Anticyclotomic main conjectures.

*(English)*Zbl 1200.11082Let \(p>3\) be prime. Let \(F\) be a totally real number field, \(M/F\) a totally imaginary quadratic extension in which all prime ideals dividing \(p\) are unramified, \(\Sigma\) a \(p\)-ordinary CM type of \(M\), \(\overline{W}\) the completion of the ring of integers in an algebraic closure of \(\mathbb Q_p\), and \(\psi: \text{Gal}(\overline{F}/M)\to \overline{W}^{\times}\) an anticyclotomic character of finite order prime to \(p\). Assume that the conductor of \(\psi\), regarded as a Hecke character, is a product of primes above \(p\) and primes that split in \(M/F\), that the local characters \(\psi_{\mathfrak P}\) are non-trivial for all \({\mathfrak P}\in \Sigma_p\), and that \(\psi\) restricted to \(\text{Gal}(\overline{F}/M(\sqrt{(-1)^{(p-1)/2}p}))\) is nontrivial. Let \(L_p^-(\psi)\) be the anticyclotomic \(p\)-adic Hecke \(L\)-function, regarded as an element of \(\overline{W}[[\Gamma_M^-]]\), where \(\Gamma_M^-\) is the Galois group of the composite of the anticyclotomic \(\mathbb Z_p\)-extensions of \(M\). Let \(X=\text{Gal}(L_{\infty}/M_{\infty}^-M(\psi))\), where \(M(\psi)\) is the fixed field of the kernel of \(\psi\) and \(L_{\infty}/M_{\infty}^-M(\psi)\) is the maximal abelian \(p\)-extension unramified outside \(\Sigma_p\). Then \(\text{Gal}(M_{\infty}^-M(\psi)/M)\) acts on \(X\) by conjugation. Let \({\mathcal F}^-(\psi)\) be the characteristic polynomial of \(X[\psi]\). The main result of the paper is that \({\mathcal F}^-(\psi)= L_p^-(\psi)\) up to a unit in \(\overline{W}[[\Gamma_M^-]]\). It was already known from work of the author [in: \(L\)-functions and Galois representations, Burns, David (ed.) et al., Based on the symposium, Durham, UK, July 19–30, 2004. Cambridge: Cambridge University Press. London Mathematical Society Lecture Note Series 320, 207–269 (2007; Zbl 1159.11023)] that \(L_p^-(\psi)\) divides \({\mathcal F}^-(\psi)\). The reverse divisibility is proved by reducing it to an integrality statement, and this is proved using generalized Eisenstein series, introduced by G. Shimura [Ann. Math. (2) 111, 313–375 (1980; Zbl 0438.12003)], [Ann. Math. 114, 127–164 (1981; Zbl 0468.10016); ibid. 569–607 (1981; Zbl 0486.10021)], on orthogonal groups of signature \((n,2)\).

The present paper extends work of K. Rubin [Invent. Math. 93, 701–713 (1988; Zbl 0673.12004)], [Invent. Math. 103, No. 1, 25–68 (1991; Zbl 0737.11030)], B. Mazur and J. Tilouine [Publ. Math., Inst. Hautes Étud. Sci. 71, 65–103 (1990; Zbl 0744.11053)], and J. Tilouine [Duke. Math. J. 59, 629–673 (1989; Zbl 0707.11079)]. The methods of the present paper are based on those of the author and J. Tilouine [Ann. Sci. Éc. Norm. Sup. 4-th series 26, 189–259 (1993; Zbl 0778.11061), Invent. Math. 117, 89–147 (1994; Zbl 0819.11047)] and the author [in: \(L\)-functions and Galois representations, Burns, David (ed.) et al., Based on the symposium, Durham, UK, July 19–30, 2004. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Series 320, 207–269 (2007; Zbl 1159.11023)]. More recent work of the author [Int. Math. Res. Not. 5, 912–952 (2009; Zbl 1193.11103)] removes the condition that the primes not above \(p\) in the conductor of \(\psi\) split completely in \(M/F\).

The present paper extends work of K. Rubin [Invent. Math. 93, 701–713 (1988; Zbl 0673.12004)], [Invent. Math. 103, No. 1, 25–68 (1991; Zbl 0737.11030)], B. Mazur and J. Tilouine [Publ. Math., Inst. Hautes Étud. Sci. 71, 65–103 (1990; Zbl 0744.11053)], and J. Tilouine [Duke. Math. J. 59, 629–673 (1989; Zbl 0707.11079)]. The methods of the present paper are based on those of the author and J. Tilouine [Ann. Sci. Éc. Norm. Sup. 4-th series 26, 189–259 (1993; Zbl 0778.11061), Invent. Math. 117, 89–147 (1994; Zbl 0819.11047)] and the author [in: \(L\)-functions and Galois representations, Burns, David (ed.) et al., Based on the symposium, Durham, UK, July 19–30, 2004. Cambridge: Cambridge University Press. Lond. Math. Soc. Lect. Note Series 320, 207–269 (2007; Zbl 1159.11023)]. More recent work of the author [Int. Math. Res. Not. 5, 912–952 (2009; Zbl 1193.11103)] removes the condition that the primes not above \(p\) in the conductor of \(\psi\) split completely in \(M/F\).

Reviewer: Lawrence C. Washington (College Park)

##### MSC:

11R23 | Iwasawa theory |

11F27 | Theta series; Weil representation; theta correspondences |

11F30 | Fourier coefficients of automorphic forms |

11F33 | Congruences for modular and \(p\)-adic modular forms |

11F41 | Automorphic forms on \(\mbox{GL}(2)\); Hilbert and Hilbert-Siegel modular groups and their modular and automorphic forms; Hilbert modular surfaces |

11F60 | Hecke-Petersson operators, differential operators (several variables) |

11F80 | Galois representations |

11G15 | Complex multiplication and moduli of abelian varieties |

11G18 | Arithmetic aspects of modular and Shimura varieties |

11R34 | Galois cohomology |

11R42 | Zeta functions and \(L\)-functions of number fields |