Selmer groups and central values of L-functions for modular forms  [ Groupes de Selmer et valeurs centrales de fonctions L de formes modulaires ]
Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1231-1276.

Dans cet article, nous construisons un système d’Euler en utilisant les cycles CM sur les variétés de Kuga–Sato au-dessus de courbes de Shimura, et montrons une relation avec les valeurs centrales de fonctions L de Rankin–Selberg associées aux formes modulaires de poids 2 et aux caractères de classes d’un corps quadratique imaginaire. Comme application, nous prouvons que la non-annulation des valeurs centrales de fonctions L de Rankin–Selberg implique la finitude des groupes de Selmer associés à la représentation galoisienne de la forme modulaire sous certaines hypothèses.

In this article, we construct an Euler system using CM cycles on Kuga–Sato varieties over Shimura curves and show a relation with the central values of Rankin–Selberg L-functions for elliptic modular forms and ring class characters of an imaginary quadratic field. As an application, we prove that the non-vanishing of the central values of Rankin–Selberg L-functions implies the finiteness of Selmer groups associated to the corresponding Galois representation of modular forms under some assumptions.

Reçu le : 2016-01-16
Révisé le : 2016-09-15
Accepté le : 2016-09-16
Publié le : 2017-05-31
DOI : https://doi.org/10.5802/aif.3108
Classification : 11F67,  11R23
Mots clés: Formes modulaires, groupes de Selmer, conjecture de Bloch–Kato
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     author = {Chida, Masataka},
     title = {Selmer groups and central values of $L$-functions for modular forms},
     journal = {Annales de l'Institut Fourier},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {3},
     year = {2017},
     pages = {1231-1276},
     doi = {10.5802/aif.3108},
     language = {en},
     url = {aif.centre-mersenne.org/item/AIF_2017__67_3_1231_0/}
}
Chida, Masataka. Selmer groups and central values of $L$-functions for modular forms. Annales de l'Institut Fourier, Tome 67 (2017) no. 3, pp. 1231-1276. doi : 10.5802/aif.3108. https://aif.centre-mersenne.org/item/AIF_2017__67_3_1231_0/

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