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On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

  • Jeanine Van Order (a1)


We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$ -adic $L$ -functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over $\text{CM}$ fields via the technique of Skinner and Urban.

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[1] Bertolini, M. and Darmon, H., Derived heights and generalized Mazur-Tate regulators. Duke Math. J. 76(1994), no. 1, 75111.
[2] Bertolini, M., Heegner points on Mumford-Tate curves. Invent. Math. 126(1996), no. 3, 413456.
[3] Bertolini, M., Iwasawa's main conjecture for elliptic curves over anticyclotomic Zp-extensions. Ann. of Math. 162(2005), no. 1, 164.
[4] Bertolini, M., A rigid-analytic Gross-Zagier formula and arithmetic applications. Invent. Math. 126(1996), no. 3, 413456.
[5] Bosch, S.,Lütkebohmert, W., and Raynaud, M., Néron models. Ergebnisse der Mathematik und Ihre Grenzgebiete (3), 21, Springer- Verlag, Berlin, 1990.
[6] Bourbaki, N., Élements de mathématique, Fasc. XXXI, Algèbre commutative, Chapitre 7: Diviseurs. Actualiés Scientifiques et Industrielles, 1314, Paris, Hermann, 1965.
[7] Boutot, J.-F. and Carayol, H. Uniformisation p-adique des courbes de Shimura: les thèoreèmes de Cerednik et Drinfeld. Courbes modulaires et courbes de Shimura (Orsay, 1987/1988). Astérisque 196-197(1991), 7, 45158.
[8] Buzzard, K., Integral models of certain Shimura curves. Duke Math. J. 87(1997), no. 3, 591612.
[9] Carayol, H., Sur la mauvaise réduction des courbes de Shimura. Compositio Math. 59(1986), no. 2, 151230.
[10] Carayol, H., Sur les représentations galoisiennes modulo l attachées aux formes modulaires. Duke. Math J.. 59(1989), no. 3, 785801.
[11] Cerednik, I. V., Uniformization of algebraic curves by discrete arithmetic subgroups of PGL2(kw) with compact quotient spaces. (Russian) Math. Sb. 100(142)(1976) no. 1, 5988; translated in Math. USSR Sbornik, 29(1976), 55-78.
[12] Cheng, C., Multiplicities of Galois representations in cohomology groups of Shimura curves over totally real field. Ph.D. thesis, Northwestern University, 2011.
[13] Cheng, C., Ihara's lemma for Shimura curves. preprint, 2011.
[14] Coates, J. and Greenberg, R., Kummer theory for abelian varieties over local fields. Invent. Math. 124(1996), no. 1-3, 129174.
[15] Cornut, C. and Vatsal, V., CM points and quaternion algebras. Doc. Math. 10(2005), 263309.
[16] Cornut, C., Nontriviality of Rankin-Selberg L-functions and CM points. In: L-functions and Galois representations, London Math. Soc. Lecture Note Ser., 320, Cambridge University Press, Cambridge, 2007, pp. 121186.
[17] Diamond, F., The Taylor-Wiles construction and multiplicity one. Invent. Math. 128(1997), no. 2, 379391.
[18] Diamond, F. and Taylor, R., Nonoptimal levels of mod l modular representations. Invent. Math. 115(1994), no. 3, 435462.
[19] Dimitrov, M., Galois representations modulo p and cohomolgy of Hilbert modular varieties. Ann. Sci. É cole Norm. Sup. (4) 38(2005), no. 4, 505551.
[20] Drinfeld, V., Coverings of p-adic symmetric domains. (Russian), Funkcional. Anal. i Prilozen. 10(1976), no. 2, 29-40; translated in Funct. Anal. Appl. 10(1976), 107115.
[21] Drinfeld, V., Elliptic modules. (Russian) Mat. Sb. (N.S.) 94(136)(1974), 594–627, 656.
[22] B. Edixhoven, Appendix in [4].
[23] Fouquet, O., Dihedral Iwasawa theory of nearly ordinary quaternionic automorphic forms. 2009,
[24] Garrett, P. B., Holomorphic Hilbert modular forms. The Wadsworth & Brooks/Cole Mathematics Series.Wadsworth & Brooks/Cole Advanced Books & Software, Pacific Grove, CA, 1990.
[25] van der Geer, G., Hilbert modular surfaces. Ergebnisse der Mathematik und ihrer Grenzgebiete (3), 16, Springer-Verlag, Berlin, 1988.
[26] Goren, E. Z., Lectures on Hilbert modular varieties and modular forms. CRM Monograph Series, 14 American Mathematical Society, Providence, RI, 2001.
[27] Grothendieck, A., ed., Groupes de monodromie en géométrie algébrique, Séminaire de Géometrie Algébrique du Bois-Marie 1967–1969. Lecture Notes in Math., 228, Springer-Verlag, Berlin-New York, 1972.
[28] Hachimori, Y. and Venjakob, O., Completely faithful Selmer groups over Kummer extensions. Doc. Math. 2003, Extra Vol., 443478.
[29] Howard, B., Bipartite Euler systems. J. Reine Angew. Math. 597(2006), 125.
[30] Howard, B., Iwasawa theory of Heegner points on abelian varieties of GL2-type. Duke Math. J. 124(2004), no. 1, 145.
[31] Ihara, Y., Shimura curves over finite fields and their rational points. In: Applications of curves over finite fields (Seattle,WA, 1997), Contemp. Math., 245, American Mathematical Society, Providence, RI, 1999, pp. 1523.
[32] Jacquet, H. and Langlands, R. P., Automorphic forms on GL(2). Lecture Notes in Mathematics, 114, Springer-Verlag, Berlin-New York, 1970.
[33] Jarvis, F., Mazur's principle for totally real fields. Compositio Math. 116(1999), no. 1, 3979.
