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Level Raising and Anticyclotomic Selmer Groups for Hilbert Modular Forms of Weight Two

Published online by Cambridge University Press:  20 November 2018

Jan Nekovář
Jan Nekovář*
Affiliation:
Université Pierre et Marie Curie (Paris 6), Institut de Mathématiques de Jussieu, Théorie des Nombres, Case 247, 4, place Jussieu, F-75252 Paris cedex 05, France email: nekovar@math.jussieu.fr
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Abstract

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In this article we refine the method of Bertolini and Darmon $\left[ \text{BD}1 \right],\,\left[ \text{BD2} \right]$ and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

Keywords

11G4011F4111G18Hilbert modular formsSelmer groupsShimura curves
Type
Research Article
Information
Canadian Journal of Mathematics , Volume 64 , Issue 3 , 01 June 2012 , pp. 588 - 668
Copyright
Copyright © Canadian Mathematical Society 2012

References

[AT] Artin, E. and Tate, J., Class field theory. Second edition, Addison-Wesley, Redwood City, 1990.Google Scholar
[BD1] Bertolini, M. and Darmon, H., A rigid analytic Gross–Zagier formula and arithmetic applications. Ann. of Math. 146(1997), 111–147. http://dx.doi.org/10.2307/2951833 Google Scholar
[BD2] Bertolini, M. and Darmon, H., Iwasawa's Main Conjecture for Elliptic Curves over Anticyclotomic Z p-extensions. Ann. of Math. 162(2005), 1–64. http://dx.doi.org/10.4007/annals.2005.162.1 Google Scholar
[Bl] Blasius, D., Hilbert modular forms and the Ramanujan conjecture. In: Noncommutative geometry and number theory, Aspects Math. E37, Vieweg, Wiesbaden, 2006, 35–56.Google Scholar
[BK] Bloch, S. and Kato, K., L-functions and Tamagawa numbers of motives. In: The Grothendieck Festschrift I, Progr. Math. 86, Birkhäuser, Boston, Basel, Berlin, 1990, 333–400.Google Scholar
[Bo] Bondarko, M. V., The generic fiber of finite group schemes; a “finite wild” criterion for good reduction of abelian varieties. (Russian) Izv. Ross. Akad. Nauk Ser. Mat. 70(2006), 21–52; English translation in: Izv. Math. 70(2006), 661–691.Google Scholar
[BLRa] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron Models. Ergeb. Math. Grenzgeb. (3) 21, Springer, 1990.Google Scholar
[BLRi] Boston, B., Lenstra, H. W., Ribet, K., Quotients of group rings arising from two-dimensional representations. C. R. Acad. Sci. Paris Sér. I 312(1991), 323–328.Google Scholar
[BC] Boutot, J.-F. and Carayol, H., Uniformisation p-adique des courbes de Shimura: les théorèmes de Čerednik et de Drinfeld. In: Astérisque 196–197, Soc. Math. de France, Paris, 1991, 45–158.Google Scholar
[BZ] Boutot, J.-F. and Zink, T., The p-adic uniformization of Shimura curves. Preprint, 1995.Google Scholar
[Br] Breuil, C., Groupes p-divisibles, groupes finis et modules filtrés. Ann. of Math. 152(2000), 489–549. http://dx.doi.org/10.2307/2661391 Google Scholar
[C1] Carayol, H., Sur les représentations p-adiques associées aux forms modulaires de Hilbert. Ann. Sci. Ecole Norm. Sup. 19(1986), 409–468.Google Scholar
[C2] Carayol, H., Sur la mauvaise réduction des courbes de Shimura. Compos. Math. 59(1986), 151–230.Google Scholar
[CR] Curtis, C. W. and Reiner, I., Methods of Representation Theory, Vol. I. Wiley, New York, 1981.Google Scholar
[Če] Čerednik, I. V., Uniformization of algebraic curves by discrete arithmetic subgroups of PGL2(kw) with compact quotient spaces. Mat. Sbornik 29(1976), 55–78.Google Scholar
[Ch] Chi, W., Twists of central simple algebras and endomorphism algebras of some abelian varieties. Math. Ann. 276(1987), 615–632. http://dx.doi.org/10.1007/BF01456990 Google Scholar
