# zbMATH — the first resource for mathematics

Kato’s Euler system and rational points on elliptic curves. I: A $$p$$-adic Beilinson formula. (English) Zbl 1317.11071
For 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.

##### 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
Full Text: