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

Delbourgo, Daniel; Ward, Tom

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

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.

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.

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