[Cd] Chida, M., Selmer groups and the central values of L-functions for modular forms. In preparation.Google Scholar
[CV1] Cornut, C. and Vatsal, V., Nontriviality of Rankin–Selberg L-functions and CM points. In: L-functions and Galois representations (Durham, July 2004), London Math. Soc. Lecture Note Ser. 320, Cambridge University Press, 2007, 121–186.Google Scholar
[CV2] Cornut, C. and Vatsal, V., CM points and quaternion algebras. Doc. Math. 10(2005), 263–309.Google Scholar
[Di] Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties. Ann. Sci. Ecole Norm. Sup. 38(2005), 505–551.Google Scholar
[Dr] Drinfeld, V. G., Coverings of p-adic symmetric domains. Funkcional. Anal. i Prilozen. 10(1976), 29–40.Google Scholar
[Ed] Edixhoven, B., Appendix to [BD1].Google Scholar
[FC] Faltings, G. and Chai, C. L., Degeneration of abelian varieties. Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990.Google Scholar
[Fo] Fontaine, J.-M., Sur certains types de représentations p-adiques du groupe de Galois d’un corps local; construction d’un anneau de Barsotti–Tate. Ann. of Math. (2) 115(1982), 529–577. http://dx.doi.org/10.2307/2007012 Google Scholar
[FH] Friedberg, S. and Hoffstein, J., Nonvanishing theorems for automorphic L-functions on GL(2). Ann. of Math. (2) 142(1995), 385–423. http://dx.doi.org/10.2307/2118638 Google Scholar
[G1] Grothendieck, A., Le groupe de Brauer III: exemples et compléments. In: Dix exposés sur la cohomologie des schémas (A. Grothendieck and N. H. Kupier, eds.), Advanced Studies in Pure Mathematics 3, North Holland, Amsterdam, 1968, 88–188.Google Scholar
[G2] Grothendieck, A., Modèles de Néron et monodromie. In: SGA 7 I, Exposé IX, Lecture Notes in Math. 288, Springer, Berlin, 1972, 313–523.Google Scholar
[Gr] Grove, L. C., Classical groups and geometric algebra. Graduate Studies in Mathematics 39, Amer. Math. Soc., Providence, 2002.Google Scholar
[He] Henniart, G., Représentations l-adiques abéliennes. In: Séminaire de Théorie des Nombres de Paris 1980/81, Progress in Math, 22, Birkhäuser Boston, Boston, MA, 1982, pp. 107–126.Google Scholar
[H] Howard, B., Bipartite Euler systems. J. Reine Angew. Math. 597(2006), 1–25. http://dx.doi.org/10.1515/CRELLE.2006.062 Google Scholar
[J] Jarvis, F., Mazur's principle for totally real fields of odd degree. Compos. Math. 116(1999), 39–79. http://dx.doi.org/10.1023/A:1000600311268 Google Scholar
[JL] Jordan, B. and R. Livné, Local diophantine properties of Shimura curves. Math. Ann. 270(1985), 235–248. http://dx.doi.org/10.1007/BF01456184 Google Scholar
[Ki] Kisin, M., Crystalline representations and F-crystals. In: Algebraic geometry and number theory, Progr. Math. 253, Birkhäuser, Boston, 2006, 459–496.Google Scholar
[Ko Lo] Kolyvagin, V. A. and D. Yu. Logachev, Finiteness of the Shafarevich–Tate group and the group of rational points for some modular abelian varieties. (Russian) Algebra i Analiz 1(1989), 171–196; English translation: Leningrad Math. J. 1(1990), 1229–1253.Google Scholar
[Ku] Kurihara, A., On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 25(1979), 277–301.Google Scholar
[La] Lang, S., Algebraic groups over finite fields. Amer. J. Math. 78(1956), 555–563. http://dx.doi.org/10.2307/2372673 Google Scholar
[Li1] Liu, T., Potentially good reduction of Barsotti–Tate groups. J. Number Theory 126(2007), 155–184. http://dx.doi.org/10.1016/j.jnt.2006.11.008 Google Scholar
[Li2] Liu, T., Torsion p-adic Galois representations and a conjecture of Fontaine. Ann. Sci. Ecole Norm. Sup. 40(2007), 633–674.Google Scholar
[Li3] Liu, T., Lattices in filtered (', N)-modules. J. Inst. Math. Jussieu, to appear.Google Scholar
