Selmer groups and central values of $L$-functions for modular forms

Chida, Masataka

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.

Classification : 11F67, 11R23
Keywords: Modular forms, Selmer groups, Bloch–Kato conjecture
Mot clés : Formes modulaires, groupes de Selmer, conjecture de Bloch–Kato

[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 ${\mathbb{Z}}_p$-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 http://dx.doi.org/10.1515/crelle-2015-0072)

[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 $\ell$ 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 $\mathrm{GL}\left(2\right)$, 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 http://front.math.ucdavis.edu/1503.02888, 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 $\ell$, 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)