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 -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 -functions implies the finiteness of Selmer groups associated to the corresponding Galois representation of modular forms under some assumptions.
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 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 de Rankin–Selberg implique la finitude des groupes de Selmer associés à la représentation galoisienne de la forme modulaire sous certaines hypothèses.
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Keywords: Modular forms, Selmer groups, Bloch–Kato conjecture
Mot clés : Formes modulaires, groupes de Selmer, conjecture de Bloch–Kato
Chida, Masataka 1
@article{AIF_2017__67_3_1231_0, author = {Chida, Masataka}, title = {Selmer groups and central values of $L$-functions for modular forms}, journal = {Annales de l'Institut Fourier}, pages = {1231--1276}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {67}, number = {3}, year = {2017}, doi = {10.5802/aif.3108}, language = {en}, url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3108/} }
TY - JOUR AU - Chida, Masataka TI - Selmer groups and central values of $L$-functions for modular forms JO - Annales de l'Institut Fourier PY - 2017 SP - 1231 EP - 1276 VL - 67 IS - 3 PB - Association des Annales de l’institut Fourier UR - https://aif.centre-mersenne.org/articles/10.5802/aif.3108/ DO - 10.5802/aif.3108 LA - en ID - AIF_2017__67_3_1231_0 ER -
%0 Journal Article %A Chida, Masataka %T Selmer groups and central values of $L$-functions for modular forms %J Annales de l'Institut Fourier %D 2017 %P 1231-1276 %V 67 %N 3 %I Association des Annales de l’institut Fourier %U https://aif.centre-mersenne.org/articles/10.5802/aif.3108/ %R 10.5802/aif.3108 %G en %F AIF_2017__67_3_1231_0
Chida, Masataka. Selmer groups and central values of $L$-functions for modular forms. Annales de l'Institut Fourier, Volume 67 (2017) no. 3, pp. 1231-1276. doi : 10.5802/aif.3108. https://aif.centre-mersenne.org/articles/10.5802/aif.3108/
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