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GENERALIZED EXPLICIT RECIPROCITY LAWS Contents 1 GENERALIZED EXPLICIT RECIPROCITY LAWS KAZUYA KATO Contents 1. Review on the classical explicit reciprocity law 4 2. Review on BdR 6 3. p-divisible groups and norm compatible systems in K-groups 16 4. Generalized explicit reciprocity law 19 5. Proof of the generalized explicit reciprocity law 27 6. Another generalized explicit reciprocity law 57 References 64 Introduction 0.1. In this paper, we give generalizations of the explicit reciprocity law to p-adically complete discrete valuation fields whose residue fields are not necessarily perfect. The main theorems are 4.3.1, 4.3.4, and 6.1.9. In subsequent papers, we will apply our gen- eralized explicit reciprocity laws to p-adic completions of the function fields of modular curves, whose residue fields are function fields in one variable over finite fields and hence imperfect, and will study the Iwasawa theory of modular forms. I outline in 0.3 what is done in this paper in the case of p-adic completions of the function fields of modular curves, comparing it with the case of classical explicit reciprocity law outlined in 0.2. I then explain in 0.4 how the results of this paper will be applied in the Iwasawa theory of modular forms, for these applications are the motivations of this paper, comparing it with the role of the classical explicit reciprocity laws in the classical Iwasawa theory and in related known theories. 0.2. The classical explicit reciprocity law established by Artin, Hasse, Iwasawa, and many other people is the theory to give explicit descriptions of Hilbert symbols. One formulation of the classical reciprocity law reviewed in §1.1 has the following form. Let K be a finite unramified extension of Qp, let Kn = K(ζpn )(n ≥ 0) where ζN for × N ≥ 1 denotes a primitive N-th root of 1, and let lim←− (Kn) be the inverse limit with n × × respect to the norm maps (Kn+1) −→ (Kn) .Thenform ≥ 0, a homomorphism × λm :lim←− (Kn) −→ Km n (see §1.1) is defined by using the theory of Hilbert symbols. The classical explicit reci- procity law reviewed in §1.1 is concerned with the explicit description of this homomor- phism by using differential forms. 1 0.3. The generalized explicit reciprocity laws Thm. 4.3.1 and Thm. 4.3.4 have the following form when specialized to p-adic completions of modular function fields. Let N ≥ 3, let Y (N) be the modular curve of level N, and let K be a p-adic completion of the function field of Y (N). (See 3.1.4 and 4.1.4 for precise meaning of this.) For n ≥ 0, n × let Yn = Y (Np ) ⊗Y (N) K. Instead of lim←− (Kn) in 0.2, we consider the inverse limit n lim←− K2(Yn) with respect to the norm maps K2(Yn+1) −→ K2(Yn). For s =(s(i))i=1,2 ∈ n N2 (N = {0, 1, 2, 3,...})andm ≥ 0, we define a homomorphism s ⊗k λm :lim←− K2(Yn) −→ O(Ym) ⊗K coLie(E) (k =2+s(1) + s(2)) n where O(Ym)istheaffine ring of Ym, E is the universal elliptic curve over K (the pull back of the universal elliptic curve over Y (N)), coLie(E) is the space of invariant dif- ferential forms on E which is of dimension one over K,and( )⊗k is the k-th tensor power as a K-vector space. This homomorphism is defined by using the Galois cohomol- ogy theory, crystalline cohomology theory, and the (k − 2)-th symmetric tensor power k−2 SymZp (TpE)ofthep-adic Tate module of E. The generalized explicit reciprocity laws 4.3.1 and 4.3.4 are, when specialized to this case, concerned with the explicit description s of this homomorphism λm by using differential forms. s Note that the target group of λm contains the space of modular forms of weight k of level Npm. 0.4. The results of this paper is applied in the paper [Ka5] as follows. I recall some known theories in 0.4.1 and 0.4.2, and then explain in 0.4.3 what will be done in the paper [Ka5]. The parallelism with the theories in 0.4.1 and 0.4.2 is the guiding principle throughout the paper [Ka5]. 