Preliminaries
Basic definitions
The actions of φ and G on ˜B are limited to endomorphisms of BK and AK for each K, and vi.
Then B+cris is a subring of B+dR which is stable under the influence of GQp; it clearly contains and we write Bcris =B+cris[1/t]. The action ofφon Binf+ extends to an action onB+cris, and we haveφ(t) =pt. Note that φ does not actually have a good extension to BdR+ ; Bcris+ is essentially the subring which φ extends out nicely). The action ofφonBcris oscillates with the effect of GK, so we have a semi-linear action ofφonDcris(V).
If dimKDcris(V) = dimQp(V), then V is said to be crystalline. Thus every crystal representation is also de Rham.). We define the GK representation Qp(k), called the kth Tate twist of Qp, with the action g·s=χcyclo(g)k·s. Then, for any GK-representation V, we define the Tate twist V as V(k) =V ⊗QpQp(k); that is, GK acts on V(k) with a given action on V times the kth power of cyclotomic character.
As V than the Rham (resp. crystalline) is, het once that(k) the Rham is (resp. crystalline), and dR(V(k)) =t−kDdR(V) (resp. crystalline).
The Bloch-Kato exponential map
The Tamagawa number conjecture for Tate motives
- The conjecture of Bloch and Kato
- The dual of exp
- The group ring Q p [G]
- The equivariant Tamagawa number conjecture
- Iwasawa cohomology
- Iwasawa cohomology of Q p (1)
- Certain quotients of H Iw 1 (K, Z p (r))
- The reciprocity law
- The operator ∇
- The map T m φ −n on A ψ=p K r−1
For any ~ GK-representation V, we can naturally identify DdR(V∗(1)) with the dual of DdR(V): there exists a non-degenerate pairing given by. Having established the period isomorphism, we turn our attention to the study of the ring structure of the group. Then for each χ∈G, we setˆ c(χ) to be the conductor of χ, and we have. 1.4) Note that since -|G|, we know that ˆG coincides with the irreducible character set of Gover Qurp, and thus the element given in (1.4) is always a unit of Qurp.
In this chapter we will develop some tools that we can use to study the equivalent conjecture of Tamagawa numbers. In section 2.1 we will define the Iwasawa cohomology and the Iwasawa algebra, and prove some results about the Iwasawa cohomology of the Tate motif. In section 2.2 we introduce the theory of (φ,ΓK)-modules and state an explicit reciprocity law for the map exp∗.
In this section we introduce some Iwasawa theory, define Iwasawa cohomology and show that we can realize the group H1(K,Zp(r)) of Theorem 1.2.1 and Conjecture 1.2.3 as a quotient of the Iwasawa cohomologyHIw1 (K,Zp) (1)). Remember we have a compatible system of pnth roots of unityζpn, and for each nwe setKn =K(ζpn). For the rest of this section, we will study the Iwasawa cohomology group HIw1 (K,Qp(1)).
If H is an abelian group, we can define the pro-pcomplement of H, written Hb, to be the inverse limit lim. We observe that since Ubis the kernel of the valuation map on Zp, it is the augmentation ideal of A(K∞). We prove the accuracy of the first sequence; the proof of the second sequence is similar.
In particular, we define the Cherbonnier-Colmez double exponential, and recall a reciprocity law for it in the case of the Tate twist. Recall that the p-adic realization of the Tate motif Qp(r) is a GK action given by g·s = χcyclo(g)r·s, and that for any GK representationV we haveV(r) =V ⊗QpQp(r) ). If we define log by the usual power series, then we have that∇logα= ∇αα for every αin B for which log converges.
If we take taker= 1, then we found that Tmφ−nP is invariant for sufficiently large n—in particular for n≥1—if P ∈Aψ=1K. Exactly the same proof works for BKψ=1 =D(Qp)ψ=1, so we recapture the special case of the statement in 2.2.2 that Tm(φ−n(Exp*V*(1)(µ))) is invariant for big enough.
Butb OEK = kJπKK, and thus Od×EK = k\JπKK× = 1 +πKkJπKK, and we have the first isomorphism.
This lemma suggests that the ideal (σ−1, δe−1, γ−1) is a principal ideal of ΛK, since we know that Ubtf ∼= ΛK and thus over ΛK is generated by a single element. In section 4.1 we will study the structure of Aψ=1F and find an element that generates Aψ=1F (1)tf. In section 4.2 we will study the map T0◦φ−n and prove the conjecture for unramified extensions of Qp.
The module P F
Using Theorem 2.2.2, we can answer this question by constructing the Cherbonnier-Colmez double exponential Exp∗Q. We want to extend this isomorphism to give an isomorphism of OFJπKlog to a subgroup ofFJπK, which motivates the following discussion. Since the map dlog is given by an integral power series, we know that logOFJπKlog ⊆PF.
So the function defined by ρ(f) = ˆf∞ is a well-defined homomorphism, and since ψ( ˆf∞) =f∞/p, its image is inPψ=pF,log−1. We know that ρ(f)∈Pψ=pF,log−1 and that the image ofρ(f) modpOFJπKis f, soρ is surjective and therefore an isomorphism. Since log is given by a power series, it respects ring homomorphism and therefore commutes with φ and ΓF.
The top row of this diagram was established in Section 4.1.2, as was the leftmost arrow in the bottom row. The other two isomorphisms in the bottom row are given in Corollary 4.1.6 of the previous section. The rightmost vertical arrow is clear because Aψ=1F (1) is the image of ∇log and Pψ=pF,log−1 is the image of log.
Now that we have this basis, we want to find a useful mapping Pψ=pF,log−1 →OFJπK.
- Computing ∇ on A ψ=1 F
- Proof of the conjecture for unramified extensions
- The vector space V /(γ 1 − 1)V
- V /(γ 1 − 1)V as a F p [G][∆]-module
Since we are studying H1(F,Qp(r)), by Theorem 2.2.2 we can take only the right projection in the top row; this corresponds to taking the coefficient tr−1 on the bottom row, or in other words using the mapping (r−1)!1 dtdr−1r−1|t=0. Although we do not produce a complete answer, we show some work towards conjectures and computational formulas for various actions of G and ΓK. We will first show that all elements are non-zero - that is, they are not in the form γ1−1.
Similarly, for each g(πK)∈D0 we have that the lowest degree term ofgisπK`+e(p−2), and`+e(p−1) is not divisible by psincedivide. Thus we can factor aγ1−1 from all gτ,i, and have a collection tehτ,i such that. Since linear combinations of elements of C0 can produce elements with arbitrary coefficients of πK in degree`-efor 1≤`≤e(p−1), p-`, and since no element of V can have minimal degree`- efor`≤e (p−1), p|`, we can assume that fK0 has no nonzero coefficients in degrees less than (p−2) + 1.
The previous section gives us a basis for V /(γ1−1)V as a vector space, but we want to find a basis as aFp[G][∆]-module. If we have such a basis, we can use Nakayama's lemma to lift it to a basis of V as aFp[G][∆]Jγ1−1K=Fp[G]JΓKK module. Thus we see that their actions, even if they can be calculated, are complicated and not very manageable.
