Non-abelian congruences between L-values of elliptic curves
Annales de l'Institut Fourier, Volume 58 (2008) no. 3, p. 1023-1055

Let E be a semistable elliptic curve over . We prove weak forms of Kato’s K 1 -congruences for the special values L1 , E / ( 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 p [[ Gal ((μ p ,Δ p )/)]], 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 . Nous démontrons des versions faibles des congruences K 1 de Kato, pour les valeurs spéciales L1 , E / ( 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 p [[ Gal ((μ p ,Δ p )/)]], 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.

DOI : https://doi.org/10.5802/aif.2377
Classification:  11R23,  11G40,  19B28
Keywords: Iwasawa theory, modular forms, p-adic L-functions
@article{AIF_2008__58_3_1023_0,
     author = {Delbourgo, Daniel and Ward, Tom},
     title = {Non-abelian congruences between $L$-values of elliptic curves},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {58},
     number = {3},
     year = {2008},
     pages = {1023-1055},
     doi = {10.5802/aif.2377},
     mrnumber = {2427518},
     zbl = {1165.11077},
     language = {en},
     url = {http://www.numdam.org/item/AIF_2008__58_3_1023_0}
}
Delbourgo, Daniel; Ward, Tom. Non-abelian congruences between $L$-values of elliptic curves. Annales de l'Institut Fourier, Volume 58 (2008) no. 3, pp. 1023-1055. doi : 10.5802/aif.2377. http://www.numdam.org/item/AIF_2008__58_3_1023_0/

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