0.4.1. The numbers 1 − α where α is a root of 1 and α ≠ 1, are called cyclotomic units and very important. (0.4.1.a) First, they are related to zeta functions via the analytic formula −1 log(|1 − α|)=− lim s (ζ≡a(N)(s)+ζ≡−a(N)(s)) s→0 where α = exp(2πia/N) =1with̸ a, N ∈ Z, N ≥ 1, and ζ≡a(N)(s) is the partial Riemann zeta function defined by ζ (s)= n−s when Re(s) > 1 ≡a(N) ! n≥1 n≡a mod N and extended by analytic continuation to a moromorphic function on C. (0.4.1.b) Secondly, as is reviewed in §1.2, the elements 1−α are related to zeta functions also in the following p-adic way. ([Iw1]): Via the explicit reciprocity law of p-adic cyclotomic fields, the elements 1−α are related to the values lim(ζ≡a(N)(s) − ζ≡−a(N)(s)). s→1 (0.4.1.c) Thirdly the elements 1 − α form an “Euler system” in the sense of Kolyvagin [Ko]. Rubin [Ru1] obtained a new proof of Iwasawa main conjecture (which was originally proved by Mazur and Wiles [MW]) by using cyclotomic units and the theory of Euler systems. 2 0.4.2. Elliptic units ([Ro]) are important numbers, which are elements of abelian ex- tensions of a quadratic imaginary field F . (0.4.2.a) First, their log(||) are related to partial Dedekind zeta functions of F via analytic formulas similar to those in (0.4.1.a). (0.4.2.b) Secondly, elliptic units are related to the value at s = 1 of the zeta function L(E, s) of an elliptic curve E having complex multiplication by OF (= the integer ring of F ), in the following p-adic way: If we apply the explicit reciprocity law of Wiles [Wi] for Lubin-Tate groups to the case where the base field is a p-adic completion of F and where the Lubin-Tate group is the p-divisible group associated to E, then elliptic units are related to L(E, 1) via this explicit reciprocity law ([CW2]). This fact was important in the works of Coates and Wiles ([CW1]) on Birch and Swinnerton-Dyer conjectures. (Cf.[dS]for(0.4.2.a)and (0.4.2.b).) (0.4.2.c) Thirdly, elliptic units form an Euler system. Rubin ([Ru2]) proved the “main conjecture” for F by using elliptic units and the theory of Euler systems. 0.4.3. In the Iwasawa theory of modular forms, we will use the elements of Beilinson in K2 of modular curves ([Bei]) in place of cyclotomic units and elliptic units. (0.4.3.a) As was shown by Beilinson [Bei], these elements are analytically related to lim s−1L(f, s) for elliptic cusp forms f of weight two, via regulator maps. s→0 (0.4.3.b) We related these elements to zeta functions in the following p-adic way: Our explicit reciprocity law 4.3.1 shows, as in the paper [Ka5], that the elements of s Beilinson are related via the homomorphisms λm in 0.3 to the values L(f, k − 1) of the zeta functions of elliptic cusp forms f of weight k =2+s(1) + s(2). The role of the Lubin-Tate group in (0.4.2.b) is played here by the p-divisible group associated to the universal elliptic curve. (A rough explanation of this (0.4.3.b) is given in 4.3.7.) (0.4.3.c) The elements of Beilinson form an Euler system as is shown in [Ka5]. In subsequent papers, we will study the Iwasawa theory of elliptic cusp forms by using the p- adic result described in (0.4.3.b) and the theory of Euler systems. The “main conjectures” in (0.4.1.c) and (0.4.2.c) were equations between a characteristic polynomial related to ideal class groups and a characteristic polynomial related to zeta functions. In [Ka4] [Ka5] and subsequant papers, for an eigen elliptic cusp form f of weight k ≥ 2, under some mild assumption on f, we will prove that the characteristic polynomial related to Selmer groups divides the characteristic polynomial related to zeta functions (that is, we prove a “half” of the “main conjecture” for f) together with finiteness results for Selmer groups. 0.5. The organization of this paper is as follows. §1. Review on the classical explicit reciprocity law. §2. Review on BdR. §3. p-divisible groups and norm compatible sustems in K-groups. §4. Generalized explicit reciprocity law. §5. Proof of the generalized explicit reciprocity law. §6. Another generalized explicit reciprocity law. In §1, we review the classical explicit reciprocity law, and review the fact that cyclotomic units are related p-adically to zeta functions via the explicit reciprocity law. In §2, we review the theory of Tate, Fontaine, Messing, Faltings and other people about the relationship between p-adic Galois representations and crystalline cohomology. The ring BdR defined by Fontaine in [Fo1] in the perfect residue field case and generalized 3 by Faltings [Fa] to the general case, plays a central role in this theory. We review in §2 neccesary things from this theory.
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