum_{n=1}^{p^m}\chi^{-1}\varepsilon(n)\exp(2\pi in/p^m)}\times\frac{L(E,\chi^{-1},1)}{\Omega_E^{\text{sign}(\chi)}}, \] where \(m=\text{ord}_p(\text{cond}(\chi\varepsilon^{-1}))\), the right hand side is considered \(p\)-adically via a fixed embedding \(\overline{\mathbb{Q}}\hookrightarrow\overline{\mathbb{Q}}_p\), \(\Omega^+_E\) (resp. \(\Omega^-_E\)) denotes the real (resp. imaginary) period of a Néron differential associated to a minimal Weierstrass equal of \(E\) over \({\mathbb{Z}}\).
The following quantities appear in the nonsemistable \(p\)-adic Birch and Swinnerton-Dyer conjecture. Aside from the Tate-Shafarevich group Ш, the Mordell-Weil group and the Tamagawa factors \(\text{Tam}_{\ell,{\mathbb{Q}}}\), the \(p\)-adic regulator \(\text{reg}_p(E/{\mathbb{Q}})\) (expressed in terms of the \(p\)-adic pairing \(\langle , \rangle_{p,{\mathbb{Q}}}\)) and the \(\ell\)-invariant \(\ell_p(E)\).
The main theorem states that under the above hypothesis
(A) The Iwasawa module \(\text{Sel}(E/F_{\infty})^{\wedge}\) is \({\mathbb{Z}}_p[[\Gamma]]\)-torsion.
(B) Denote \(L_p^{\text{alg}}\) a generator of the characteristic ideal of \(\text{Sel}(E/F_{\infty})^{\wedge}\), \(\kappa\) a cyclotomic character at \(p\) and \(s\in{\mathbb{Z}}_p\). If \(\text{ Ш}(E/{\mathbb{Q}})_{p^{\infty}}\) is infinite or \(\langle , \rangle_{p,{\mathbb{Q}}}\) is degenerate then \(\text{ord}_{s=0}(\kappa^s(L_p^{\text{alg}}))>r_E=\text{rank}(E({\mathbb{Q}}))\). If \(\text{ Ш}(E/{\mathbb{Q}})_{p^{\infty}}\) is finite and \(\langle , \rangle_{p,{\mathbb{Q}}}\) is non-degenerate the previous inequality is indeed an equality and \[ \frac 1{r!}\left(\frac{d^{r_E}}{ds^{r_E}} \kappa^s(L_p^{\text{alg}})\right)_{s=0} \sim\;\ell_p(E/{\mathbb{Q}}) \times\frac{\text{reg}_p(E/{\mathbb{Q}})\times\#\text{ Ш}(E/{\mathbb{Q}})_{p^{\infty}}\times\prod_{\ell}\text{Tam}_{\ell,{\mathbb{Q}}}}{\#E({\mathbb{Q}})_{\text{tor}}^2} , \] where \(a\sim b\) means \(\text{ord}_p(a)=\text{ord}_p(b)\).
(C) \(L_p^{\text{alg}}\) divides in \({\mathbb{Z}}_p[[\Gamma]][\mu_p,p^{-1}]\) the branch of \(L_p^{an}\) fixed by \(\Lambda\).
(D) \(\text{ord}_{s=0}(\kappa^s(L_p^{\text{alg}}))\geq r_E\).
In the potentially supersingular case (A) is false because \(\text{Sel}(E/F_{\infty})^{\wedge}\) is often pseudoisomorphic to \({\mathbb{Z}}_p [[\Gamma]]\) which is free. (B) plus (C) will imply (D). Formulae of type in (B) were first proven in the good ordinary case by Perrin-Riou and Schneider and in the multiplicative case by Jones.

MSC:
11G40 \(L\)-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
11G07 Elliptic curves over local fields
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