[L1] Longo, M., Euler systems obtained from congruences between Hilbert modular forms. Rend. Sem. Math. Univ. Padova 118(2007), 1–34.Google Scholar
[L2] Longo, M., On the Birch and Swinnerton–Dyer conjecture for modular elliptic curves over totally real fields. Ann. Inst. Fourier 56(2006), 689–733. http://dx.doi.org/10.5802/aif.2197 Google Scholar
[L3] Longo, M., Anticyclotomic Iwasawa Main Conjecture for Hilbert modular forms. Preprint, 2007.Google Scholar
[LRV] Longo, M., Rotger, V. and Vigni, S., Special values of L-functions and the arithmetic of Darmon points. arxiv:1004.3424.Google Scholar
[LV1] Longo, M. and Vigni, S., On the vanishing of Selmer groups for elliptic curves over ring class fields. J. Number Theory 130(2010), 128–163. http://dx.doi.org/10.1016/j.jnt.2009.07.004 Google Scholar
[LV2] Longo, M. and Vigni, S., Quaternion algebras, Heegner points and the arithmetic of Hida families. Manuscr. Math., to appear.Google Scholar
[Ma] Mazur, B., Local flat duality. Amer. J. Math. 92(1970), 343–361. http://dx.doi.org/10.2307/2373327 Google Scholar
[Mi1] Milne, J. S., Étale cohomology. Princeton University Press, Princeton, 1980.Google Scholar
[Mi2] Milne, J. S., Arithmetic duality theorems. Perspect. Math. 1, Academic Press, Boston, 1986.Google Scholar
[Mi3] Milne, J. S., Canonical models of (mixed) Shimura varieties and automorphic vector bundles. In: Automorphic forms, Shimura varieties, and L-functions, Vol. I (Ann Arbor, MI, 1988), Perspect. Math. 10, Academic Press, Boston, 1990, 283–414.Google Scholar
[Mo] Momose, F., On the C-adic representations attached to modular forms. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 28(1981), 89–109.Google Scholar
[Mu] Mumford, D., An analytic construction of degenerating curves over complete local rings. Compos. Math. 24(1972), 129–174.Google Scholar
[N1] Nekovář, J., The Euler system method for CM points on Shimura curves. In: L-functions and Galois representations (Durham, July 2004), London Math. Soc. Lecture Note Ser. 320, Cambridge University Press, 2007, 471–547.Google Scholar
[N2] Nekovář, J., Selmer complexes. Astérisque 310(2006), Soc. Math. de France, Paris.Google Scholar
[N3] Nekovář, J., On the parity of ranks of Selmer groups III. Doc. Math. 12(2007), 243–274; Erratum: Doc. Math. 14(2009), 191–194.Google Scholar
[PW] Pollack, R. and T.Weston, On anticyclotomic μ-invariants of modular forms. Compos. Math., to appear.Google Scholar
[Pr] Prasad, D., Some applications of seesaw duality to branching laws. Math. Ann. 304(1996), 1–20. http://dx.doi.org/10.1007/BF01446282 Google Scholar
[R] Rajaei, A., On the levels of mod Hilbert modular forms. J. Reine Angew. Math. 537(2001), 33–65. http://dx.doi.org/10.1515/crll.2001.058 Google Scholar
[Ra1] Raynaud, M., Spécialisation du foncteur de Picard. Inst. Hautes Etudes Sci. Publ. Math. 38(1970), 27–76.Google Scholar
[Ra2] Raynaud, M., Schémas en groupes de type (p, … , p). Bull. Soc. Math. France 102(1974), 241–280.Google Scholar
[Ri1] Ribet, K., On C-adic representations attached to modular forms. Invent. Math. 28(1975), 245–275. http://dx.doi.org/10.1007/BF01425561 Google Scholar
[Ri2] Ribet, K., Galois action on division points of Abelian varieties with real multiplications. Amer. J. Math. 98(1976), 751–804. http://dx.doi.org/10.2307/2373815 Google Scholar
[Ri3] Ribet, K., Galois representations attached to eigenforms with nebentypus. In: Modular forms in one variable V (Serre, J.-P. and Zagier, D. B., eds.), Lecture Notes in Math. 601, Springer, Berlin, 1977, 17–52.Google Scholar
