With cedram.org English français Home Overview Search for an article Submit a paper Informations for the authors Subscription Limited access - RUCHE Table of contents for this issue | Previous article Daniel Delbourgo; Tom WardNon-abelian congruences between $L$-values of elliptic curves(Congruences non-abeliennes entres les valeurs $L$ des courbes elliptiques)Annales de l'institut Fourier, 58 no. 3 (2008), p. 1023-1055, doi: 10.5802/aif.2377 Article PDF | Reviews MR 2427518 | Zbl 1165.11077 Class. Math.: 11R23, 11G40, 19B28Keywords: Iwasawa theory, modular forms, $p$-adic $L$-functions Résumé - AbstractLet $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $L\bigl (1, E/\mathbb{Q}( \mu _{p^n},\@root p^n \of {\Delta } )\bigr ) .$ More precisely, we show that they are true modulo $p^{n+1}$, rather than modulo $p^{2n}$. Whilst not quite enough to establish that there is a non-abelian $L$-function living in $K_1\bigl ( \mathbb{Z}_p[[ \rm {Gal} ( \mathbb{Q}( \mu _{p^\infty },\!\!\@root p^\infty \of {\Delta } )/\mathbb{Q}) ]] \bigr )$, they do provide strong evidence towards the existence of such an analytic object. For example, if $n=1$ these verify the numerical congruences found by Tim and Vladimir Dokchitser. Bibliography[1] Thanasis Bouganis, $L$-functions of elliptic curves and false Tate curve extensions, thesis, University of Cambridge, 2005 [2] Thanasis Bouganis & Vladimir Dokchitser, “Algebraicity of $L$-values for elliptic curves in a false Tate curve tower”, Math. Proc. Cambridge Philos. Soc. 142 (2007) no. 2, p. 193-204 Article |  MR 2314594 |  Zbl pre05152948[3] John Coates, Takako Fukaya, Kazuya Kato, Ramdorai Sujatha & Otmar Venjakob, “The $\rm GL_2$ main conjecture for elliptic curves without complex multiplication”, Publ. Math. Inst. Hautes Études Sci. (2005) no. 101, p. 163-208 Numdam |  MR 2217048 |  Zbl 1108.11081[4] Koji Doi, Haruzo Hida & Hidenori Ishii, “Discriminant of Hecke fields and twisted adjoint $L$-values for $\rm GL(2)$”, Invent. Math. 134 (1998) no. 3, p. 547-577 Article |  MR 1660929 |  Zbl 0924.11035[5] T. Dokchitser & V. Dokchitser, “Computations in non-commutative Iwasawa theory”, Proc. Lond. Math. Soc. (3) 94 (2007) no. 1, p. 211-272, With an appendix by J. Coates and R. Sujatha Article |  MR 2294995 |  Zbl pre05129594[6] Vladimir Dokchitser, “Root numbers of non-abelian twists of elliptic curves”, Proc. London Math. Soc. (3) 91 (2005) no. 2, p. 300-324, With an appendix by Tom Fisher Article |  MR 2167089 |  Zbl 1076.11042[7] Volker Dünger, “$p$-adic interpolation of convolutions of Hilbert modular forms”, Ann. Inst. Fourier (Grenoble) 47 (1997) no. 2, p. 365-428 Cedram |  MR 1450421 |  Zbl 0882.11025[8] Shai Haran, “$p$-adic $L$-functions for modular forms”, Compositio Math. 62 (1987) no. 1, p. 31-46 Numdam |  MR 892149 |  Zbl 0618.10027[9] Kazuya Kato, “$K_1$ of some non-commutative completed group rings”, $K$-Theory 34 (2005) no. 2, p. 99-140 Article |  MR 2180109 |  Zbl 1080.19002[10] Alexey A. Panchishkin, Non-Archimedean $L$-functions of Siegel and Hilbert modular forms, Lecture Notes in Mathematics 1471, Springer-Verlag, 1991  MR 1122593 |  Zbl 0732.11026[11] Jean-Pierre Serre, “Sur les représentations modulaires de degré $2$ de ${\rm Gal}(\overline{\bf Q}/{\bf Q})$”, Duke Math. J. 54 (1987) no. 1, p. 179-230 Article |  MR 885783 |  Zbl 0641.10026[12] Goro Shimura, “Corrections to: “The special values of the zeta functions associated with Hilbert modular forms” [Duke Math. J. 45 (1978), no. 3, 637–679”, Duke Math. J. 48 (1981) no. 3 Article |  Zbl 0394.10015[13] Glenn Stevens, “Stickelberger elements and modular parametrizations of elliptic curves”, Invent. Math. 98 (1989) no. 1, p. 75-106 Article |  MR 1010156 |  Zbl 0697.14023[14] J. Tate, Number theoretic background, Automorphic forms, representations and $L$-functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), Part 2, Proc. Sympos. Pure Math., XXXIII, Amer. Math. Soc., 1979, p. 3–26  MR 546607 |  Zbl 0422.12007 © Annales de L'Institut Fourier - ISSN (électronique) : 1777-5310