Hostname: page-component-8448b6f56d-wq2xx Total loading time: 0 Render date: 2024-04-19T00:57:45.668Z Has data issue: false hasContentIssue false
  • English
  • Français

On the anticyclotomic Iwasawa main conjecture for modular forms

Part of: Discontinuous groups and automorphic forms Algebraic number theory: global fields

Published online by Cambridge University Press:  27 November 2014

Masataka Chida  and
Ming-Lun Hsieh
Masataka Chida
Affiliation:
Department of Mathematics, Kyoto University, Kitashirakawa Oiwake-cho, Sakyo-ku, Kyoto 606-8502, Japan email chida@math.kyoto-u.ac.jp
Ming-Lun Hsieh
Affiliation:
Department of Mathematics, National Taiwan University, No. 1, Sec. 4, Roosevelt Road, Taipei 10617, Taiwan email mlhsieh@math.ntu.edu.tw
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We generalize the work of Bertolini and Darmon on the anticyclotomic main conjecture for elliptic curves to modular forms of higher weight.

Keywords

Iwasawa main conjecturemodular forms

MSC classification

Primary: 11R23: Iwasawa theory
Secondary: 11F67: Special values of automorphic $L$-series, periods of modular forms, cohomology, modular symbols
Type
Research Article
Information
Compositio Mathematica , Volume 151 , Issue 5 , May 2015 , pp. 863 - 897
Copyright
© The Authors 2014 

References

Bertolini, M. and Darmon, H., Derived heights and generalized Mazur–Tate regulators, Duke Math. J. 76 (1994), 75–111.CrossRefGoogle Scholar
Bertolini, M. and Darmon, H., Iwasawa’s main conjecture for elliptic curves over anticyclotomic ℤp-extensions, Ann. of Math. (2) 162 (2005), 1–64.CrossRefGoogle Scholar
Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives, in The Grothendieck Festschrift, Vol. I, Progress in Mathematics, vol. 86 (Birkhäuser, Boston, 1990), 333–400.Google Scholar
Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron models, Ergebnisse der Mathematik und ihrer Grenzgebiete (3), vol. 21 (Springer, Berlin, 1990).CrossRefGoogle Scholar
Boutot, J.-F. and Carayol, H., Uniformisation $p$-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld, Astérisque (1991) no. 196–197, 7, 45–158 (1992), Courbes modulaires et courbes de Shimura (Orsay, 1987/1988).Google Scholar
Carayol, H., Sur les représentations l-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), 409–468.CrossRefGoogle Scholar
Čerednik, I. V., Uniformization of algebraic curves by discrete arithmetic subgroups of PGL2(k w) with compact quotient spaces, Mat. Sb. 100(142) (1976), 59–88; 165.Google Scholar
Chida, M., Selmer groups and central values of $L$-functions for modular forms, Preprint (2013),http://www.math.kyoto-u.ac.jp/∼chida/Selmer.pdf.Google Scholar
Chida, M. and Hsieh, M.-L., Special values of anticyclotomic $L$-functions for modular forms, submitted, Preprint (2012), arXiv:1204.2427.Google Scholar
Diamond, F. and Taylor, R., Nonoptimal levels of mod l modular representations, Invent. Math. 115 (1994), 435–462.CrossRefGoogle Scholar
Drinfeld, V. G., Coverings of p-adic symmetric domains, Funkcional. Anal. i Priložen. 10 (1976), 29–40.Google Scholar
Greenberg, R., Iwasawa theory for elliptic curves, in Arithmetic theory of elliptic curves (Cetraro, 1997), Lecture Notes in Mathematics, vol. 1716 (Springer, Berlin, 1999), 51–144.CrossRefGoogle Scholar
Hida, H., On p-adic Hecke algebras for GL(2) over totally real fields, Ann. of Math. (2) 128 (1988), 295–384.CrossRefGoogle Scholar
Ihara, Y., Shimura curves over finite fields and their rational points, in Applications of curves over finite fields (Seattle, WA, 1997), Contemporary Mathematics, vol. 245 (American Mathematical Society, Providence, RI, 1999), 15–23.CrossRefGoogle Scholar
Longo, M. and Vigni, S., Quaternion algebras, Heegner points and the arithmetic of Hida families, Manuscripta Math. 135 (2011), 273–328.CrossRefGoogle Scholar
Nekovář, J., Selmer complexes, Astérisque 310 (2006).Google Scholar
Nekovář, J., Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two, Canad. J. Math. 64 (2012), 588–668.CrossRefGoogle Scholar
Pollack, R. and Weston, T., On anticyclotomic 𝜇-invariants of modular forms, Compositio Math. 147 (2011), 1353–1381.CrossRefGoogle Scholar
Ribet, K. A., On modular representations of Gal(Q∕Q) arising from modular forms, Invent. Math. 100 (1990), 431–476.CrossRefGoogle Scholar
Ribet, K. A., Images of semistable Galois representations, Pacific J. Math. (1997), 277–297.CrossRefGoogle Scholar
Rubin, K., Euler systems, Annals of Mathematics Studies, vol. 147 (Princeton University Press, Princeton, NJ, 2000).Google Scholar
Skinner, C. and Urban, E., The Iwasawa main conjectures for GL2, Invent. Math. 195 (2014), 1–277.CrossRefGoogle Scholar
Vatsal, V., Special values of anticyclotomic L-functions, Duke Math. J. 116 (2003), 219–261.CrossRefGoogle Scholar