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Anticyclotomic main conjectures. (English) Zbl 1200.11082
Let \(p>3\) 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 \(M/F\).

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
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