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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
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[1] A. A. Beilinson, Higher regulators of modular curves, in Applications of Algebraic K-theory to Algebraic Geometry and Number Theory, Part I, II (Boulder, Colo., 1983), Contemporary Mathematics, Vol. 55, American Mathematical Society, Providence, RI, 1986, pp. 1–34.
[2] M. Bertolini, H. Darmon and K. Prasanna, Generalised Heegner cycles and p-adic Rankin L-series, Duke Mathematical Journal 162 (2013), 1033–1148. · Zbl 1302.11043
[3] M. Bertolini and H. Darmon, Hida families and rational points on elliptic curves, Inventiones Mathematicae 168 (2007), 371–431. · Zbl 1129.11025
[4] M. Bertolini and H. Darmon, Kato’s Euler system and rational points on elliptic curves II: The explicit reciprocity law, in preparation. · Zbl 1317.11071
[5] M. Bertolini and H. Darmon, Kato’s Euler system and rational points on elliptic curves III: The conjecture of Perrin-Riou, in preparation. · Zbl 1317.11071
[6] M. Bertolini, H. Darmon and V. Rotger, Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series, submitted. · Zbl 1325.14034
[7] A. Besser, Syntomic regulators and p-adic integration. I. Rigid syntomic regulators, in Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), Israel Journal of Mathematics 120 (2000), 291–334. · Zbl 1001.19003
[8] A. Besser, Syntomic regulators and p-adic integration. II. K 2 of curves, in Proceedings of the Conference on p-adic Aspects of the Theory of Automorphic Representations (Jerusalem, 1998), Israel Journal of Mathematics 120 (2000), 335–359. · Zbl 1001.19004
[9] K. Bannai and G. Kings, p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure, American Journal of Mathematics 132 (2010), 1609–1654. · Zbl 1225.11075
[10] S. J. Bloch, Higher Regulators, Algebraic K-theory, and Zeta Functions of Elliptic Curves, CRM Monograph Series, Vol. 11, American Mathematical Society, Providence, RI, 2000. · Zbl 0958.19001
[11] F. Brunault, Valeur en 2 de fonctions L de formes modulaires de poids 2: théorème de Beilinson explicite, Bulletin de la Société Mathématique de France 135 (2007), 215–246. · Zbl 1207.11059
[12] F. Brunault, Régulateurs p-adiques explicites pour le K 2 des courbes elliptiques, Actes de la Conférence ”Fonctions L et Arithmétique”, Publ. Math. Besançon Algèbre Théorie Nr., Lab. Math. Besançon, Besançon, 2010, pp. 29–57.
[13] R. F. Coleman and E. de Shalit, p-adic regulators on curves and special values of p-adic L-functions, Inventiones Mathematicae 93 (1988), 239–266. · Zbl 0655.14010
[14] R. F. Coleman, A p-adic Shimura isomorphism and p-adic periods of modular forms, in p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemporary Mathematics, Vol. 165, American Mathematical Society, Providence, RI, 1994, pp. 21–51.
[15] P. Colmez, Fonctions L p-adiques, Séminaire Bourbaki, Vol. 1998/99, Astérisque No. 266 (2000), Exp. No. 851, 3, 21–58.
[16] P. Colmez, La conjecture de Birch et Swinnerton-Dyer p-adique, (French) Astérisque No. 294 (2004), ix, 251–319. · Zbl 1094.11025
[17] H. Darmon and V. Rotger, Diagonal cycles and Euler systems I: A p-adic Gross-Zagier formula, Annales Scientifiques de l’École Normale Supérieure, to appear. · Zbl 1356.11039
[18] M. Gealy, On the Tamagawa number conjecture for motives attached to modular forms, PhD Thesis, California Institute of Technology, 2006.
[19] H. Hida, Elementary Theory of L-functions and Eisenstein Series, London Mathematical Society Student Texts, Vol.26, Cambridge University Press, Cambridge, 1993. · Zbl 0942.11024
[20] K. Kato, p-adic Hodge theory and values of zeta functions of modular forms, in Cohomologies p-adiques et applications arithmétiques. III, Astérisque No. 295 (2004), ix, 117–290.
[21] K. Kitagawa, On standard p-adic L-functions of families of elliptic cusp forms, in p-adic Monodromy and the Birch and Swinnerton-Dyer Conjecture (Boston, MA, 1991), Contemporary Mathematics, Vol. 165, American Mathematical Society, Providence, RI, 1994, pp. 81–110.
[22] M. Niklas, Rigid syntomic regulators and the p-adic L-function of a modular form, Regensburg PhD Thesis, 2010, available at http://epub.uni-regensburg.de/19847/
[23] B. Mazur, J. Tate and J. Teitelbaum, On p-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Inventiones Mathematicae 84 (1986), 1–48. · Zbl 0699.14028
[24] B. Perrin-Riou, Fonctions L p-adiques d’une courbe elliptique et points rationnels, Annales de l’Institut Fourier (Grenoble) 43 (1993), 945–995. · Zbl 0840.11024
[25] B. Perrin-Riou, Théorie d’Iwasawa des représentations p-adiques sur un corps local, with an appendix by Jean-Marc Fontaine, Inventiones Mathematicae 115 (1994), 81–161. · Zbl 0838.11071
[26] G. Shimura, The special values of the zeta functions associated with cusp forms, Communications on Pure and Applied Mathematics 29 (1976), 783–804. · Zbl 0348.10015
[27] G. Shimura, On a class of nearly holomorphic automorphic forms, Annals of Mathematics 123 (1986), 347–406. · Zbl 0593.10022
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