Non-abelian congruences between L-values of elliptic curves
[Congruences non-abeliennes entres les valeurs L des courbes elliptiques]
Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 1023-1055.

Soit E une courbe elliptique définie sur . Nous démontrons des versions faibles des congruences K1 de Kato, pour les valeurs spéciales L1,E/(μpn,Δpn). Plus précisément, nous vérifions que les congruences sont vraies modulo pn+1, plutôt que modulo p2n. Bien que ça ne suffise pas pour établir l’existence d’une fonction L p-adique qui vit dans K1p[[Gal((μp,Δp)/)]], 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 . We prove weak forms of Kato’s K1-congruences for the special values L1,E/(μpn,Δpn). More precisely, we show that they are true modulo pn+1, rather than modulo p2n. Whilst not quite enough to establish that there is a non-abelian L-function living in K1p[[Gal((μp,Δp)/)]], 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.

DOI : 10.5802/aif.2377
Classification : 11R23, 11G40, 19B28
Keywords: Iwasawa theory, modular forms, p-adic L-functions
Mot clés : théorie d’Iwasawa, formes modulaires, fonctions L p-adiques
Delbourgo, Daniel 1 ; Ward, Tom 2

1 Monash University School of Mathematical Sciences Victoria 3800 (Australia)
2 University of Nottingham School of Mathematical Sciences Nottingham NG7 2RD (United Kingdom)
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Delbourgo, Daniel; Ward, Tom. Non-abelian congruences between $L$-values of elliptic curves. Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 1023-1055. doi : 10.5802/aif.2377. http://archive.numdam.org/articles/10.5802/aif.2377/

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  • Bouganis, Thanasis Non-abelian p-adic L-functions and Eisenstein series of unitary groups – the CM method, Annales de l'Institut Fourier, Volume 64 (2014) no. 2, pp. 793-891 | DOI:10.5802/aif.2866 | Zbl:1396.11124
  • Kim, Dohyeong p-adic L-functions over the false Tate curve extensions, Mathematical Proceedings of the Cambridge Philosophical Society, Volume 155 (2013) no. 3, pp. 483-498 | DOI:10.1017/s0305004113000431 | Zbl:1286.11177
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  • Bouganis, Thanasis Non-Abelian congruences between special values of L-functions of elliptic curves: the CM case, International Journal of Number Theory, Volume 7 (2011) no. 7, pp. 1883-1934 | DOI:10.1142/s179304211100468x | Zbl:1279.11107

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