Selmer groups and central values of L-functions for modular forms
Annales de l'Institut Fourier, Volume 67 (2017) no. 3, pp. 1231-1276.

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.

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.

Published online:
DOI: 10.5802/aif.3108
Classification: 11F67, 11R23
Keywords: Modular forms, Selmer groups, Bloch–Kato conjecture
Mot clés : Formes modulaires, groupes de Selmer, conjecture de Bloch–Kato
Chida, Masataka 1

1 Mathematical Institute Tohoku University 6-3, Aramaki Aza-Aoba, Aoba-ku Sendai 980-8578 (Japan)
License: CC-BY-ND 4.0
Copyrights: The authors retain unrestricted copyrights and publishing rights
     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 = {}
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  -
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
%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.

[1] Bertolini, Massimo; Darmon, Henri René Heegner points on Mumford–Tate curves, Invent. Math., Volume 126 (1996) no. 3, pp. 413-456 | DOI

[2] Bertolini, Massimo; Darmon, Henri René Iwasawa’s main conjecture for elliptic curves over anticyclotonic p -extensions, Ann. Math., Volume 162 (2005) no. 1, pp. 1-64 | DOI

[3] Besser, Amnon CM cycles over Shimura curves, J. Algebr. Geom., Volume 4 (1995) no. 4, pp. 659-691

[4] Besser, Amnon On the finiteness of Ш for motives associated to modular forms, Doc. Math., J. DMV, Volume 2 (1997) no. 1, pp. 31-46

[5] Bloch, Spencer; Kato, Kazuya L-functions and Tamagawa numbers of motives, The Grothendieck Festschrift, Vol. 1 (Progress in Mathematics), Volume 86, Birkhäuser, Boston, MA, 1990, pp. 333-400

[6] Boston, Nigel; Lenstra, Hendrik W.jun.; Ribet, Kenneth A. Quotients of group rings arising from two-dimensional representations, C. R. Acad. Sci. Paris Sér. I Math., Volume 312 (1991) no. 4, pp. 323-328

[7] Boutot, Jean-François; Carayol, Henri Uniformisation p-adique des courbes de Shimura: les théorèmes de Cerednik et de Drinfeld, Astérisque, Volume 196–197 (1991), pp. 45-158

[8] Chida, Masataka; Hsieh, Ming-Lun Special values of anticyclotomic L-functions for modular forms (to appear in J. reine angwe. Math., available at

[9] Chida, Masataka; Hsieh, Ming-Lun On the anticyclotomic Iwasawa main conjecture for modular forms, Compos. Math., Volume 151 (2015) no. 5, pp. 863-893 | DOI

[10] Deligne, Pierre La fourmule de Picard–Lefschetz, SGA 7 II, Exposé XV (Lecture Notes in Mathematics), Volume 340, Springer, 1973, pp. 165-196

[11] Deligne, Pierre Le formalisme des cycles evanescents, SGA 7 II, Exposé XIII (Lecture Notes in Mathematics), Volume 340, Springer, 1973, pp. 82-115

[12] Diamond, Fred The Taylor–Wiles construction and multiplicity one, Invent. Math., Volume 128 (1997) no. 2, pp. 379-391 | DOI

[13] Diamond, Fred; Taylor, Richard Non-optimal levels of mod l modular representations, Invent. Math., Volume 115 (1994) no. 3, pp. 435-462 | DOI

[14] Fu, Lei Étale cohomology theory, Nankai Tracts in Mathematics, 13, World Scientific Publishing Co. Pte. Ltd., 2011, ix+611 pages

[15] Hung, Pin-Chi On the non-vanishing mod of central L-values with anticyclotomic twists for Hilbert modular forms, J. Number Theory, Volume 173 (2017), pp. 170-209 | DOI

[16] Illusie, Luc On semistable reduction and the calculation of nearby cycles, Geometric aspects of Dwork’s theory, Volume I, II, Walter de Gruyter GmbH & Co. KG, Berlin, 2004, pp. 785-803

[17] Iovita, Adrian; Spiess, Michael Derivatives of p-adic L-functions, Heegner cycles and monodromy modules attached to modular forms, Invent. Math., Volume 154 (2003) no. 2, pp. 333-384 | DOI

[18] Jacquet, Hervé Michel; Langlands, Robert P. Automorphic forms on GL (2), Lecture Notes in Mathematics, 114, Springer, Berlin, 1970, vii+548 pages

[19] Jannsen, Uwe Continuous étale cohomology, Math. Ann., Volume 280 (1988) no. 2, pp. 207-245 | DOI

[20] Jannsen, Uwe Mixed motives and algebraic K-theory, Lecture Notes in Mathematics, 1400, Springer, 1990, xiii+246 pages

[21] Jordan, Bruce W.; Livné, Ron Integral Hodge theory and congruences between modular forms, Duke Math. J., Volume 80 (1995) no. 2, pp. 419-484 | DOI

[22] Kato, Kazuya p-adic Hodge theory and values of zeta functions of modular forms, Astérisque, Volume 295 (2004), pp. 117-290

[23] Kings, Guido; Loeffler, David; Zerbes, Sarah Livia Rankin–Eisenstein classes and explicit reciprocity laws (Preprint available at, to appear in Camb. J. Math.)

[24] Loeffler, David; Zerbes, Sarah Livia Rankin-Eisenstein classes in Coleman families, Res. Math. Sci., Volume 3 (2016) (Paper no 29, 53 pp., electronic only) | DOI

[25] Longo, Matteo On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields, Ann. Inst. Fourier, Volume 56 (2006) no. 3, pp. 689-733 | DOI

[26] Longo, Matteo; Vigni, Stefano On the vanishing of Selmer groups for elliptic curves over ring class fields, J. Number Theory, Volume 130 (2010) no. 1, pp. 128-163 | DOI

[27] Nekovář, Jan Kolyvagin’s method for Chow groups and Kuga-Sato varieties, Invent. Math., Volume 107 (1992) no. 1, pp. 99-125 | DOI

[28] Nekovář, Jan On the p-adic height of Heegner cycles, Math. Ann., Volume 302 (1995) no. 4, pp. 609-686 | DOI

[29] Nekovář, Jan p-adic Abel-Jacobi maps and p-adic heights, The Arithmetic and Geometry of Algebraic Cycles, (Banff, Canada, 1998) (CRM Proc. and Lect. Notes 24), Amer. Math. Soc, Providence, R.I., 2000, pp. 367-379

[30] Nekovář, Jan Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two, Can. J. Math., Volume 64 (2012) no. 3, pp. 588-668 | DOI

[31] Nekovář, Jan; Nizioł, Wiesława Syntomic cohomology and p-adic regulators for varieties over p-adic fields, Algebra Number Theory, Volume 10 (2016) no. 8, pp. 1695-1790 | DOI

[32] Nizioł, Wiesława On the image of p-adic regulators, Invent. Math., Volume 127 (1997) no. 2, pp. 375-400 | DOI

[33] Rajaei, Ali On the levels of mod l Hilbert modular forms, J. Reine Angwe. Math., Volume 537 (2001), pp. 33-65

[34] Saito, Takeshi Weight spectral sequences and independence of , J. Inst. Math. Jussieu, Volume 2 (2003) no. 4, pp. 583-634 | DOI

[35] Scholl, Anthony James Motives for modular forms, Invent. Math., Volume 100 (1990) no. 2, pp. 419-430 | DOI

[36] Taylor, Richard On the meromorphic continuation of degree two L-functions, Doc. Math., J. DMV, Volume Extra Vol. (2006), pp. 729-779 (electronic)

Cited by Sources: