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

  • Jan Nekovář (a1)

Abstract

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.

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References

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[AT] Artin, E. and Tate, J., Class field theory. Second edition, Addison-Wesley, Redwood City, 1990.
[BD1] Bertolini, M. and Darmon, H., A rigid analytic Gross–Zagier formula and arithmetic applications. Ann. of Math. 146(1997), 111147. http://dx.doi.org/10.2307/2951833
[BD2] Bertolini, M. and Darmon, H., Iwasawa's Main Conjecture for Elliptic Curves over Anticyclotomic Z p-extensions. Ann. of Math. 162(2005), 164. http://dx.doi.org/10.4007/annals.2005.162.1
[Bl] Blasius, D., Hilbert modular forms and the Ramanujan conjecture. In: Noncommutative geometry and number theory, Aspects Math. E37, Vieweg, Wiesbaden, 2006, 3556.
[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, 333400.
[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), 2152; English translation in: Izv. Math. 70(2006), 661–691.
[BLRa] Bosch, S., Lütkebohmert, W. and Raynaud, M., Néron Models. Ergeb. Math. Grenzgeb. (3) 21, Springer, 1990.
[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), 323328.
[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, 45158.
[BZ] Boutot, J.-F. and Zink, T., The p-adic uniformization of Shimura curves. Preprint, 1995.
[Br] Breuil, C., Groupes p-divisibles, groupes finis et modules filtrés. Ann. of Math. 152(2000), 489549. http://dx.doi.org/10.2307/2661391
[C1] Carayol, H., Sur les représentations p-adiques associées aux forms modulaires de Hilbert. Ann. Sci. Ecole Norm. Sup. 19(1986), 409468.
[C2] Carayol, H., Sur la mauvaise réduction des courbes de Shimura. Compos. Math. 59(1986), 151230.
[CR] Curtis, C. W. and Reiner, I., Methods of Representation Theory, Vol. I. Wiley, New York, 1981.
[Če] Čerednik, I. V., Uniformization of algebraic curves by discrete arithmetic subgroups of PGL2(kw) with compact quotient spaces. Mat. Sbornik 29(1976), 5578.
[Ch] Chi, W., Twists of central simple algebras and endomorphism algebras of some abelian varieties. Math. Ann. 276(1987), 615632. http://dx.doi.org/10.1007/BF01456990
[Cd] Chida, M., Selmer groups and the central values of L-functions for modular forms. In preparation.
[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, 121186.
[CV2] Cornut, C. and Vatsal, V., CM points and quaternion algebras. Doc. Math. 10(2005), 263309.
[Di] Dimitrov, M., Galois representations modulo p and cohomology of Hilbert modular varieties. Ann. Sci. Ecole Norm. Sup. 38(2005), 505551.
[Dr] Drinfeld, V. G., Coverings of p-adic symmetric domains. Funkcional. Anal. i Prilozen. 10(1976), 2940.
[Ed] Edixhoven, B., Appendix to [BD1].
[FC] Faltings, G. and Chai, C. L., Degeneration of abelian varieties. Ergeb. Math. Grenzgeb. (3) 22, Springer, Berlin, 1990.
[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), 529577. http://dx.doi.org/10.2307/2007012
[FH] Friedberg, S. and Hoffstein, J., Nonvanishing theorems for automorphic L-functions on GL(2). Ann. of Math. (2) 142(1995), 385423. http://dx.doi.org/10.2307/2118638
[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, 88188.
[G2] Grothendieck, A., Modèles de Néron et monodromie. In: SGA 7 I, Exposé IX, Lecture Notes in Math. 288, Springer, Berlin, 1972, 313523.
[Gr] Grove, L. C., Classical groups and geometric algebra. Graduate Studies in Mathematics 39, Amer. Math. Soc., Providence, 2002.
[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. 107126.
[H] Howard, B., Bipartite Euler systems. J. Reine Angew. Math. 597(2006), 125. http://dx.doi.org/10.1515/CRELLE.2006.062
[J] Jarvis, F., Mazur's principle for totally real fields of odd degree. Compos. Math. 116(1999), 3979. http://dx.doi.org/10.1023/A:1000600311268
[JL] Jordan, B. and R. Livné, Local diophantine properties of Shimura curves. Math. Ann. 270(1985), 235248. http://dx.doi.org/10.1007/BF01456184
[Ki] Kisin, M., Crystalline representations and F-crystals. In: Algebraic geometry and number theory, Progr. Math. 253, Birkhäuser, Boston, 2006, 459496.
[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), 171196; English translation: Leningrad Math. J. 1(1990), 1229–1253.
[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), 277301.
[La] Lang, S., Algebraic groups over finite fields. Amer. J. Math. 78(1956), 555563. http://dx.doi.org/10.2307/2372673
[Li1] Liu, T., Potentially good reduction of Barsotti–Tate groups. J. Number Theory 126(2007), 155184. http://dx.doi.org/10.1016/j.jnt.2006.11.008
[Li2] Liu, T., Torsion p-adic Galois representations and a conjecture of Fontaine. Ann. Sci. Ecole Norm. Sup. 40(2007), 633674.
[Li3] Liu, T., Lattices in filtered (', N)-modules. J. Inst. Math. Jussieu, to appear.
[L1] Longo, M., Euler systems obtained from congruences between Hilbert modular forms. Rend. Sem. Math. Univ. Padova 118(2007), 134.
[L2] Longo, M., On the Birch and Swinnerton–Dyer conjecture for modular elliptic curves over totally real fields. Ann. Inst. Fourier 56(2006), 689733. http://dx.doi.org/10.5802/aif.2197
