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Characteristic Elements in Noncommutative Iwasawa Theory

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Characteristic Elements in Noncommutative Iwasawa Theory Characteristic Elements in Noncommutative Iwasawa Theory Meinen Eltern Contents Introduction 1 General Notation and Conventions 9 1. p-adic Lie groups 11 2. The Iwasawa algebra - a review of Lazard’s work 13 3. Iwasawa-modules 18 4. Virtual objects and some requisites from K-theory 24 5. The relative situation 28 5.1. Non-commutative power series rings 29 5.2. The Weierstrass preparation theorem 32 5.3. Faithful modules and non-principal reflexive ideals 33 6. Ore sets associated with group extensions 36 6.1. The direct product case 40 6.2. The semi-direct product case 42 6.3. The GL2 case 43 7. Twisting 45 7.1. Twisting of Λ-modules 45 7.2. Evaluating at representations 50 7.3. Equivariant Euler-characteristics 52 8. Descent of K-theory 57 9. Characteristic elements of Selmer groups 65 9.1. Local Euler factors 66 9.2. The characteristic element of an elliptic curve over k∞ 67 9.3. The false Tate curve case 68 9.4. The GL2-case 71 10. Towards a main conjecture 71 Appendix A. Filtered rings 74 Appendix B. Induction 76 References 79 Introduction Let p be a prime number, which, for simplicity, we shall always assume odd. The goal of non-commutative Iwasawa theory is to extend Iwasawa theory over Zp- extensions to the case of p-adic Lie extensions. Thus it might be useful to recall briefly some main aspects of the classical theory: Let k be a finite extension of Q, and write kcyc for the cyclotomic Zp-extension of k. We put Γ = G(kcyc/k). The main conjecture for an elliptic curve E over k relates the Γ-module structure of the Pontryagin dual ∨ X(kcyc) = Xf (E/kcyc) = (Selp∞ (E/kcyc)) of the Selmer group Selp∞ (E/kcyc) of E over kcyc to the p-adic L-function of E which in turn interpolates the values at 1 of the complex Hasse-Weil L-function of E twisted by Artin characters. More precisely, assuming that E has good ordinary reduction at all places Sp above p Mazur’s conjecture predicts that X(kcyc) is a Λ(Γ)-torsion module. Here we write Λ(G) = Zp[[G]] for the Iwasawa algebra of a p-adic Lie group G and we recall that Λ(Γ) can be identified with the formal power series ring Zp[[X]] depending on the choice of a topological generator of Γ. Using the structure theory of finitely generated modules over an integrally closed domain, one assigns to X(kcyc) its characteristic ideal char(X(kcyc)) = (fX(kcyc)) which is generated by a unique polynomial fX(kcyc) in Λ(Γ) = Zp[[X]], which is a power of p times a distinguished polynomial. The existence of a p-adic L-function LE of E is conjectured in general, but it has only been proven in special cases, e.g. if E is already defined over Q (with good ordinary reduction at p) and k is a finite abelian extension of Q [37] or if E is an elliptic curve, which is defined over its field of complex multiplication K and k is any abelian extension of K. In any case, we view the p-adic L-function as a p-adic measure and thus as an element of Λ(Γ). The main conjecture in the ordinary case asserts that, up to a unit in Λ(Γ), the p-adic L-function LE in Λ(Γ) agrees with fX(kcyc), i.e. LEΛ(Γ) = fX(kcyc)Λ(Γ). The reason for studying the main conjecture is that it is essentially the only known general procedure for studying the exact formulae of number theory (for example, the Birch and Swinnerton-Dyer conjecture for E over k in this case). One has 2 INTRODUCTION to evaluate LE on the analytic side while, on the Iwasawa module side, one has to consider Galois descent. More precisely, evaluation of fX(kcyc) corresponds to calculating the Γ-Euler characteristic Y (−1)i χ(Γ,X(kcyc)) := (#Hi(Γ,X(kcyc))) i≥0 of X(kcyc) and one obtains for the p-adic absolute values −1 −1 |LE(0)|p = |fX(kcyc)(0)|p = χ(Γ,X(kcyc)) if fX(kcyc)(0) is non-zero or equivalently if χ(Γ,X(kcyc)) is defined, i.e. if Hi(Γ,X(kcyc)) is finite for i = 0 and i = 1. Thus, if the value L(E, 1) at 1 of the Hasse-Weil L-function of E is non-zero, one obtains as a corollary from the main conjecture a special case of the conjecture of Birch and Swinnerton-Dyer using the explicit determination of χ(Γ,X(kcyc)) = ρp(E/k) by Perrin-Riou [44] and Schneider [47], as is explained in more detail in [53, p.592ff]. Here ρp(E/k) denotes the p-Birch and Swinnerton-Dyer constant, see section 9.3. Let us now replace