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Daniel Delbourgo; Tom Ward
Non-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, 19B28
Keywords: Iwasawa theory, modular forms, $p$-adic $L$-functions

Résumé - 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.

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