[L3] Longo, M., Anticyclotomic Iwasawa Main Conjecture for Hilbert modular forms. Preprint, 2007.
[LRV] Longo, M., Rotger, V. and Vigni, S., Special values of L-functions and the arithmetic of Darmon points. arxiv:1004.3424.
[LV1] Longo, M. and Vigni, S., On the vanishing of Selmer groups for elliptic curves over ring class fields. J. Number Theory 130(2010), 128163. http://dx.doi.org/10.1016/j.jnt.2009.07.004
[LV2] Longo, M. and Vigni, S., Quaternion algebras, Heegner points and the arithmetic of Hida families. Manuscr. Math., to appear.
[Ma] Mazur, B., Local flat duality. Amer. J. Math. 92(1970), 343361. http://dx.doi.org/10.2307/2373327
[Mi1] Milne, J. S., Étale cohomology. Princeton University Press, Princeton, 1980.
[Mi2] Milne, J. S., Arithmetic duality theorems. Perspect. Math. 1, Academic Press, Boston, 1986.
[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, 283414.
[Mo] Momose, F., On the C-adic representations attached to modular forms. J. Fac. Sci. Univ. Tokyo, Sect. IA Math. 28(1981), 89109.
[Mu] Mumford, D., An analytic construction of degenerating curves over complete local rings. Compos. Math. 24(1972), 129174.
[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, 471547.
[N2] Nekovář, J., Selmer complexes. Astérisque 310(2006), Soc. Math. de France, Paris.
[N3] Nekovář, J., On the parity of ranks of Selmer groups III. Doc. Math. 12(2007), 243274; Erratum: Doc. Math. 14(2009), 191–194.
[PW] Pollack, R. and T.Weston, On anticyclotomic μ-invariants of modular forms. Compos. Math., to appear.
[Pr] Prasad, D., Some applications of seesaw duality to branching laws. Math. Ann. 304(1996), 120. http://dx.doi.org/10.1007/BF01446282
[R] Rajaei, A., On the levels of mod Hilbert modular forms. J. Reine Angew. Math. 537(2001), 3365. http://dx.doi.org/10.1515/crll.2001.058
[Ra1] Raynaud, M., Spécialisation du foncteur de Picard. Inst. Hautes Etudes Sci. Publ. Math. 38(1970), 2776.
[Ra2] Raynaud, M., Schémas en groupes de type (p, … , p). Bull. Soc. Math. France 102(1974), 241280.
[Ri1] Ribet, K., On C-adic representations attached to modular forms. Invent. Math. 28(1975), 245275. http://dx.doi.org/10.1007/BF01425561
[Ri2] Ribet, K., Galois action on division points of Abelian varieties with real multiplications. Amer. J. Math. 98(1976), 751804. http://dx.doi.org/10.2307/2373815
[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, 1752.
[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), 4362. http://dx.doi.org/10.1007/BF01457819
[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), 7594.
[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), 185194. http://dx.doi.org/10.1017/S0017089500006170
[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), 431476. http://dx.doi.org/10.1007/BF01231195
[Ru] Rubin, K., Euler systems. Ann. of Math. Stud. 147, Princeton University Press, Princeton, 2000.
[S] Sah, C.-H., Automorphisms of finite groups. J. Algebra 10(1968), 4768. http://dx.doi.org/10.1016/0021-8693(68)90104-X
[Sa] Saito, H., On Tunnell's formula for characters of GL(2). Compos. Math. 85(1993), 99108.
[Sc] Scharlau, W., Quadratic and Hermitian Forms. Grundlehren Math.Wiss. 270, Springer, Berlin, 1985.
[Sk] Skinner, C., A note on p-adic Galois representations attached to Hilbert modular forms. Doc. Math. 14(2009), 241258.
[Ta] Tate, J., p-divisible groups. In: Proceedings of a conference on local fields (Driebergen, 1966), Springer, Berlin, 1967, 158183.
[T1] Taylor, R., On Galois representations associated to Hilbert modular forms. Invent. Math. 98 (1989), 265280. http://dx.doi.org/10.1007/BF01388853
[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, 333340.
[TZ] Tian, Y. and Zhang, S.-W., in preparation.
[Tu] Tunnell, J., Local “-factors and characters of GL2. Amer. J. Math. 105(1983), 12771308. http://dx.doi.org/10.2307/2374441
[V] Varshavsky, Y., P-adic uniformization of unitary Shimura varieties II. J. Differential Geom. 49(1998), 75113.
[VZ] Vasiu, A. and Zink, T., Boundedness results for finite flat group schemes over discrete valuation rings of mixed characteristics. arxiv:0911.2474.
[Vi] Vigneras, M.-F., Arithmétique des algèbres de quaternions. Lecture Notes in Math. 800, Springer, Berlin, 1980.
[W1] Waldspurger, J.-L., Quelques propriétés arithmétiques de certaines formes automorphes sur GL(2). Compos. Math. 54(1985), 121171.
[W2] Waldspurger, J.-L., Sur les valeurs de certains fonctions L automorphes en leur centre de symétrie. Compos. Math. 54(1985), 173242.
[W3] Waldspurger, J.-L., Correspondances de Shimura et quaternions. Forum Math. 3(1991), 219307. http://dx.doi.org/10.1515/form.1991.3.219
[YZZ] Yuan, X., Zhang, S.-W. and Zhang, W., Heights of CM points I. Gross–Zagier formula. Preprint, 2008.
[Zh1] Zhang, S.-W., Heights of Heegner points on Shimura curves. Ann. of Math. (2) 153(2001), 27147. http://dx.doi.org/10.2307/2661372
[Zh2] Zhang, S.-W., Arithmetic of Shimura curves and the Birch and Swinnerton–Dyer conjecture. Sci. China Math. 53(2010), 573592. http://dx.doi.org/10.1007/s11425-010-0046-2
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