Abstract
This article is the first in a series devoted to Kato’s Euler system arising from p-adic families of Beilinson elements in the K-theory of modular curves. It proves a p-adic Beilinson formula relating the syntomic regulator (in the sense of Coleman-de Shalit and Besser) of certain distinguished elements in the K-theory of modular curves to the special values at integer points ≥ 2 of the Mazur-Swinnerton-Dyer p-adic L-function attached to cusp forms of weight 2. When combined with the explicit relation between syntomic regulators and p-adic étale cohomology, this leads to an alternate proof of the main results of [Br2] and [Ge] which is independent of Kato’s explicit reciprocity law.
Similar content being viewed by others
References
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.
M. Bertolini, H. Darmon and K. Prasanna, Generalised Heegner cycles and p-adic Rankin L-series, Duke Mathematical Journal 162 (2013), 1033–1148.
M. Bertolini and H. Darmon, Hida families and rational points on elliptic curves, Inventiones Mathematicae 168 (2007), 371–431.
M. Bertolini and H. Darmon, Kato’s Euler system and rational points on elliptic curves II: The explicit reciprocity law, in preparation.
M. Bertolini and H. Darmon, Kato’s Euler system and rational points on elliptic curves III: The conjecture of Perrin-Riou, in preparation.
M. Bertolini, H. Darmon and V. Rotger, Beilinson-Flach elements and Euler systems I: syntomic regulators and p-adic Rankin L-series, submitted.
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.
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.
K. Bannai and G. Kings, p-adic elliptic polylogarithm, p-adic Eisenstein series and Katz measure, American Journal of Mathematics 132 (2010), 1609–1654.
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.
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.
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.
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.
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.
P. Colmez, Fonctions L p-adiques, Séminaire Bourbaki, Vol. 1998/99, Astérisque No. 266 (2000), Exp. No. 851, 3, 21–58.
P. Colmez, La conjecture de Birch et Swinnerton-Dyer p-adique, (French) Astérisque No. 294 (2004), ix, 251–319.
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.
M. Gealy, On the Tamagawa number conjecture for motives attached to modular forms, PhD Thesis, California Institute of Technology, 2006.
H. Hida, Elementary Theory of L-functions and Eisenstein Series, London Mathematical Society Student Texts, Vol.26, Cambridge University Press, Cambridge, 1993.
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.
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.
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/
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.
B. Perrin-Riou, Fonctions L p-adiques d’une courbe elliptique et points rationnels, Annales de l’Institut Fourier (Grenoble) 43 (1993), 945–995.
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.
G. Shimura, The special values of the zeta functions associated with cusp forms, Communications on Pure and Applied Mathematics 29 (1976), 783–804.
G. Shimura, On a class of nearly holomorphic automorphic forms, Annals of Mathematics 123 (1986), 347–406.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Bertolini, M., Darmon, H. Kato’s Euler system and rational points on elliptic curves I: A p-adic Beilinson formula. Isr. J. Math. 199, 163–188 (2014). https://doi.org/10.1007/s11856-013-0047-2
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11856-013-0047-2