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

[Congruences non-abeliennes entres les valeurs $L$ des courbes elliptiques]

Delbourgo, Daniel ; Ward, Tom

Annales de l'Institut Fourier, Tome 58 (2008), p. 1023-1055 / Harvested from Numdam

Résumé

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\bigl (1, E/\mathbb{Q}( \mu _{p^n},\@root p^n \of {\Delta } )\bigr ) . $ 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\bigl ( \mathbb{Z}_p[[ \rm {Gal}( \mathbb{Q}( \mu _{p^\infty },\!\!\@root p^\infty \! \of {\Delta } )/\mathbb{Q}) ]] \bigr ) , $ 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.

Abstract

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\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.

Détail

Publié le : 2008-01-01

MR 2427518 | Zbl 1165.11077 | 1 citation dans Numdam

DOI : https://doi.org/10.5802/aif.2377
Classification:  11R23,  11G40,  19B28
Mots clés: théorie d’Iwasawa, formes modulaires, fonctions $L$ $p$-adiques

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