Non-abelian congruences between $L$-values of elliptic curves

Delbourgo, Daniel; Ward, Tom

Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 1023-1055.

Soit $E$ une courbe elliptique définie sur $\mathbb{Q}$. Nous démontrons des versions faibles des congruences $K_1$ de Kato, pour les valeurs spéciales $L\left(1,E/\mathbb{Q}({\mu }_{p^n},\sqrt[p^n]{\Delta })\right).$ Plus précisément, nous vérifions que les congruences sont vraies modulo $p^{n+1}$, plutôt que modulo $p^{2n}$. Bien que ça ne suffise pas pour établir l’existence d’une fonction $L$ $p$-adique qui vit dans $K_1\left({\mathbb{Z}}_p\left[\left[\mathrm{Gal}(\mathbb{Q}({\mu }_{{\mathrm{p}}^{\infty }},\sqrt[{\mathrm{p}}^{\infty }]{\Delta })/\mathbb{Q})\right]\right]\right),$ elles fournissent beaucoup d’indices de l’existence de cet objet analytique. Par exemple, si $n=1$ les congruences trouvées numériquement par Tim et Vladimir Dokchitser sont vraies.

Let $E$ be a semistable elliptic curve over $\mathbb{Q}$. We prove weak forms of Kato’s $K_1$-congruences for the special values $L\left(1,E/\mathbb{Q}({\mu }_{p^n},\sqrt[p^n]{\Delta })\right).$ 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 $K_1\left({\mathbb{Z}}_p\left[\left[\mathrm{Gal}(\mathbb{Q}({\mu }_{{\mathrm{p}}^{\infty }},\sqrt[{\mathrm{p}}^{\infty }]{\Delta })/\mathbb{Q})\right]\right]\right)$, 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.

[1] Bouganis, Thanasis $L$-functions of elliptic curves and false Tate curve extensions, University of Cambridge (2005) (Ph. D. Thesis)

[2] Bouganis, Thanasis; Dokchitser, Vladimir Algebraicity of $L$-values for elliptic curves in a false Tate curve tower, Math. Proc. Cambridge Philos. Soc., Volume 142 (2007) no. 2, pp. 193-204 | DOI | MR

[3] Coates, John; Fukaya, Takako; Kato, Kazuya; Sujatha, Ramdorai; Venjakob, Otmar The ${\mathrm{GL}}_2$ main conjecture for elliptic curves without complex multiplication, Publ. Math. Inst. Hautes Études Sci. (2005) no. 101, pp. 163-208 | Numdam | MR | Zbl

[4] Doi, Koji; Hida, Haruzo; Ishii, Hidenori Discriminant of Hecke fields and twisted adjoint $L$-values for $\mathrm{GL}\left(2\right)$, Invent. Math., Volume 134 (1998) no. 3, pp. 547-577 | DOI | MR | Zbl

[5] Dokchitser, T.; Dokchitser, V. Computations in non-commutative Iwasawa theory, Proc. Lond. Math. Soc. (3), Volume 94 (2007) no. 1, pp. 211-272 (With an appendix by J. Coates and R. Sujatha) | DOI | MR

[6] Dokchitser, Vladimir Root numbers of non-abelian twists of elliptic curves, Proc. London Math. Soc. (3), Volume 91 (2005) no. 2, pp. 300-324 (With an appendix by Tom Fisher) | DOI | MR | Zbl

[7] Dünger, Volker $p$-adic interpolation of convolutions of Hilbert modular forms, Ann. Inst. Fourier (Grenoble), Volume 47 (1997) no. 2, pp. 365-428 | DOI | Numdam | MR | Zbl

[8] Haran, Shai $p$-adic $L$-functions for modular forms, Compositio Math., Volume 62 (1987) no. 1, pp. 31-46 | Numdam | MR | Zbl

[9] Kato, Kazuya $K_1$ of some non-commutative completed group rings, $K$-Theory, Volume 34 (2005) no. 2, pp. 99-140 | DOI | MR | Zbl

[10] Panchishkin, Alexey A. Non-Archimedean $L$-functions of Siegel and Hilbert modular forms, Lecture Notes in Mathematics, 1471, Springer-Verlag, Berlin, 1991 | MR | Zbl

[11] Serre, Jean-Pierre Sur les représentations modulaires de degré $2$ de $\mathrm{Gal}(\overline{\mathbf{Q}}/\mathbf{Q})$, Duke Math. J., Volume 54 (1987) no. 1, pp. 179-230 | DOI | MR | Zbl

[12] Shimura, Goro 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., Volume 48 (1981) no. 3, pp. 697 | DOI | Zbl

[13] Stevens, Glenn Stickelberger elements and modular parametrizations of elliptic curves, Invent. Math., Volume 98 (1989) no. 1, pp. 75-106 | DOI | EuDML | MR | Zbl

[14] Tate, J. 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., Providence, R.I., 1979, pp. 3-26 | MR | Zbl