kcyc by some p-adic Lie extension k∞ of k, i.e. a Galois exten- sion such that its Galois group G = G(k∞/k) is a p-adic Lie group. Moreover, in order to simplify our discussion, we assume that G is a pro-p group and has no elements of order p. The first main example arises by adjoining the coordinates of all p-power division points E(p) on E (but note the technically nasty fact that we may then have to replace k by a finite extension to ensure that G is pro-p). If E does not admit complex multiplication, G is an open subgroup of GL2(Zp) via the representation on the Tate module of E by a celebrated result of Serre [50]. As a second important example we shall also discuss the “false Tate curve” where k∞ arises as trivializing extension of a p-adic representation which is analogous to that of the local Galois representation associated with a Tate elliptic curve. th More precisely, we assume that k contains the group µp of p roots of unity and × then k∞ is obtained by adjoining to kcyc the p-power roots of an element in k which is not a root of unity. By Kummer theory, its Galois group is isomorphic ∼ to the semidirect product G(k∞/k) = Zp(1) o Zp where the action is given by the cyclotomic character. Again we consider the Selmer group of an elliptic curve over k with good ordinary reduction at Sp as above. Conjecturally, the dual X(k∞) = Xf (E/k∞) of the Selmer group Selp∞ (E/k∞) of E over k∞ is again a Λ(G)-torsion module [8] and in the above cases this can be shown under certain conditions ([9],[22]) using Kato’s deep theorem on Euler systems. INTRODUCTION 3 Now the crucial algebraic question needed to be answered to formulate a main conjecture in general is the following: Can one assign to any finitely generated Λ(G)-torsion module M, a characteristic element, FM say, lying in the field of fractions of Λ(G) having the following property: if M has finite G-Euler characteristic, then FM (0) exists and is equal, up to a p-adic unit, to χ(G, M), and similarly for all twists of M by Artin characters of G? We should mention that on the p-adic analytic part concerning p-adic L-functions adapted to this setting practically nothing is known and thus we will mainly discuss what one might expect only on the algebraic side of the picture that has been sketched. In the following we describe several attempts and problems in the quest of such characteristic elements. A major achievement in non-commutative Iwasawa theory is certainly the struc- ture theorem on torsion Iwasawa modules proved by J. Coates, P. Schneider and R. Sujatha [10]. It says roughly that every finitely generated Λ(G)-torsion mod- ule M decomposes - up to pseudo-isomorphism - into the direct sum of cyclic modules Λ(G)/Li, i = 1, . , r, where Li denote reflexive left ideals of Λ(G) r M M ≡ Λ(G)/Li (modulo pseudo-null) i=1 (see 3.6 for the precise statement and the needed conditions on G). It was aston- ishing to obtain this strong (non-commutative) result almost totally parallel to the above mentioned structure theory over integrally closed (commutative) do- mains. Unfortunately, it turned out that the Euler characteristic is not invariant under pseudo-isomorphisms in general (see 3.7, 8.8) and reflexive left ideals need not be principal by an example of D. Vogel (see 5.13). Due to this dilemma it is still not clear which role this structure result plays in the above picture that is expected. Going back to commutative Iwasawa theory once again we recall that the char- acteristic polynomial fM of a Λ(Γ)-torsion module M can also be interpreted as an element in the relative K-group ∼ × × K0(Λ(Γ),Q(Γ)) = Q(Γ) /Λ(Γ) associated to the ring homomorphism Λ(Γ) → Q(Γ), where Q(Γ) denotes the field of fractions of Λ(Γ). Here, for any ring R, we write R× for the group of units. As observed by Kato, this interpretation can be described nicely using the determi- nant functor of Knudsen and Mumford [32]. We should remark that this relative K-group is naturally isomorphic to the Grothendieck group K0(Λ(Γ)-modtor) of the (abelian) category Λ(Γ)-modtor of finitely generated Λ(Γ)-torsion modules. 4 INTRODUCTION In the general situation of p-adic Lie groups (with our general assumption) there exists also a skew field of fractions Q(G) of Λ(G) and from the long exact sequence of G- respectively K-theory one obtains again an identification ∼ ∼ × × × × K0(Λ(G)-modtor) = K0(Λ(G),Q(G)) = Q(G) /[Q(G) ,Q(G) ]Λ(G) .
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