## Kyoto Journal of Mathematics

### The cyclotomic Iwasawa main conjecture for Hilbert cusp forms with complex multiplication

#### Abstract

We deduce the cyclotomic Iwasawa main conjecture for Hilbert modular cusp forms with complex multiplication from the multivariable main conjecture for CM number fields. To this end, we study in detail the behavior of the $p$-adic $L$-functions and the Selmer groups attached to CM number fields under specialization procedures.

#### Article information

Source
Kyoto J. Math., Volume 58, Number 1 (2018), 1-100.

Dates
Received: 23 July 2015
Revised: 21 April 2016
Accepted: 19 July 2016
First available in Project Euclid: 15 February 2018

Permanent link to this document
https://projecteuclid.org/euclid.kjm/1518685212

Digital Object Identifier
doi:10.1215/21562261-2017-0018

Mathematical Reviews number (MathSciNet)
MR3776280

Zentralblatt MATH identifier
06873129

#### Citation

Hara, Takashi; Ochiai, Tadashi. The cyclotomic Iwasawa main conjecture for Hilbert cusp forms with complex multiplication. Kyoto J. Math. 58 (2018), no. 1, 1--100. doi:10.1215/21562261-2017-0018. https://projecteuclid.org/euclid.kjm/1518685212

#### References

• [1] A. Ash and G. Stevens, Modular forms in characteristic $\ell$ and special values of their $L$-functions, Duke Math. J. 53 (1986), 849–868.
• [2] D. Blasius, On the critical values of Hecke $L$-series, Ann. of Math. (2) 124 (1986), 23–63.
• [3] D. Blasius, “Hilbert modular forms and the Ramanujan conjecture,” in Noncommutative Geometry and Number Theory, Aspects Math. E37, Vieweg, Wiesbaden, 2006, 35–56.
• [4] D. Blasius and J. D. Rogawski, Motives for Hilbert modular forms, Invent. Math. 114 (1994), 55–87.
• [5] J.-L. Brylinski and J.-P. Labesse, Cohomologie d’intersection et fonctions $L$ de certaines variétés Shimura, Ann. Sci. Éc. Norm. Supér. (4) 17 (1984), 361–412.
• [6] H. Carayol, Sur les représentations $l$-adiques associées aux formes modulaires de Hilbert, Ann. Sci. Éc. Norm. Supér. (4) 19 (1986), 409–468.
• [7] J. Coates and R. Sujatha, Fine Selmer groups of elliptic curves over $p$-adic Lie extensions, Math. Ann. 331 (2005), 809–839.
• [8] A. Dabrowski, $p$-adic $L$-functions of Hilbert modular forms, Ann. Inst. Fourier (Grenoble) 44 (1994), 1025–1041.
• [9] P. Deligne, “Valeurs de fonctions $L$ et périodes d’intégrales,” with an appendix by N. Koblitz and A. Ogus, in Automorphic Forms, Representations, and $L$-Functions (Proc. Sympos. Pure Math., Oregon State Univ., Corvallis, Ore., 1977), II, Proc. Sympos. Pure Math. 33, Amer. Math. Soc., Providence, 1979, 313–346.
• [10] M. Dimitrov, Automorphic symbols, $p$-adic $L$-functions and ordinary cohomology of Hilbert modular varieties, Amer. J. Math. 135 (2013), 1117–1155.
• [11] S. S. Gelbart, Automorphic Forms on Adèle Groups, Ann. of Math. Stud. 83, Princeton Univ. Press, Princeton, 1975.
• [12] R. Greenberg, On the Iwasawa invariants of totally real number fields, Amer. J. Math. 98 (1976), 263–284.
• [13] R. Greenberg, On the structure of certain Galois groups, Invent. Math. 47 (1978), 85–99.
• [14] R. Greenberg, “Iwasawa theory and $p$-adic deformations of motives,” in Motives, II (Seattle, WA, 1991), Proc. Sympos. Pure Math. 55, Amer. Math. Soc., Providence, 1994, 193–223.
• [15] R. Greenberg, On the structure of certain Galois cohomology groups, Doc. Math. 2006, Extra Vol., 335–391.
• [16] R. Greenberg, Surjectivity of the global-to-local map defining a Selmer group, Kyoto J. Math. 50 (2010), 853–888.
• [17] R. Greenberg, “On the structure of Selmer groups,” in Elliptic Curves, Modular Forms and Iwasawa Theory, Springer Proc. Math. Stat. 188, Springer, Cham, 2016, 225–252.
• [18] H. Hasse, Über die Klassenzahl abselscher Zahlkörper, Springer, Berlin, 1985.
• [19] G. Henniart, “Représentations $\ell$-adiques abéliennes,” in Seminar on Number Theory, Paris 1980–81 (Paris, 1980/1981), Progr. Math. 22, Birkhäuser Boston, Boston, 1982, 107–126.
• [20] H. Hida, On $p$-adic Hecke algebras for $\mathrm{GL}_{2}$ over totally real fields, Ann. of Math. (2) 128 (1988), 295–384.
• [21] H. Hida, “Nearly ordinary Hecke algebras and Galois representations of several variables,” in Algebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, Baltimore, Md., 1989, 115–134.
• [22] H. Hida, Hilbert Modular Forms and Iwasawa Theory, Oxford Math. Monogr., Oxford Univ. Press, Oxford, 2006.
• [23] H. Hida and J. Tilouine, Anti-cyclotomic Katz $p$-adic $L$-functions and congruence modules, Ann. Sci. Éc. Norm. Supér. (4) 26 (1993), 189–259.
