Construction of a homomorphism concerning Euler systems for an elliptic curve
The aim of this paper is to generalize the main results contained in the work [\textit{M. Kurihara}, Invent. Math. 149, No. 1, 195--224 (2002; Zbl 1033.11028)] comparing Euler systems and what the author here calls ``admissible systems''. In the quoted work Kurihara constructed a pairing on cohomology groups attached to an elliptic curve \(E/\mathbb{Q}\) with values in group rings and studies the behavior of this pairing in the cyclotomic \(\mathbb{Z}_p\)-extension of \(\mathbb{Q}\) where \(p\) is a prime of good, supersingular reduction for \(E\). By fixing one explicit argument, one deduces from the pairing a group homomorphism \[ \mathcal{P}_{N}:H^1(\mathbb{Q}_{p}(\zeta_N),V_pE)\longrightarrow \mathbb{Q}_p[\mathrm{Gal}(\mathbb{Q}(\zeta_N)/\mathbb{Q})]\tag{1} \] and the main goal of the paper under review is to compare the Euler system belonging to the left-hand side of (1) constructed by \textit{K. Kato} in [Astérisque 295, 117--290 (2004; Zbl 1142.11336)] to the system of modular elements in the group rings appearing in (1) constructed by \textit{B. Mazur} and \textit{J. Tate} [ Duke Math. J. 54, 711--750 (1987; Zbl 0636.14004)], together with some integrality property of the above morphism: this is done in a greater generality than Kurihara's but with similar techniques. More precisely, let \(E/\mathbb{Q}\) be an elliptic curve and let \(p\) be an odd prime of good, supersingular reduction for \(E\). Denoting by \(T_pE\) the \(p\)-adic Tate module of \(E\) and by \(V_pE\) the representation \(T_pE\otimes\mathbb{Q}_p\), \textit{K. Kato} defined in [loc. cit.] an Euler system \((z_N)_{N\geq 1}\) of elements \(z_N\in H_{\acute{e}t}^1(\mathbb{Z}[\mu_N,1/S],V_pE)\). This means that, for every prime number \(q\), the following relation holds \[ \text{Norm}_{qN/N}(z_{qN})=\begin{cases} z_N & q\mid N \\F_q(\text{Frob}_q^{-1})z_n & q \nmid N,\end{cases} \tag{2} \] where \(F_q(T)=1-(a_q/q)T+(\varepsilon_q/q)T^2\in\mathbb{Q}[T]\) is the polynomial whose coefficient \(\varepsilon_q\) is \(1\) or \(0\) according as \(q\) has good or bad reduction, and \(a_q\) is the \(q\)-th Fourier coefficient of the normalized cusp form corresponding to \(E\). On the other hand, for every integer \(N\), let \(\mathcal{G}_N\) be the Galois group \(\mathrm{Gal}(\mathbb{Q}(\zeta_N)/\mathbb{Q})\); \textit{B. Mazur} and \textit{J. Tate} defined in [loc. cit.] what they call ``modular elements'', that are elements \(\theta_N\in\mathbb{Q}[\mathcal{G}_N]\) which satisfy \[ \pi_{qN/N}(\theta_{qN})=\begin{cases} a_q\theta_N-\varepsilon_q\nu_{N/\frac{N}{q}}(\theta_\frac{N}{q}) & q\mid N \\F_q(\mathrm{Frob}_q^{-1})z_n & q \nmid N\end{cases}\tag{3} \] where \(\pi_{qN/N}:\mathbb{Q}[\mathcal{G}_{qN}]\rightarrow\mathbb{Q}[\mathcal{G}_N]\) is the natural projection and \(\nu_{N/\frac{N}{q}}:\mathbb{Q}[\mathcal{G}_\frac{N}{q}]\rightarrow\mathbb{Q}[\mathcal{G}_N]\) is the trace map. Although the definitions of the two systems are different in nature, both objects can be ``evaluated'' against characters \(\chi\) of \(\mathcal{G}_N\) to interpolate the special values \(L(E,\chi,1)\) compare Theorem 12.5 of [\textit{K. Kato}, loc. cit.] with Section 1.4 of [\textit{B. Mazur} and \textit{J. Tate}, loc. cit.]. One of the main results of the paper under review is the Theorem {3.6} For every \(N\geq 1\) we have \(\mathcal{P}_{N}(z_N)=\theta_N\). More generally, the author calls a system of elements in \(\mathbb{Q}_p[\mathcal{G}_N]\) for all \(N\geq 1\) satisfying the axioms (3) an ``admissible system''. Then he proves Theorem {3.4} If \((w_N)_{N\geq 1}\) is an Euler system, then \(\big(\mathcal{P}_{N}(w_N)\big)_{N\geq 1}\) is an admissible system. Moreover, in Theorem 4.1 the author proves an integrality statement regarding the above theorem: namely, he shows that if the \(\mathbb{F}_p(\zeta_N)\)-rational \(p\)-torsion of the reduction of \(E\pmod{p}\) is trivial, then Theorem \(3.4\) holds also when replacing \(V_pE\) with \(T_pE\) (\textit{resp.} \(\mathbb{Q}[\mathcal{G}_N]\) with \(\mathbb{Z}_p[\mathcal{G}_N]\)) in the above definitions of Euler (resp. admissible) system. The paper falls in four sections, the \(0\)-th being a general introduction. In the first section the author fixes notations and defines Euler and admissible systems. In section \(2.\) the map \(\mathcal{P}_N\) is constructed and its main properties are studied (observe that in [\textit{M. Kurihara} and \textit{R. Pollack}, \(L\)-functions and Galois representations. Lond. Math.Soc. Lect. Note Series 320, 300--332 (2007; Zbl 1148.11029)] the map \(\mathcal{P}_{N}\) is what in the work under review is called \(\mathcal{P}_{p^N}\)) and in section \(3.\) the comparison between the notions of Euler and admissible system is performed. Finally, in the fourth section, the integral version of the map is analyzed through a divisibility property involving the logarithm of the formal group \(\hat{E}\) and proven in [\textit{T. Honda}, J. Math. Soc. Japan 22, 213--246 (1970; Zbl 0202.03101)].
scientific article; zbMATH DE number 5604247