[34] Jarvis, F., Level lowering for modular mod l representations over totally real fields. Math. Ann. 313(1999), no. 1, 141160.
[35] Jordan, B.W. and Livné, R. A., Local Diophantine properties of Shimura curves. Math. Ann. 270(1985), no. 2, 235248.
[36] Kato, K., p-adic Hodge theory and values of zeta functions of modular forms. Cohomologies p-adiques et applications arithmétiques. III. Astérisque 295(2004), ix, 117290.
[37] Katz, N. M. and Mazur, B., Arithmetic moduli of elliptic curves. Annals of Mathematics Studies, 108, Princeton University Press, Princeton, NJ, 1985.
[38] Kisin, M., Moduli of finite flat group schemes, and modularity. Ann. of Math. 170(2009), no. 3, 10851180.
[39] Kurihara, A., On some examples of equations defining Shimura curves and the Mumford uniformization. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 25(1979), no. 3,277300.
[40] Ling, S., Shimura subgroups of Jacobians of Shimura curves. Proc. Amer. Math. Soc. 118(1993), no. 2, 385390.
[41] Longo, M., Anticyclotomic Iwasawa's main conjecture for Hilbert modular forms. Commentarii Mathematici Helvetici, to appear.
[42] Longo, M., Euler systems obtained from congruences between Hilbert modular forms. Rend. Semin. Mat. Univ. Padova 18(2007), 134.
[43] Longo, M., On the Birch and Swinnerton-Dyer conjecture for modular elliptic curves over totally real fields. Ann. Inst. Fourier (Grenoble) 56(2006), no. 3, 689733.
[44] Mazur, B. and Rubin, K., Kolyvagin systems. Mem. Amer. Math. Soc. 168(2004), no. 799.
[45] Mazur, B. and Wiles, A., Class fields of abelian extensions of Q. Invent. Math. 76(1984), no. 2, 179330.
[46] Milne, J. S., Arithmetic duality theorems. Perspectives in Mathematics, 1. Academic Press, Boston, MA, 1986.
[47] Morita, Y., Reduction modulo p of Shimura curves. Hokkaido Math. J. 10(1981), no. 2, 209238.
[48] Mumford, D., An analytic construction of degenerating curves over complete local rings Compos. Math. 24(1972), 129174.
[49] Nekovar, J., Level raising and anticyclotomic Selmer groups for Hilbert modular forms of weight two. Canad. J. Math., to appear.
[50] Pollack, R. and Weston, T., On anticyclotomic μ-invariants of modular forms. Compositio Math., to appear.
[51] Rajaei, A., On the levels of mod l Hilbert modular forms. J. Reine Angew. Math. 537(2001), 3365.
[52] Raynaud, M., Spécialization du foncteur de Picard. Inst. Hautes Études Sci. Publ. Math. 38(1970), 2776.
[53] Ribet, K., Bimodules and abelian surfaces. In: Algebraic number theory, Adv. Stud. Pure. Math., 17, Academic Press, Boston, MA, 1989, pp. 359407.
[54] Ribet, K., On modular representations of Gal(Q/Q) arising from modular forms. Invent. Math. 100(1990), no. 2, 431476.
[55] Serre, J.-P., Propriétés galoisiennes des points d’ordre fini des courbes elliptiques. Invent. Math. 15(1972), no. 4, 259331.
[56] Shimura, G., The special values of zeta functions associated with Hilbert modular forms. Duke Math. J. 45(1978), no. 3, 637679.
[57] Skinner, C. and Urban, E., The Iwasawa main conjectures for GL2.
[58] Tate, J. T., Global class field theory. In: Algebraic Number Theory (Proc. Instructional Conf., Brighton, 1965), Thompson, Washington, D.C., 1967, pp. 162203.
[59] Tate, J. T., Relations between K2 and Galois cohomology. Invent Math. 36(1976), 257274.
[60] Taylor, R., On Galois representations associated to Hilbert modular forms. Invent. Math. 98(1989), no. 2, 265280.
[61] Van Order, J., On the dihedral Euler characteristics of Selmer groups of abelian varieties. arxiv:abs/1112.3825
[62] Van Order, J., On the quaternionic p-adic L-functions associated to Hilbert modular eigenforms. arxiv:abs/1112.3821.
[63] Vatsal, V., Special values of anticyclotomic L-functions. Duke Math. J. 116(2003), no. 2, 219261.
[64] Varshavsky, Y., P-adic uniformization of unitary Shimura varieties. Inst. Hautes Études Publ. Math. 87(1998), 57–119.
[65] Varshavsky, Y., P-adic uniformization of unitary Shimura varieties. II. J. Differential Geom. 49(1998), no. 1, 75113.
[66] Vignéras, M.-F., Arithmétique des algèbres des quaternions. Lecture Notes in Mathematics, 800, Springer, Berlin, 1980.
[67] Waldspurger, J.-P., Sur les valeurs de certaines fonctions L automorphes en leur centre de symétrie. Compositio Math. 54(1985), no. 2, 173242.
[68] Wan, X., Ph.D. Thesis, Princeton University, in progress.
[69] Wan, X., On ordinary λ-adic representations associated to modular forms. Invent. Math. 94(1988), no. 3, 529573.
[70] Yuan, X., Zhang, S.-W., and W.|Zhang, Heights of CM points I: Gross-Zagier formula.
[71] Zhang, S.-W., Gross-Zagier formular for GL2. Asian J. Math. 5(2001), no. 2, 183290.
[72] Zhang, S.-W., Heights of Heegner points on Shimura curves. Ann. of Math. 153(2001), no. 1, 27147.
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On the Dihedral Main Conjectures of Iwasawa Theory for Hilbert Modular Eigenforms

  • Jeanine Van Order (a1)


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