Heegner points and Beilinson-Kato elements: a conjecture of Perrin-Riou (Q2078855)

Let \(E\) be an elliptic curve defined over \(\mathbb{Q}\), and let \(p \geq 3\) be a prime such that \(E\) has semistable reduction at \(p\). The Euler system constructed by \textit{K. Kato} [Astérisque 295, 117--290 (2004; Zbl 1142.11336)] provides, at the bottom layer, a global Galois cohomology class \[ \zeta_E^{\mathrm{Kato}} \in H^1(\mathbb{Q}, V_p(E)), \] where \(V_p(E) = T_p(E) \otimes_{\mathbb{Z}_p} \mathbb{Q}_p\), and \(T_p(E)\) is the \(p\)-adic Tate module of \(E\). In this article, the authors are primarily interested in the situation where \(L(E,1)=0\). In this case, Kato's explicit reciprocity law implies that the local Galois cohomology class \(\mathrm{res}_p(\zeta_E^{\mathrm{Kato}})\) belongs to \(H^1_f(\mathbb{Q}_p, V_p(E))\), and therefore defines a local point in \(E(\mathbb{Q}_p) \otimes \mathbb{Q}_p\). Perrin-Riou has conjectured that this local point is a prescribed element in the natural image of the group of rational points \(E(\mathbb{Q}) \otimes \mathbb{Q}_p\). The authors settle this conjecture by proving the existence of a global point \(P\) in \(E(\mathbb{Q})\) such that \begin{itemize} \item[1.] The point \(P\) has infinite order if and only if \(L(E,s)\) has a simple zero at \(s=1\); \item[2.] The following equality holds in \(\mathbb{Q}_p\) up to multiplication by a non-zero rational number: \[ \log_{\omega_E}(\mathrm{res}_p(\zeta_E^{\mathrm{Kato}})) = \log^2_{\omega_E}(P), \] where \(\omega_E\) is the Néron differential associated to a global minimal Weierstraß equation for \(E\), and \(\log_{\omega_E} \colon E(\mathbb{Q}_p) \to \mathbb{Q}_p\) is the corresponding \(p\)-adic Lie group logarithm. \end{itemize} The authors actually state and prove a natural generalisation of Perrin-Riou's conjecture for modular forms of weight \(k \geq 2\) and level \(\Gamma_0(N)\) which are \(p\)-semistable, which implies the above theorem in the case of elliptic curves. The point \(P\) above is obtained using a Heegner point associated to an imaginary quadratic field \(K\). In order to relate \(P\) and \(z_E^{\mathrm{Kato}}\), the authors use the Beilinson-Flach elements, which serve as a bridge between the two invariants. The Beilinson-Flach elements are global Iwasawa cohomology classes with values in the tensor product of two Coleman families of \(p\)-adic modular forms. The proof of the main theorem of the article relies on an explicit reciprocity law for these Beilinson-Flach elements, extending earlier work of \textit{M. Bertolini} et al. [J. Algebr. Geom. 24, No. 3, 569--604 (2015; Zbl 1328.11073); Camb. J. Math. 5 (1), 1--122 (2017); Res. Math. Sci. 3 (2016)].