• [24] H. Hida and J. Tilouine, On the anticyclotomic main conjecture for CM fields, Invent. Math. 117 (1994), 89–147.
• [25] M.-L. Hsieh, Eisenstein congruence on unitary groups and Iwasawa main conjecture for CM fields, J. Amer. Math. Soc. 27 (2014), 753–862.
• [26] K. Kato, “$p$-adic Hodge theory and values of zeta functions of modular forms,” in Cohomologies $p$-adiques et applications arithmetiques, III, Astérisque 295, Soc. Math. France, Paris, 2004, 117–290.
• [27] N. M. Katz, $p$-adic $L$-functions for CM fields, Invent. Math. 49 (1978), 199–297.
• [28] J.-P. Labesse and R. P. Langlands, $L$-indistinguishability for $\mathrm{SL}(2)$, Canad. J. Math. 31 (1979), 726–785.
• [29] J. I. Manin, Non-Archimedean integration and $p$-adic Jacquet-Langlands $L$-functions (in Russian), Uspehi Mat. Nauk. 31, no. 1 (1976), 5–54.
• [30] B. Mazur, J. Tate, and J. Teitelbaum, On $p$-adic analogues of the conjectures of Birch and Swinnerton-Dyer, Invent. Math. 84 (1986), 1–48.
• [31] B. Mazur and A. Wiles, Class fields of abelian extensions of $\mathbb{Q}$, Invent. Math. 76 (1984), 179–330.
• [32] T. Miyake, On automorphic forms on $\mathrm{GL}_{2}$ and Hecke operators, Ann. of Math. (2) 94 (1971), 174–189.
• [33] C. P. Mok, The exceptional zero conjecture for Hilbert modular forms, Compos. Math. 145 (2009), 1–55.
• [34] J. Neukirch, A. Schmidt, and K. Wingberg, Cohomology of Number Fields, Grundlehren Math. Wissen. 323, Springer, Berlin, 2000.
• [35] T. Ochiai, Euler system for Galois deformations, Ann. Inst. Fourier (Grenoble) 55 (2005), 113–146.
• [36] T. Ochiai, Several variables $p$-adic $L$-functions for Hida families of Hilbert modular forms, Doc. Math. 17 (2012), 807–849.
• [37] T. Ochiai and K. Prasanna, Two-variable Iwasawa theory for Hida families with complex multiplication, in preparation.
• [38] M. Ohta, On the zeta function of an abelian scheme over the Shimura curve, Japan J. Math. (N.S.) 9 (1983), 1–25.
• [39] A. A. Panchishkin, Motives over totally real fields and $p$-adic $L$-functions, Ann. Inst. Fourier (Grenoble) 44 (1994), 989–1023.
• [40] B. Perrin-Riou, Groupe de Selmer d’une courbe elliptique à multiplication complexe, Compos. Math. 43 (1981), 387–417.
• [41] K. A. Ribet, “Galois representations attached to eigenforms with nebentypus,” in Modular Functions of One Variable, V (Proc. Second Internat. Conf., Univ. Bonn, Bonn, 1976), Lecture Notes in Math. 601, Springer, Berlin, 1977, 17–51.
• [42] D. E. Rohrlich, On $L$-functions of elliptic curves and cyclotomic towers, Invent. Math. 75 (1984), 409–423.
• [43] D. E. Rohrlich, $L$-functions and division towers, Math. Ann. 281 (1988), 611–632.
• [44] K. Rubin, The “main conjectures” of Iwasawa theory for imaginary quadratic fields, Invent. Math. 103 (1991), 25–68.
• [45] K. Rubin, Euler Systems, Ann. of Math. Stud. 147, Princeton Univ. Press, Princeton, 2000.
• [46] N. Schappacher, Periods of Hecke Characters, Lecture Notes in Math. 1301, Springer, Berlin, 1988.
• [47] J.-P. Serre, Abelian $l$-Adic Representations and Elliptic Curves, Res. Notes Math. 7, A K Peters, Wellesley, Mass., 1998.
• [48] G. Shimura, The special values of the zeta functions associated with Hilbert modular forms, Duke Math. J. 45 (1978), 637–679.
• [49] G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, reprint of the 1971 original, Publ. Math. Soc. Japan 11, Princeton Univ. Press, Princeton, 1994.
• [50] C. Skinner and E. Urban, The Iwasawa main conjecture for $\mathrm{GL}_{2}$, Invent. Math. 195 (2014), 1–277.
• [51] G. Stevens, The cuspidal group and special values of $L$-functions, Trans. Amer. Math. Soc. 291, no. 2 (1985), 519–550.
• [52] J. Tate, “Duality theorems in Galois cohomology over number fields,” in Proc. Internat. Congr. Mathematicians (Stockholm, 1962), Inst. Mittag-Leffler, Djursholm, 1963, 288–295.
• [53] R. Taylor, On Galois representations associated to Hilbert modular forms, Invent. Math. 98 (1989), 265–280.
• [54] R. Taylor, “On Galois representations associated to Hilbert modular forms, II,” in Elliptic Curves, Modular Forms, and Fermat’s Last Theorem (Hong Kong, 1993), Ser. Number Theory 1, Int. Press, Cambridge, Mass., 1995, 185–191.
• [55] L. C. Washington, Introduction to Cyclotomic Fields, 2nd ed., Grad. Texts in Math. 83, Springer, New York, 1997.
• [56] A. Wiles, On $p$-adic representations for totally real fields, Ann. of Math. (2) 123 (1986), 407–456.
• [57] A. Wiles, On ordinary $\lambda$-adic representations associated to modular forms, Invent. Math. 94 (1988), 529–573.
• [58] J.-P. Wintenberger, Structure galoisienne de limites projectives d’unités locales, Compos. Math. 42 (1980/1981), 89–103.