[Ri4] Ribet, K., Galois representations attached to eigenforms with nebentypus. In: Modular forms in one variable V (Serre, J.-P. and Zagier, D. B., Twists of modular forms and endomorphisms of abelian varieties. Math. Ann. 253(1980), 43–62. http://dx.doi.org/10.1007/BF01457819 Google Scholar
[Ri5] Ribet, K., Galois representations attached to eigenforms with nebentypus. In: Modular forms in one variable V (Serre, J.-P. and Zagier, D. B., Division fields of Abelian varieties with complex multiplication.Mém. Soc.Math. France (2) 2(1980), 75–94.Google Scholar
[Ri6] Ribet, K., Galois representations attached to eigenforms with nebentypus. In: Modular forms in one variable V (Serre, J.-P. and Zagier, D. B., On C-adic representations attached to modular forms II. Glasgow Math. J. 27(1985), 185–194. http://dx.doi.org/10.1017/S0017089500006170 Google Scholar
[Ri7] Ribet, K., Galois representations attached to eigenforms with nebentypus. In: Modular forms in one variable V (Serre, J.-P. and Zagier, D. B., On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100(1990), 431–476. http://dx.doi.org/10.1007/BF01231195 Google Scholar
[Ru] Rubin, K., Euler systems. Ann. of Math. Stud. 147, Princeton University Press, Princeton, 2000.Google Scholar
[S] Sah, C.-H., Automorphisms of finite groups. J. Algebra 10(1968), 47–68. http://dx.doi.org/10.1016/0021-8693(68)90104-X Google Scholar
[Sa] Saito, H., On Tunnell's formula for characters of GL(2). Compos. Math. 85(1993), 99–108.Google Scholar
[Sc] Scharlau, W., Quadratic and Hermitian Forms. Grundlehren Math.Wiss. 270, Springer, Berlin, 1985.Google Scholar
[Sk] Skinner, C., A note on p-adic Galois representations attached to Hilbert modular forms. Doc. Math. 14(2009), 241–258.Google Scholar
[Ta] Tate, J., p-divisible groups. In: Proceedings of a conference on local fields (Driebergen, 1966), Springer, Berlin, 1967, 158–183.Google Scholar
[T1] Taylor, R., On Galois representations associated to Hilbert modular forms. Invent. Math. 98 (1989), 265–280. http://dx.doi.org/10.1007/BF01388853 Google Scholar
[T2] Taylor, R., On Galois representations associated to Hilbert modular forms II. In: Elliptic Curves, Modular Forms and Fermat's Last Theorem (Coates, J. and Yau, S. T., eds.), International Press, 1997, 333–340.Google Scholar
[TZ] Tian, Y. and Zhang, S.-W., in preparation.Google Scholar
[Tu] Tunnell, J., Local “-factors and characters of GL2. Amer. J. Math. 105(1983), 1277–1308. http://dx.doi.org/10.2307/2374441 Google Scholar
[V] Varshavsky, Y., P-adic uniformization of unitary Shimura varieties II. J. Differential Geom. 49(1998), 75–113.Google Scholar
[VZ] Vasiu, A. and Zink, T., Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristics. arxiv:0911.2474.Google Scholar
[Vi] Vigneras, M.-F., Arithmétique des algèbres de quaternions. Lecture Notes in Math. 800, Springer, Berlin, 1980.Google Scholar
[W1] Waldspurger, J.-L., Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2). Compos. Math. 54(1985), 121–171.Google Scholar
[W2] Waldspurger, J.-L., Sur les valeurs de certains fonctions L automorphes en leur centre de symétrie. Compos. Math. 54(1985), 173–242.Google Scholar
[W3] Waldspurger, J.-L., Correspondances de Shimura et quaternions. Forum Math. 3(1991), 219–307. http://dx.doi.org/10.1515/form.1991.3.219 Google Scholar
[YZZ] Yuan, X., Zhang, S.-W. and Zhang, W., Heights of CM points I. Gross–Zagier formula. Preprint, 2008.Google Scholar
[Zh1] Zhang, S.-W., Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153(2001), 27–147. http://dx.doi.org/10.2307/2661372 Google Scholar
[Zh2] Zhang, S.-W., Arithmetic of Shimura curves and the Birch and Swinnerton–Dyer conjecture. Sci. China Math. 53(2010), 573–592. http://dx.doi.org/10.1007/s11425-010-0046-2Google Scholar