no. 3
Non-abelian congruences between L-values of elliptic curves
[Congruences non-abeliennes entres les valeurs L des courbes elliptiques]
Delbourgo, Daniel1 ; Ward, Tom2
1 Monash University School of Mathematical Sciences Victoria 3800 (Australia)
2 University of Nottingham School of Mathematical Sciences Nottingham NG7 2RD (United Kingdom)
Annales de l'Institut Fourier, Tome 58 (2008) no. 3, pp. 1023-1055.

Soit E une courbe elliptique définie sur ℚ. Nous démontrons des versions faibles des congruences K 1 de Kato, pour les valeurs spéciales L1 , E / ℚ ( μ p n , Δ p n ). 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 ℤ p [[ Gal (ℚ(μ p ∞ ,Δ p ∞ )/ℚ)]], 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 ℚ. We prove weak forms of Kato’s K 1 -congruences for the special values L1 , E / ℚ ( μ p n , Δ p n ). 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 ℤ p [[ Gal (ℚ(μ p ∞ ,Δ p ∞ )/ℚ)]], 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. https://aif.centre-mersenne.org/articles/10.5802/aif.2377/

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