Kato’s Euler system and rational points on elliptic curves. I: A \(p\)-adic Beilinson formula.

*(English)*Zbl 1317.11071For a modular elliptic curve \(E/\mathbb Q\), Kato’s construction of an Euler system arising from \(p\)-adic families of Beilinson elements in the \(K\)-theory of modular curves in [K. Kato, in: Cohomologies \(p\)-adiques et applications arithmétiques (III). Paris: Société Mathématique de France. 117–290 (2004; Zbl 1142.11336)] yields a global class \(\kappa\in H^1(\mathbb Q,V_p(E))\), where \(V_p(E)\) is the \(p\)-adic Galois representation attached to \(E\). Kato’s reciprocity law implies that \(\kappa\) is crystalline, hence belongs to the \(p\)-adic Selmer group of \(E\), precisely when the Hasse-Weil \(L\)-series \(L(E,s)\) vanishes at \(s = 1\). In this case, B. Perrin-Riou [Ann. Inst. Fourier 43, No. 4, 945–995 (1993; Zbl 0840.11024)] conjectures that the image \(\mathrm{res}_p(\kappa)\) in \(H_f^1(\mathbb Q_p,V_p(E))\) is non-zero if and only if \(L'(E,1)\) is non-zero, and predicts a precise relation between the logarithm of \(\mathrm{res}_p(\kappa)\) and the formal group logarithm of a global point in \(E(\mathbb Q)\). The present article is the first in a series of three, whose ultimate goal is the demonstration of Perrin-Riou’s conjecture. The main step here is a proof of a \(p\)-adic Beilinson formula relating the syntomic regulators (in the sense of R. Coleman and E. de Shalit [Invent. Math. 93, No. 2, 239–266 (1988; Zbl 0655.14010)] and A. Besser [Isr. J. Math. 120, Part B, 291–334 (2000; Zbl 1001.19003); ibid. 120, Part B, 335–359 (2000; Zbl 1001.19004)]) of certain distinguished elements in the \(K\)-theory of modular curves to the special values at integer points \(\geq 2\) of the Mazur and Swinnerton-Dyer \(p\)-adic \(L\)-function attached to a cusp form \(f\) of weight 2. The authors’ proof is independent from Kato’s reciprocity law (which can be re-derived from it), and makes heavy use of results and techniques of H. Darmon and V. Rotger [Ann. Sci. Éc. Norm. Supér. (4) 47, No. 4, 779–832 (2014; Zbl 1356.11039)]. It is based on a direct evaluation of the \(p\)-adic Rankin \(L\)-function attached to a Hida family interpolating \(f\). From the factorization of this \(p\)-adic Rankin \(L\)-function into a product of two Mazur-Kitagawa \(p\)-adic \(L\)-functions, the \(p\)-adic Beilinson formula then follows. In strong analogy with the complex formula, it relates in a precise explicit way the product \(L_p(f,\chi_1, 2)\cdot L^*(f,\chi_2,1)\) to the regulator \(\mathrm{reg}_p\{E{_l,\chi} , E_l(\chi_1,\chi_2)\}(\eta^{\mathrm{ur}}_f)\). Notations: \(L_p(f,\chi_2,1) = \tau(\chi_2)\Omega_f^\varepsilon L(f,\chi_2,1)\), where \(\tau(\chi_2)\) is a Gauss sum, \(\varepsilon = \chi_2(-1)\) and \(\Omega_f^\varepsilon\) is a complex period; \(\chi_1, \chi_2\) is a pair of primitive Dirichlet characters with coprime conductors \(N_1\), \(N_2\), \(N := N_1\cdot N_2\), \(\chi^{-1}= \chi_1\cdot \chi_2\) is even; \(E_l(\chi_1, \chi_2)\) is an Eisenstein series, and \(E_l(\chi)\) is a certain normalized Eisenstein series as in [H. Hida, Elementary theory of \(L\)-functions and Eisenstein series. Cambridge: Cambridge University Press (1993; Zbl 0942.11024)]; \(\eta^{\mathrm{ur}}_f\) is a certain element constructed from an antiholomorphic differential as in Darmon-Rotger [loc. cit.], corollary 2.13 given modular units \(u_1, u_2\) in \(O(Y_1(N))^*\), one has the Steinberg symbol \(\{u_1, u_2 \}\in K_2(Y_1(N)),\) and one can construct the \(p\)-adic regulator \(\mathrm{reg}_p\{u_1, u_2 \}\in H^1_{dR}(X_1(N))\) of Coleman-de Shalit, in the description given by Besser [Zbl 1001.19004]; then \(\mathrm{reg}_p\{E_l,\chi,E_l(\chi_1, \chi_2)\}(\eta^{ur}_f)\) is by definition the value of a Poincaré pairing. Note that one can choose \(\chi_2\) so that the factor \(L^*(f,\chi_2,1)\) does not vanish, in which case the \(p\)-adic Beilinson formula above expresses the value of \(L_p(f, \chi_1, l)\) at a point outside the range of classical interpolation. When combined with the explicit relation between syntomic regulators and \(p\)-adic étale cohomology, it leads also to an alternate definition of the \(p\)-adic étale regulator, and gives an alternate proof (independent of Kato’s reciprocity law) of the results of Brunault (for \(l = 2\)) and Gealy (for \(l \geq 2\)) on the special values of the Mazur and Swinnerton-Dyer \(p\)-adic \(L\)-function.

Reviewer: Thong Nguyen Quang Do (Besançon)

##### MSC:

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

11G05 | Elliptic curves over global fields |

11F67 | Special values of automorphic \(L\)-series, periods of automorphic forms, cohomology, modular symbols |

11R23 | Iwasawa theory |

##### Citations:

Zbl 1142.11336; Zbl 0840.11024; Zbl 0942.11024; Zbl 1001.19004; Zbl 0655.14010; Zbl 1001.19003; Zbl 1356.11039
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\textit{M. Bertolini} and \textit{H. Darmon}, Isr. J. Math. 199, Part A, 163--188 (2014; Zbl 1317.11071)

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##### References:

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